Superconductor

The most popular posts so far

Our marketing people were just looking at some sort of web site analytics tool and showed me the relative popularity of the posts on this blog so far. I found the results somewhat surprising. The most popular post by far (in terms of page views) was "How to get a strong password." The second most popular was "Hyperelliptic curves." The fact that I wouldn't have guessed this at all probably explains why I don't work in marketing. Both of these were 10 to 20 times more popular than the typical post.

Fun with Algorithms

It looks like there's actually an international conference dedicated to having fun with algorithms. This is the Fun with Algorithms conference, and it has been held every three years since 1998.

Implementing the Tate pairing can be fun. Maybe I'll submit a paper on that when this conference is held in 2013. Maybe I'll talk about implementing it in Lisp, like I did on one of Voltage's Jack Bauer Days a few months ago.

#voltagelive Voltage Customer Summit Video

Data-centric security for a data-centric world - #voltagelive 2011 in NYC

New innovation and emerging technology brings with it opportunities for streamlining costs, eliminating hurdles for end users and reducing risks to the business. However, implementing game changing solutions can be unique to your environment, policies and processes.

That's why I invite you to join Voltage Security at its first customer summit in New York City on November 9, 2011. The summit will focus on data-centric security and will feature top Voltage customers such as Amex, Wells Fargo, State Street and others, who will discuss how they implemented encryption projects for mail, data and payments. Also presenting will be Eric Ouellet, research vice president with Gartner Group, who is currently developing new analyses of how companies use encryption. The summit features customers talking to customers—at last count this includesAmerican Express, BJ's Wholesale Club, Citigroup, Deutsche Bank, Fidelity Investments, JPMorgan Chase, UBS, State Street Bank andWells Fargo. The goal of the summit is to enable Voltage customers to network with each other and pick up valuable best practices. If you're interested in attending, please visit www.voltage.com/live.

The theme is 'Data-centric Security for a Data-centric World,' it's an area of huge attention. Data is the lifeblood of industry, commerce and leisure, every business and every transaction. That's why protecting it is such a serious and difficult responsibility. 

Here's a quick scan of the Hot Topics we have on tap at Voltage Security Live 2011:

Cloud Data Security Data-centric Encryption Ecommerce Security

Email Encryption Mobile Data Security Payment Security

There's no question that every company continues to face a long serious of challenges related to these topics. The conference is designed to tackle specific issues and help formulate achievable solutions. The areas to be covered include:

• How to fund and integrate a data-centric strategy into your overall security program
• Best practices for data-centric encryption based on real-world implementation at a Fortune 50 Bank
• How to roll out encryption projects successfully across the organization and end-user community
• Successful phases for fast and non-disruptive implementation—what you need to do before during and after an implementation
• Elements of key management architecture and design
• The role of cloud and mobile data-centric security

Voltage Security Live 2011 will bring together the brightest minds in our field, all with considerable experience. There will be representatives from teams responsible for implementation, as well as enterprise and security architects looking for, and developing, best practices for data-centric encryption. 

The sessions will cover customer project case studies addressing issues such as how to maximize end-user adoption for your B2C implementations and implementing data-centric encryption projects, while the Customer Track focuses on panel discussions and presentations on topics such as protecting outsourced data, eDiscovery and Archiving and securing application emails. There's also an Architecture Track, featuring panel discussions and presentations on topics such as key management architecture, security policy, enterprise applications and the web services API and scalable design considerations. And there's the Security Panel, with a discussion and general Q&A featuring leaders from the security community—Gartner Security Analyst, Encryption Architects, QSAs. 

There's going to be a broad cross-section of security specialists attending, but some executives will find it particularly enlightening: CXOs and security leaders responsible for security strategy and programs; VPs/Directors responsible for security implementation; and architects responsible for security and application and enterprise architecture. If you have one of these roles, we think this conference is exactly right for you. 

We know there are constant demands on your time - we hope to see you there.

Register at www.voltage.com/live

Webcast - What's New in Crypto for z/OS?

Our marketing people are having another of their fine webcasts. This one tells you about what sort of new encryption technology is available for protecting data on z/OS mainframes. It's sponsored by IBM and will take place at 10 a.m. (Pacific) on November 15, 2011 and will last for 60 miniutes. If this sounds interesting, you can sign up for this event here.

Here's an overview of what this webcast will cover:

Mainframes remain the core of business-critical operations in most of the world's largest and most successful enterprises, including those in banking, insurance, healthcare, and retail. But as IT management will attest, there can be issues with complexity when it comes to securing critical information on the mainframes running on z/OS.

Implementing an encryption solution can be disruptive to business operations and require hundreds of lines of code to acquire and store keys and perform cryptographic operations. Adding to the complexity are the deep expertise and highly specialized knowledge needed to keep all the moving parts working. Not only that, roughly 80% of z/OS customers use CICS, yet many vendors offer no developer abstraction to make it easy to encrypt from CICS transactions.

In this webcast we'll discuss proven strategies for easily implementing data security in the z/OS environment.

We'll also cover:

• Extending the proven security and defenses of the mainframe to protect information assets against new world threat to data 
• Pros and cons of various types of encryption and key management
• Achieving system interoperability with a data security platform approach
• Simplified key management that virtually eliminates key management issues
• A native z/OS encryption implementation that's easy to use with CICS or any z/OS environment

Telling the future with Cleverbot

It looks like we won't be using anything except elliptic curves or hyperelliptic curves for a while. At least that's what the almost-certainly-authoritative Cleverbot thinks will happen.

  

Voltage Customer Summit #VoltageLive - Only 23 Spaces left

 *** Only 23 spaces left ***

Voltage Security invites you to "Voltage Security Live 2011" at Bridgewaters in New York City on November 9, 2011. This customer summit will focus on data-centric security and will feature several leading Voltage customers, such as American Express, Wells Fargo, State Street and others, who will discuss how they have implemented encryption projects for email encryption, data-centric encryption and end-to-end payment encryption. Also presenting will be Eric Ouellet, research vice president with Gartner Group, who is currently working on a new analysis of how companies use encryption. The goal of the summit is to enable Voltage customers to network with each other and pick up valuable best practices.

Stop Press: Thanks to our sponsors - Coalfire, OpenPath, Teradata and Thales, we are able to offer registration for eligible particpants at no cost.  Register now at www.voltage.com/live

Join Voltage customers including: ADP, American Express, Bank of America, AT&T, Citigroup, Deloitte, Deutsche Bank, Elavon, Fidelity Investments, Heartland, JPMorgan Chase, McGraw-Hill, UBS and Wells Fargo. 
.
Highlights of the agenda include:

• CxOs Panel – Business dynamics for data-centric encryption security – How to get your security project funded
• Key Note – Eric Ouellet, Vice President Research, Gartner Group                      
• How to maximize customer adoption – Kim Mroczkowski, Wells Fargo
• 4. How to structure a data-centric encryption project – Emily Mossberg, Deloitte
• 5. “Birds of a Feather” Networking lunch
• 6. Tracks: Customer and Best Practices – American Express, State Street, Thales, PwC, Coalfire 
• 7. Security Leadership Panel – Gartner Group, State Street, American Express, Wells Fargo

Stop Press: Thanks to our sponsors - Coalfire, OpenPath, Teradata and Thales, we are able to offer registration for eligible particpants at no cost.  Register now atwww.voltage.com/live 

 

A Simpler Approach to Encrypting z/OS Data

To some, mainframes are seen as dinosaurs, technology that is obsolete or should be. However, this veteran platform has shown its resilience in enterprise computing for a reason. The benefits it offers to the corporate infrastructure - extreme scalability, high throughout, high availability - are matchless.

However, as IT executives responsible for running mainframes or other platforms with z/OS can attest - there are issues regarding complexity. For example, traditional encryption solutions can require hundreds of lines of code to acquire and store keys and perform cryptographic operations. And that isn't even the biggest problem; it's the knowledge required of all the moving parts of an application to ensure effective operation. And of course, with all that code and other complexities, there's more room for error. 

That's why developing an encryption solution based on Format-Preserving Encryption to address the mainframe environment was an interesting challenge - and it's been met. 

In response to customer requests, Voltage has now developed command line tools and a simple API that dramatically reduces the number of lines of code needed - from hundreds to just three. Now, companies of all sizes benefit from using Format-Preserving Encryption on the mainframe, including the ability to "encrypt here, decrypt there." This helps avoid the ASCII/EBCDIC issues that plague most cross-platform encryption solutions.

But that's not all. Engineers at IBM Hursley - the software development lab in Winchester, U.K., home to many of these technologies - point out that 80% of z/OS customers use CICS, the transaction server that's critical for mainframe operation. Unfortunately, many vendors still don't support CICS. Even IBM, which provides advanced encryption interfaces built into z/OS, has no developer abstraction to make it easy to encrypt within CICS environments. Traditional solutions for encrypting data rely on POSIX facilities, which are incompatible with CICS. 

Now, in a major step forward for cryptographic operations encompassing CICS and other z/OS environments, Voltage is introducing z/Protect, part of the Voltage SecureData product family. z/Protect provides even higher application data-level abstraction, requiring just one line of code to accomplish what used to take hundreds. More to the point, no one likes to mess with mainframe applications, and the z/Protect approach means even fewer modifications. 

To learn more about this innovative approach, which perfectly complements IBM's advanced cryptography on z/OS, here's a presentation introducing Voltage SecureData z/Protect.

Ate pairing or ate pairing?

The ate pairing can be very useful for implementing pairing-based cryptography because it can be much faster that either the Tate or Weil pairing. It's what all of our shipping products are moving to for that very reason.

Note that the first letter of "ate" isn't capitalized while the first letters of "Tate" and "Weil" are. That's because the Tate pairing is named after John Tate and the Weil pairing is named after André Weil. The ate pairing isn't named after anyone, however, so the word "ate" isn't capitalized.

And that's a good thing. It turns out that if you capitalize it, you end up with the Greek goddess Ate. Here's how the Encyclopedia Mythica describes her:

The Greek personification of infatuation, the rash foolishness of blind impulse, usually caused by guilt and leading to retribution. The goddess of discord and mischief, she tempted man to do evil, and then lead him to ruin. She once even managed to entrap Zeus, but he hurled her down from the Olympus. Now she wanders the earth, as a kind of avenging spirit, but still working her mischief among mankind. Her sisters, the Litai, follow her and repair the damage she has wrought to mortals.

So never capitalize "ate." That may give you a pairing that you really don't want to use.

JPMorgan Chase awards Voltage Security for Data-centric Encryption Innovation

At the J.P. Morgan Technology Innovation Symposium, yesterday afternoon, JPMorgan Chase inducted Voltage Security into its Innovation Hall of Fame in front of hundreds of Silicon Valley executives. 

Only two vendors were selected in this year's awards which recognize top emerging technology vendors for business impact, measured in terms of driving value for the firm, disruptiveness of technology and the overall quality of the partnership. Voltage was selected by top IT executives at JPMorgan Chase for its innovative data-centric encryption approach for protecting structured and unstructured data across datacenters, the cloud and mobile devices.

 

"In an environment of ever-increasing threats, secure communications are critical to our business and our clients." 

- Guy Chiarello, Global CIO of JPMorgan Chase.
 

"Voltage's stateless key management technology is enabling JPMorgan Chase to roll out secure communications on a global scale with an excellent time-to-market." 

-Anish Bhimani, Chief Information Risk Officer of JPMorgan Chase. 

   

http://www.voltage.com/pressreleases/PR111005-Voltage-JPMC-Innovation-Award.htm 

 

Visualizing the Common Criteria

Since it's so much fun to run data visualiaztion applications on random stuff, I thought that I'd see what happens when I made a word cloud visualization of the Common Criteria using Wordle. Here's one version of it. Just looking at it gives me unsettling flashbacks to when we did an EAL2 evaluation of SecureMail a while ago.

World Record for finding the largest Non-trivial Gigantic Prime

 Posted by Terence Spies, CTO of Voltage Security:  

Just a quick note to congratulate one of our intrepid engineers, Tom Wu.  He's now a world record holder for finding the largest Non-trivial Gigantic Prime (which is also the largest "generalized repunit" prime.)  There's a large community of people competing to find really, really large primes using various mathematical techniques -- a competition that suits people that work on things like cryptography.  The primes that these folks are working with are even more gigantic than the primes that are used in cryptography.  In Tom's case, he got his record by proving that  (34120^11311-1)/34119 is prime (which is to say, it can't be evenly divided by any number but 1 and itself.)  How big is that number?  It's over 51,000 digits long.  If you want to check Tom's work, you are welcome to do some trial divisions on the number, which I've included below.  Just to get you started, I've verified that 2 and 5 don't work.

The last four digits are 6521 --  having word wrap issues with the number :)

2148113144331884020240740202538593736542798642541159883242933148518626062734696

7482251874816305000213886141475725482006177265158884624210764636520825941743374

0020655220012483597468525750693374008686770236185285560156433674204900545125358

8535626600953884925605871782665547527990639894731154430466901982685718729660486

4881369886141453004098981298912124269668420975657731805167769706658865564400487

2985048301786874040776399408593887318771648233738526328648257086879429582825753

5618052464284594947182552118866969637409623368759754058413544349424479429944238

1453366100204018476267730436220497916402494405788738528648736021078435253136151

7221171963014964372932655229548939162418946825005859899350224301930274634753020

0684494954828308831152192742278833971826220347054551466100595325861149025154876

3672199795312801583498409563729827948921032581308182964332339144176566665042149

6537476986749576942866749228558450645686991802785539256065294514437338404574296

0581331753921690498721832447164340888676652634002213991227881527449919104387357

7714112701020723693173437557360353486408207016861410061847705906827947328787099

0338587646357879171075768166706869979077383366132722341443467314751436477687326

8564320727999910848074754684093502669318849084214843691619574370967153540046164

3890040248522310477784199951373598788743549851799177851422951669850913577026632

7820776247256276324832854904768964216408291993892546368828111262651880527178492

4021471609879802918453502709224570818417375789564100533842372182327010737429302

9341677402308184080347222587930987514937877627473191227970699649139039394165726

0708385520691526113694936166004577577587114224437644076543936490785659381680141

5314277165257570973696286109859929694209323103262593801973661493029696184313897

5546093983815718627771060159699461773993711699051722659420789323215165168321628

0379610663704747481331009787998105998355988455151229732167683133905343230173260

5617068761973449453255106017837632886918193790195500113756840561556821567564844

5358359527969139956774404068939754551340847875237626391196005991239633640169576

2088320476292796243829829014042499596669808217003372152782129396290076673055685

6452733557890821404400870808625349053504209190097099116990783830306482148977057

0280224065596773811777869210042210847401271222070182068117602977590148539491228

4451319429831469022723515804952432670980977526596413589668913565483977974421666

7174734678177820726963610251276935412253869260133088544966591485080753611111001

4377471951016900302640125969316898630940161576777345391199434550942960896638971

0870484768518403596900938161019780588657271082778844823998204467180624324989472

7424420192457838263181659119047403732641695091450638991917135466056363340328975

8846397152835772968589118385280277442372426828842245397667969034235044482750272

5397103180516682859669954313525846361092620590787635141369496849623427305651094

0193640006617275290500922032553439908799972027428157522053460023990380732452904

7375097692009070816560425843881364842548899463026226603114587053702147128948491

0953954109860130287303937422540579234078371135834293368631752859878503376855040

2031465643811943173485054028093243916123479562705438024735962264847848507906629

8337452721700531869177195514935268723433083101419822250104822536439662115520634

1324153566030257061395725562639787753978764416251574410801376842835170296038166

6069517951668931899334188623751084964492633344850737150625248310189657737396964

6224365199270425061095651586786711752323344806649546571830785135844655544581851

9943584207838127622962304413211338939235014402029218119152027464410553087571535

9819509437383929400693822006993787283557395270865543089843265309082398968964910

4609897120068583286073543433387986427592490172970911980491147698394154333527971

0647137885608315574415563949234625618695411457357634968244288027311495818896405

7099543581303185435785494831411001523198973273933363760529547696865774759243387

2470137034383252526665218498934966513258487103890264921396303017730499110431555

7784578164256645248036805645498504810691105493123261898444260781144332383147727

0390860677712890318832473355818918903108275518278576398728080503296982698580209

0325493861271224628817678474066424469522292108166190961795660400693541183393732

2018771919593990208099122718700831116186802832791784784647287106532282338636933

6177371936516082323169983682860868309709483726038296022401110212900197683157490

6317967003390999043709396972917997382500962535677915362119357165770410445837442

7012246311969283737666209187585178607123107967345990874244114932034790391686369

6575845493506958188188799398586631064769035820397923640012422074554812074958650

7786549783342061339081669620788694009196009973335611774918351654244575349837388

5967531175110142755172490721273772079975257592575476591143555498009488198294136

0426246020837422162594831118973440195517576675496270644015096257924828894175919

9503175512404080078879168240584310357544642252280241570235498972457527696402423

2845009331733707733553401458342575248377620548689571186506143135603188310280351

3383600976595532774164205259990282685470623352229126949069782524524239995594035

9138531745199785543315508542916721300740703646298958592474356324366656916468064

1310527469342728171608904921878538542833387102944038740561316088932013626775413

4962643552639832526836903106282079505703127879363222897276551672405243897844670

4572255635890328027198080051503255081578867967160883868806065955044756142650096

1192632219815828990795201170373804965149526975804464328597606374128417263209284

8452818460407651502442461215891079363939692876214993744737571153021443464313764

0228123634084636032801936043649733646528148308540040894244006381588700863713297

4605156414149314697482321261139617081427156455501369186369293822894006675796264

6644873303929724487381169448010520606528527718100430290445698655379939105434853

8521393110096944750788726100733132939630983719158549375405459490933813261312242

5105172758256560393849304845781185748383436452834826184026690560461243094064592

1202564101811263829307932244791696540661731421804674558011579725584308277390346

8886436647443740946169115083638970730800676436320246617012033348500952726295560

3277571029168434594205521639310148100371766284538023895656250879482499803321445

8957399029132779068325694142796470328864354798306513041214205846123145789607268

0119131001959259207114176430960777073080439400091214671722046947284559720821596

9235778369956086652964828726007642186125913349646995958071084726390989327793489

2769490010872738710707040978675601351627979607449339019148643138914669842790910

5069671420172177667618671532010199084101351160872147750133393921975716674567637

2875351676005755516427557379868615511941551326940023030018327103767273759511694

9792879600205731487989868123007979581830135272743837373111868069688300900565954

3192204082808218380114328104365906680020083986420725741814865336674308370285236

7096020643636182685582919290727146075719331034135086743836254942816389873325136

3191562303902262036116660130903871611322405540622229470297813026841511221606249

7634724991367284079893338464380437520783928407700300631778700412491384111051230

1500407073629558033456078796650886303138616037033045149803502645366108846309445

9250132860973078634558163467860590401104140577059023058628769566097807011711022

2696478363371515138316994902474220623329984562097783141971316698645785644326170

6601908395479610073941108735222708024886451738779593887486585540333633259229887

2813495970453973807836568079951264466748827012856061718902361263108625026646630

1682103053724459537775358196018056505704126078609243356101248154612007665989799

7426286161488229795633815319438709904609470717825563572888251172714057633637366

1813802351445826623358872655012909705896001791161621086715937589626378889940253

3942529979367743908611190114653770971627733031428424457973968719204202125392287

8257972099565762354542266121255776039410642466889721489084395263697907839147572

9812730781792535159454619181391549058633601921369054903942431790638025358570386

8958387451596173547218395126698510716866646289839775950839846633032592257608990

2622911882511576074156443311850393700879656751683713093076379641609822731290315

7052069933674095873180478762968599743025056664813923197435913873844263682531817

7771956886680197293709034047521671323682206271430604917364101875697961905187716

2411503063617368755864752448934103587750209733203053375404317369184529426421374

1200109771024216714528647376905203456097683627155186581118175408517658430679457

0563435753230030641260589410622808320370528661793347330130807612040586770388578

1515460045647130438073157536702418281396469266861913940461052517148057590219901

3974504207422162540319444086683745403630618160858287323058430650568956932874917

2517955256151516516656576106984405371892212074034791998741291454436402004138023

4404984563849519817670300352509988849291016910426194411645965111395177756584094

5704951940961217290283545092574923239463890686203737339158768267004123786389540

1424056158835347716003170816485123421624251026827493878212978490891175207786431

3097605420952306321549316579499803890900635559957436984611341505638314443824201

7868054970745531599598790239798562304723737831635862859648003885078322193763194

6479760872189885715792501621743200607245653211726445067409419189628251446064124

7860508892861396135664819753143964861444591262662890974072572975989700036192059

0776281014520407799420734843630667360807968853856861884763512265203382843236722

8179747723979952835667221156317645517832241044679704158374673291021012042079496

9358839589636590448791405416026915975340840925159105124465656000752489677630267

0666189415687615252646020801504187853622979705478576214033100903269006494051036

8714743816078282114690988833396145603269479777367204381584445784968265203115459

5195291159466244420992789779516859598031747943208485741982640020822857016099208

0108970224167962726701775218638428609029776292817158517677424002195677300684332

9235019024901188924482953315260942158146069198964164326300800129273849500463083

0224226543512470375927753142484077398000339775220444585574016023694476661651278

6447273004109386930190166875491657127352872457301519983708153632531985822161687

8049936234005094888211179399062106935640792952435687549667767955241911996363085

9848086319141823841113910798453836788196178725546646985298442125223133948783712

4816542607023471028520778186894633367187959400024450112155823420057511217439181

1113496952102348891388756107507308468229186906200613572926093737873984855786572

2708111442705938786394328220617070697754902330809701383537942949048425730015392

2606114622222613450497580810040627866828403080124469641165160909422552720178880

1227959834295380992862272217074068890699937821433775023139723961916044361118849

5558371887188231447454384764548054334305855950441873655425336350838911688394640

3044476592750599370085419502659297508523289605767879381589071919426128689929655

2599032244115565101629745345036327443597655683643092402549260085145580000196965

7436991250136071766972998112848471957712562381631716978671643152430150631149649

0549209854448753978939775730752224181807678848533551024266555530949921046658530

5054169824682105662802534942886835704222893970559978110553868120279727876916227

2945472309482568037832186729433760594278223266020375840067702699433748759277800

6591494944448791840714807130757370694639362773479760039563677979488315945893648

0119693743208828816942433934426300444325010003190752155241714774994395575599769

5786450096716783976481037039600084863142245798508543444564800064265310480589478

2596634169303430864895915142853660571343645083631598112152666254070412535121894

0945332119271475559424835640320831603554808105235099059696202754320833523279645

7840710738780935471616634843666606767496577163573282019781407188965905422841844

2334524966955966397103297301946868193400085030440639511249871818471031527063989

0956622675632894153371442867573220239745419037761140988772041708136900146134527

7428053823581580745793303867242248596950949311334627489763508019196097536859355

5366434207983893549858418682721321627598492914293654554486841281782129129348048

2178547609514398745090731452939161399064034653829739286770095968353822811349124

4655865699559419588388126419868214491231737549906286010140620834000244477408987

8734676683664560288337681340594217136452569183428682722224092889736678587265433

1945451193821384587852280826589661383327709712749741674261082841572771785231145

4382816100599540467405304034976689963276068362205710072786367698599341491484975

9144779437646443298201708157103435011512225052686685937712060992916147400255564

5735205419793751417481487729167498960375248763720563131448849933207570390642197

7421839828571448187383754234200574804028669947666020675807748671343815114356001

3768289985154733264976729301970138207243226377211475686963762433517652853461566

0024381109240923294626500577418704639597156754566726105039269474665015269517416

2501936712737122663959693599766510102348164036278402432533756438165442727390401

1642592939267583031489411455165324817614514898640389835546695547324199111968497

8764953461644164750961640493747800322188216796927701831917951451485836978789208

3495885082932476613421074173926286679532445562815128994773671600394973613651483

6344133838818189940207757156030648321382078704650126655321739826249475555720983

2232556127380528683746399393777788054937258399588605327000827021908239855502536

3966760038582499179109017939990047289984288400166894806832304190617167740917664

1852145447439869719782802934987783493177365581304922065433900500723284026922392

6206067417559518486515860848712736477055822406621711995665858688462399557194526

3007500702419665204158832769627992141411521836841891883492878023271103719349417

8344421583832454113957661871087702069417239688911002192802045287263726544894629

1497661561398536923881521220902147652782884265569579556957967925186889874327649

4022637298044011877154403599467099645091404931239563076637638763213213742216554

9587207469430701273550768878348958246730037514648701255820302018834550751555985

6133989098393821346662003413158717245357630078452561312535850009659458990911145

9830997995249057222648726947775307552229226775700738517122782672614618153249330

9968648949490245586189441961187160123933096389706410734106710838222397859599170

1273122535910765612838276793143122147672058900964559544291956050638827125639169

3199881196822218904520586527018191564265069250269814940767061011753130771678824

9454859711111515511464132015224098140573007822536894262437731108820139735925640

3040686391996721087642604451460839899579170597240963668455700902387770890463675

4143828843558648744421793863462116078956102183468564618716376082122470939970994

4297404245986086613827938286672822193712293425080899751728785447288506914448267

4976923130035865523398870483981752037622238962661780834279217348377821764030780

8957406640803489401611276748845785633165451751937808530964258176260036492188523

3636879939377855691553066169754567311435819562588161714301195220410147645812962

2141783222075792110594580027190190364170638254442171385183189813236016521973378

7423596928543750504829639286763228394804223431692648522813275714024590207341686

8994446414502596182249154838808342521008765459409320863867269518914642491373302

8633422387361273929388080546018634398947097864847234963063296629987825916366581

8744620339578498502132977128620606660487057362769137751063459358661081739503650

6929545960871575873647036333559958714892711037742189293934108778184943813652805

5795080031974986663048641394734901784293379059383572295875224670393128370148246

0686291401148140851397199085732275694111615897799103981865118971096024051176175

5229057258594278141978100734058516282171994898182775607402254339186726887265403

9754633353490293507586111371621329542834985260541294610791097891576793929848354

9941717618176887402980851875272683589309763906307994188238001824379177552773117

4097363507658540667351715376361844114888280871001969330801856706899768693795570

5282789827379767514886577442164159518487116428184087184624015987350870407559551

7497621262427293912659114660192411129017019069121742242173424719916227439008461

5331290925552783124993248952049149749228864645985386187672709736641807677659064

9353094823879208294615638086319280867999868704801630443424460938280708092231875

3290435669790320554803123537221450528963399211864795680283303681777283883402108

6574891941288719299997132870121416210526546879270877038392860530173305410270354

8851479381768542921232941035282722164919070919888929008448335935333502170264442

0984113817216954281820948345686685307586160595091249420528871435522368772337458

9410603143263341253471750406529583405623764960405594631367703386810457441672852

1735885143636971101066684150142442320729033333538495638071116649970500177075423

0798794701927735362870690163851439200628142180141074251516107666911462586479695

7788061500138337405846360698731060365552589209442876808811744402256661026503969

2089942101050368295113676426172785920425259172694948530211931235221108257856333

1765127551609618385179924882244085404378889164374746644033934860374618640112301

3497580429378562423755236398872968926806300491342281641215955032748362881519861

6415812049088151645309142190216602307639102812965643764983156853218108744016531

5420523619106115209462924386803173447548706432648055976070779065820103711947502

8738146614186665422970642344824586221051140169890161848775994917201618777988112

2525811157773175439947037848285892815692135477473651640922514875305109160529159

9134483762724042466224965994299728700977727728276794625932431786019399763907431

2595992593845416207251556468205682297002800523375164402899678763769093408614113

9965537847287218173566296579459608927883599934935445765290562799427114013470698

5026394297611632611341157539578528514859113499472133801241999877137395706897551

8678451110268183289498319494513331689052008321295232276276853659856688809501846

2958000244882971719596028131551470712047357888835835309973393906135099417694817

2273600875119176498337855891033152338956329973312409112099646058780418788750864

8453845432673855409146574587275076787507251181013698202257290575998241050819452

3293054961327506171125331841321579756530397636246950888284450519416975127783425

9446713950905812813684666254538954574240477094887719011763001358350290002470210

1641999795058879399949962423359592709779584935130101457524438401430363988216186

5469825544060931661221311249034436664894896145510129120181421374032463170049917

3143747653424886026183295617371450043235838813979222148281114109175064628008965

2592299171634731563402776102946229273958278149811675106901928697296550845927339

4163540357877479553545720365471108449107069564616429249045841028543972775806611

9075433903230210879002470144260422591582712208071509410862958391020313654069096

7968971288035398037276960126235688981022740848631167302982570465700569179623403

1503672714388508142873981062076262625672778302352060778967740152924373311643951

6030291461202569790931220338619630527934384799532706482636285362100379259943298

9836066870983308408372811805913103758781828661190397030188094412622261610527975

0509405808049512527786711809203058870935633009806226870335111363682869855455160

6083467220988695791686309003957801298579047444776786949305509670393340546061694

0950464863510060840072274017506076874931863667936838154533074714681516963040358

5955144328141796181384285765041416996821743218845670430861814417258368303429055

8652639434771131579438369756443706622595738046083841100165400799893590982207035

3336838602584316731104434398519406622950632810474757371153218703878853187058138

9433064289599238386335703063615895244666361851081532993349768013589525682322063

1418851342739965331982649857171301009327631529576862021193535322737949674847987

9752038093973670626543870464145154114084441873744094495700299173341746219425103

3102796182460410018866885432649733243384065061656542359142829770239122699987143

6798287760725879473027290279381822443743412922239690683871336562029934806743992

7916681365625207889180882122362488960403830926111771978342780649477093127783429

5862414239167235230636538835103413088254717078461232474326635702356581390510965

8020495650269396369480086759359233395899542372831957124136624288483010676672366

9101274974710800170650439107027708027502523791200782525704543230116059637877789

3944489064322076363937260752421479067296019304126309423969850239840152458991237

7440010772907607411707052436979670204733196458688716071740105524362569543392181

0584472673363291472010553416989990832459073804973042733285876459652998345885394

3914956763334038110710058504175223686638471326453275704385485014065282284121309

4287064459622915064195955905725275847213070738329312496201518647613599611031191

3900036386278236983336963092411751433528723567807931283370554180167943163051807

4936673237067521091013750329810292793977802092248377833541972256078464621890087

2022481505204821273840220161835640981986073478614209906397436690878016460657452

6809391968845167910710826174193791560706600283284991332963579500815418337138368

9465420069175401185514037088689175474832705759773849356188884588528947196449650

2150428487733552971175062374838357964984597717759517491511269348395298355499345

7691264276838676313105724580371048468605088938159410597026932063687673403166421

0390226939550501450039575586992105047907654502869079677637762135439864572114651

2128158142015865726564829651238195970951009966944224852421233511178910975707355

0864180475348211326685669397735997463444897348247444394476799764750383876581880

8341910013977665571064092664094548114499182289681766831271250623096502232400451

3729570921614686741786736726946948072014666720136766199414191342299968472216331

6515833856446053358439068897611050719992678158596464788679297830726315721993281

8125317667341346781148632551393430224155807640355560319385141475777520204626610

9892232047602918218820779889287960816123959237833899945372249874865794806330704

1909013931019216624088683208632711313717761484323416287676235296212707417134556

0968293468510253620095046081125467919810827241862670375688910866198475360131619

1926351150938636485266471864013694718080940843337338509681403770440512275618015

3318757696998963183851540351295008160382347079319866243352079609240456531121777

4235805658604449769528496732586351317576192042073180855371926057168509183775071

2010269843855668804726973186652766521621388645728272038246799299497448567842507

5420804075012303591099577931873624082880455052272541634584670352299425171075771

1868061425011154287902563711933542835117970696882691505333653436783311037371224

8085712356565245392808662065082637044815107707963389398532365140720414747479885

6089167411573508746909300292330292459746226396343508952816998048522255092504232

0715898931611045976802195817051818325997182902576165768983551809705013734973324

1909301024331672165245492873576686703751351577276632137432940565739466439841933

36226351485850295389563216642694511034102398106695053965418379363686927298516155899330554516758722022522468240241533442048060732629578012508539642915727700565162981577933043944660114471061662315055564178786740809242905020879352157918806515781970662915885789638946906300498581957236354869371857719880909683488382609846101235486575073611198520117484367734617717414908520938018413174674404216165389463181802407410103187994797409146891717736112820846390306629254953165820978137025417773374959756228633246993047590075337912929546637745598347364545777939925464014326067645462637414191417359178753419236285455982161805277434230830583516471931330423904593693850141167684369176449095196353220738392120005708973699762309212744556739433650241922292542383390740476412593712561729299155457948342772434268103467543354301111734837238528240537023983491950507052672232648335406809995090122982204164041501499856545476368393156531285452886914158637814487719937972382000229518806538953102488075505664075567823189836871756276153247350167669408482568123284123893045982857471957265723397983096611441508361610332648742089592817875555326803158318648175435420610470301904888669364453680952547968065862314744773658083558979161543844648800016345313068553296246660126428782467347543333176808811986485623947140972232356888395727858210484301730894721532444517844319359527871281422942339258560494734002258824860641437106076487733292273309030710729821698182199818405958356966120613653785453049200272886438400329202273441924568730343236080548660601315707598141941049675297875972911115903553578049520128306548839406691789188751304072577631085319260236976893288924130435679886287124694538735384647173017698831423252766438393514780494226897034407781030925617778683940496275101316812000547567770394629554776200877370414763580991686580153526700879901811365863199447571491347719969109698560222483679860513161880564298900034745334325190828940128906306444348747702743710471719160223697310188290758572043554689957357902295105662998183306784899631902052349421698914285422327814631565718370777377484516459574363048142096732525494180800676565281190800545996591594916857133595699432564343197624023687506236503528646207418838663962300220421991992723382021550800705763638597860815373985270006729338027190912821360181527443475993355598232969550280295969175118439377189401003247310356167250747252883075650484896954222808444974693272308353019106892176023394389439301360523858692509776940700961033335992400758476503707515209338859169188532116028575654496730687226118859918256539210554362403009116833662194172776361261304541105794485842127722422737950550149966560849142155657680496736347990665010666991911765685498482266929213216249584878552333049748317301569972062220377118369513468436827661056949895264194360563171488335251082975250568393623139965829540668004824848743062015606277041138165062667320959861700299107608954969370012850836153230659927758130208555821449074925805197147725240742965793107004854572582753251962557971331967601602561088408125134437393479829024156436045116468557986358437911748145290432617790743777115072409115254271490709636257388397170371174053110176360407931356825327075673215182693984112574796881059798482735664210695269604066514975167051683025545865472485703879402067245230308663168421898545240580131471928472221495144796535055791115751043205483603224803589564220851117025073812218301805151358225769440855422268643456507824499154733318368511581898938308935223064688487156789070607010216479644478886091515893593862911169092977363345099182054728921651810060713394497865365181316563755598181917547144754604922857891944675541463420643909617563392284319426485071709398627656222742633602001547053324692684513006317652346726283842536718436534717650223034393932542927797102931039370440114186825953884917519933504266045300643945663213110993826776827888083584466722577338081802522321472072149272894418727963925587981728084820546921439203549203443347342466387042278862069872148237790546208871200961546508407869857090521572775432904327500149059178453343369772584850638587208217917687741246162032028184239603948674397081500796223097127525530406654346688790232551905845701524936374869317691908990594698194799279633583192381019505704582849165288729666878767643916649794829167185272148703608729053530557373074867058022271663557808898119933628867839311751100644187543909469734179939089325904204905039311193584689457245616514223368034969034697856934939281249273025313897455903475091530498549009826240386484824110552580812498308698090531278370306011201982277821464657748565532519239282381343342298694473269592958133756315386992363566919611394326442851978664885378009524704245190041598221133047281921075727937136305512944012146315037482764528767340149942507223863910983657836581675563263042679503934419881159424802296458484795905098841484255818739919796181681116110645425352636899383227708451754586882104077657638487737316068139043761567337709401093304484573666119521922762819624524763232474747379989573491594007984024569482911143990358789514026573112158873240158689068597296432377831116006469214846033010758294470978902214112541296358960660195527641244065851968716282795233111422220021644888377322091805368548948860456114289079477333921641893466827996821130090281952265682327514845872400627492478787217106405327934061340006298127038459872496918915506203840251187853946265528369824990903991367700225945566785208740293851110015030211294409772215880373804635710227293576068375010768099365145243005310331872372336097465813485216735183893403267810604564171145644160683682273737231752476473596014458874459984550269785677696797341920650796636002249110981634762040450821109582821023362202650146667561807650330214103318924677979172487705227873781833357673789858982499770753298968577752166039074244082247302346792357595163598399369888498208415856163401978965878671070445316446454707002635527096081399727188121297059967690147343812348095214023755208293187157859899147889061890236636772594200413602833481993585438131428180765202851630350919504906268799676095048507300576755660816436662333274651872502058089342427034272192999988876124118611033724106216075176410345732312286655353827981376579863667079192953537185261587535684242708499179796227833679089749772392564413891255021960360972912040845910374668412392878222119434282838069815720528020025103325913655028001348926854683342921195268315711609901920599847699140603950836024756607927328517839135166493151811820590734447142969905757276003415785584986820705388735563950186211068952434977822468113703414337947963977196285729967275519244358134908640962289963818442436146354171935680275538881687580465738410819136186117031532558299671655643874259799718525660259853660151549738704126833476912196854997304326214759636278109752704859271246383306135086109664283539023707374262177288982677498319010225735211561434590991905471430863009834869552156643162410895491132695886587804371491178765735693550247815573915500991738493663928264637826258213366785365026463463839500537437134933675280562488878595759694976433577988328433842661706589428747156571117135999674936444551580258714486587258385991303844329901427645521440591947903667941319289660253859884254561620088125028052644974421456233409235718772498129833149397112383371079683271552290430258219060408230627476230144212672600444871314883322767765766091056317180646626817737369884039041148771459442299758323139830421332839838789265556071819609550654666601713958375801906074618714984976006421235702717758350486273749274370760077144404040220528058262497939395102663369613380439628759545174185879222650362418049908832887243236665761331458383321899717194958660755312635064254664872675983864235973712089389557418265616392488021596151797224135482105308986389962030366904851813771046935415498730782406381405479407447245754868653158118715287352976507671661958139378778576720022991004237807210894348583985443532241276112070567747222256165616205710788653945519828213935462044147661749938753930525805399815066227453201650388549101768649444921267008365042046870557500008823041951554890675904529900449696956480965989539696817120078770059792925781287086329263515453693954915158075113034251847772044183810791519975394751748689898558643184755405049918895166279673309822931228293858836566706111387755782303153570817649245833032333667842054213757871730349932868292661584780873597066086136442389500475912839216088119841108486391662347062079731254672902674899029699518214058301200750777457011564438182922511442521001995984268256138869931497593880179638256871879617036192483407270454061630517638158243184932640074214448300636762825787790465815764668563496463913308208132649995698735373141415390555852291483484500815988321185085562036902145706789438017907028808993756697737206455275037792438934509173298108120513315559906171631198421284029587594147719363885827441478167145011032769468120924289202232778393912078642897540227099609458372421165141758866287932646170989185314106431149591700094802514858897396699041758339425844464240751057119983473322446616721212147223992580174334557007350960927881867315943334332642940695491969768994404326500028357416077645018208735522571909743009983949514969295187942127329404378677160832336519960825318142197155416340731851260645919302725137544555071001587138361865913869610020046813964288029628639325661944045018679990069117482062667408540078159665476792627727609464378827651542989195295835403415358975420864630649750305992709504305141593468292092307592752869922835502177491274209137425439200221874228995506897015156680511844355698459518412040452728567718750564796432436799348432449300656838463943674982675989477173707352258231743887543890370831923847776897456520636369969092162091745184280276681337290313540310668113227494245314959225865818545519011541766570779681148761193626210642580789249778213667294624434882544083197407549937774724839437798731587141866021779391252395615312782332443779641927867803117304489103054464862680262795383267726188610971192940894213790537294071416726730205471594883721433228376368504612954199281875682722354008368008829216104419411714695859365833919253478762063071643602977518389063782720656357223263241216313572361820175847504844252982987057874164220068040521672328469836464592306793563154877571296188609627034496460131947166538446212919012254668810716374688399318031589528394929563408334799865216916243791743292626558266244742675075043141932261617981129892565665403574343308622614146084017230663331431137651925433876315986267153634604550662950415824747502480466200610391051247440726964692808094793279111690973369958080448664084602168747329814256346908188025350988408755023149023602234381202521094110880045708602978279330585186682370543005483078934421374866586755895231208160088337158986419693355203319013239353078282774833577690956059207864500518287095906947620924029949931753175874435562143765874716124766415127825824615964432848815965950113554908986881667925222069954770436458538642679415426223849782999352673372300095587692034470424817584920835606592005416232142163063811860675226296801628422602496931550282926839192741178956088321254006700753736528365299660401666186057191762832685319951394146399607257198492487552743966275236823358167652345263736845328484728962813506911677057297344263147565172028990923218154374743977986972557154364071686473729120755158098972466164764364174723133739831308749148351607578967167663731642496027598324205449254589569774347797934529348095761991902758597274492714453270008449468177717198558778352897227881317343727862576226683933781009530996463859261721029999232400182859683045569675689735564094220702015200532046757936625767824609574931301140459239519684061215107082750501896927776713568402432626024600037023933362926950572772013241524813551821903918397206414228549609670803975191716443967844951716388728998794031421473735272231299350398015910385361092729651151821698880417945410904078093868101292729686432317134463009920469994588036725518880142735026353443813377718027829505406020624209320516950636677115883541647482379301527827030253893878228533896114934272449887723804178647377070069808576675658862297453768426962115387027831563204856272369832797357524987369951239975394955849991577244396810640025116294721467113309859207527611647172656504666617827100723436255716296340560241791965508380968356492760826619030380149527386679096483387561985627580858698213417544448939324693779903856883193047224074969472304371204815924557386789016238955669917078387526743293422471110798098307919295149698351690292060051768664517835400139440707737153058708374942067914479625653489576221046689102009583275990204420727805525315247763297778390457107854590999922504706533957000281201223037333439582075941217389294007004066228407908176981361152483226774502966345419401520839294164812371688599617061478145851244472595740106111531126050972692112692324644536696638829969755610748492196140526907371579847164359718930654380117174564116306831745525782917696293720614897329567329847914295782004996339608616869335314666310016440449847214080731141333305155228943212236108983815659854488788078415177433713420510131880751373939562070224033826755521057164550382364221770661807737043059665863333551478726681347075391418811989255578959247961513921243511530125570650428018856937528494461104532764853810609396585001355653985407411971100475390919632964463672445959557457316002789630701017761321356552901619335099334034371455919806718070050888430056215415504982059082680608785695024278673325597797460244564709641721764953930807121092176788023261067425229978329080484937119885452370363051174931016571226844501107817257430825095424413699735724942085067083284971345452271827117283080408199610648148305501801351699625666460747552644253733890102992155243068719740706302340978296277916446265891939505815036700437934710820597209891819380845803824373489405580453012602245923228588711964946433066030953971971940726184440457797771365704779180310842476894514626961919420313211174669240180339308707450745435849908238566480931820239396730844939834159539835520365267652153109493744024602572710936711447676793298001499843222206198367461362347443037872840867921337938513402921174599532803337419028941655713258044768482911085148208410745269128322164000964905588586025196626195346493383740849886376838798884171705552284357465574582902339807327385681157197969227967377019522131080539543034046120480759590211097940468197260495377003414278259886498284246796924083372699586047446369939350821679452021759406025102905584703517564709383595844548101684484394096722500733676149770342361688603190659208887448890673918839047238846825182209476344686168575662725171573788089428415878135155838571682175854021220641275734074531752025264273573353362002692709626613363628040612095017751241392114250780529989276473540848610640853332598425521767056764449412150944751983885702497659471745443317315326762070293557486490566097599542027753335469792061948019822613893652258138329386322030221086341121027358266736646396884636053705080988724657400690447840464461461655773380677100421272040268300850789844436092907857017920831684079866210083194383283959542451315166154938620443885757999713343989481730906306101467777016786953480987235750765706642409603195947957572164509632624718647282469393536094494559725835337508503161948059663336909847500227766196098215716445354027542069363013940360081057556406094725833098044011921095111108671389310110869292437654624015532758230108133532483194377417408982878993872197598487636503136242605945485141963728200823295338695822957359295406286581242623104959501964437473377721541066580951762907166567392524118942474796051531937120784344465538392838353702462237955463684895450508205402401766767538326373746046937439965119262653567034347184780064145230921129638556652684502897579905585208928585747724767509459453630062651344263778526708169731103463469029300549829916910486447092931869667139450365273176011129537329641226603894387230876234252863587032037294497909614671567591683038432511824686524393452013042528020324099278170984025125521954167858727517775421062571076622388296439715949837814459985000596806352968776586416979499735952173317368678825428347211164933730040751336059936782202087844841661115151355920999466237763185723650898196669757691556653256399954282915008382136400871203749859120576135422447241406715382248054714493907863050043660697414965962952153768688260782042348988535877235813817006037728000053860045017391983732828315009252255823745973350136670517193303793462190677305622791707032085838152898331912985755306438140782184066321762736305366258795271643685148227730986303639279998134896895177801704036216738218942642541095164479750959275399314287907236027708919481514446366700606740920457583015564213511500237179555539207028906525729971887457720492627272333012936945586431353293995986086638853410256025735957214693070600978906817314309405811762362647805826201329803497706023697984322142490797201509076645723680744743760476438459885290221154928780008701480693132998096463728871468846529674424122480294810088090593385769565974020112928573277939654479610870501248910980516161842145414681810570331478220295814616402947621143869657158784949113210174909599176265259116881606619023047918591671916880454181950910455139278813495233693220129047496879813104666178141108257981223074602515165936108175532839367254471605681047640818037797698555820832865575935084003380946863079979636705097178341959975234839106924745231810116701586366793200263785493953669561284714416081071612984016961273884785877704390477890088462915555691321683623723043487135348091016616511316510035238831270323996684545268032474574284123215803511240071514405463231630469826196547378293619390955186259855212638119522846507810897154078372754183885811424719364576921949646824350068876579032210791641021131920630733608839649462176499897417861015856267768691931182039332923004777396758404408101058061490665025352442920366950965737565579295993434743105014801137196283595650517307072305753392537882118467715935402561622556346903484861807204197074943579823558720947272780562150121633107652627568217122424455581933819865763943843606201823031155661068612796389108707758140625457955977607784518889768164365895835165157243764471408892405990796916673993962308391218968902957296521000029309182566898209208945162519417333450570063600926170169113983411002667135613587737038014009789266977344001875787684281485389372490401242709340836484070459274890823294938304170696679269615170432896626513086550016120050411794015064919839385679533397813534980509393593012690876051466924587473255370907705384096837539201031683226354816964154869720683490137460066238752601189952812216067293883173598288343738093144582197602508866027726486708285705911662123743368797444239280166476156979981828306808523110290453999237961253260646560567425774495149330285178346375919575603036431313930654474046718837011635745479058589055951229520208681379876315249567689557138251414168058852838594331604091561886338990005568744687710659749699580878689293355608312084175972332131656848090506755766581670037222661859960725695360356399660013482223980773176236114774758931973387262229256426038277792432369061226882382250359037486444503062809578240862862334769483279111345584571646296784782672411266449778715671619918520472464022978399132448196019813007415223189425246929863126117412585362994226091034321052785837802983674785310237697470617544476684545268032474574284123215803511240071514405463231630469826196547378293619390955186259855212638119522846507810897154078372754183885811424719364576921949646824350068876579032210791641021131920630733608839649462176499897417861015856267768691931182039332923004777396758404408101058061490665025352442920366950965737565579295993434743105014801137196283595650517307072305753392537882118467715935402561622556346903484861807204197074943579823558720947272780562150121633107652627568217122424455581933819865763943843606201823031155661068612796389108707758140625457955977607784518889768164365895835165157243764471408892405990796916673993962308391218968902957296521000029309182566898209208945162519417333450570063600926170169113983411002667135613587737038014009789266977344001875787684281485389372490401242709340836484070459274890823294938304170696679269615170432896626513086550016120050411794015064919839385679533397813534980509393593012690876051466924587473255370907705384096837539201031683226354816964154869720683490137460066238752601189952812216067293883173598288343738093144582197602508866027726486708285705911662123743368797444239280166476156979981828306808523110290453999237961253260646560567425774495149330285178346375919575603036431313930654474046718837011635745479058589055951229520208681379876315249567689557138251414168058852838594331604091561886338990005568744687710659749699580878689293355608312084175972332131656848090506755766581670037222661859960725695360356399660013482223980773176236114774758931973387262229256426038277792432369061226882382250359037486444503062809578240862862334769483279111345584571646296784782672411266449778715671619918520472464022978399132448196019813007415223189425246929863126117412585362994226091034321052785837802983674785310237697470617544476684545268032474574284123215803511240071514405463231630469826196547378293619390955186259855212638119522846507810897154078372754183885811424719364576921949646824350068876579032210791641021131920630733608839649462176499897417861015856267768691931182039332923004777396758404408101058061490665025352442920366950965737565579295993434743105014801137196283595650517307072305753392537882118467715935402561622556346903484861807204197074943579823558720947272780562150121633107652627568217122424455581933819865763943843606201823031155661068612796389108707758140625457955977607784518889768164365895835165157243764471408892405990796916673993962308391218968902957296521000029309182566898209208945162519417333450570063600926170169113983411002667135613587737038014009789266977344001875787684281485389372490401242709340836484070459274890823294938304170696679269615170432896626513086550016120050411794015064919839385679533397813534980509393593012690876051466924587473255370907705384096837539201031683226354816964154869720683490137460066238752601189952812216067293883173598288343738093144582197602508866027726486708285705911662123743368797444239280166476156979981828306808523110290453999237961253260646560567425774495149330285178346375919575603036431313930654474046718837011635745479058589055951229520208681379876315249567689557138251414168058852838594331604091561886338990005568744687710659749699580878689293355608312084175972332131656848090506755766581670037222661859960725695360356399660013482223980773176236114774758931973387262229256426038277792432369061226882382250359037486444503062809578240862862334769483279111345584571646296784782672411266449778715671619918520472464022978399132448196019813007415223189425246929863126117412585362994226091034321052785837802983674785310237697470617544476684545268032474574284123215803511240071514405463231630469826196547378293619390955186259855212638119522846507810897154078372754183885811424719364576921949646824350068876579032210791641021131920630733608839649462176499897417861015856267768691931182039332923004777396758404408101058061490665025352442920366950965737565579295993434743105014801137196283595650517307072305753392537882118467715935402561622556346903484861807204197074943579823558720947272780562150121633107652627568217122424455581933819865763943843606201823031155661068612796389108707758140625457955977607784518889768164365895835165157243764471408892405990796916673993962308391218968902957296521000029309182566898209208945162519417333450570063600926170169113983411002667135613587737038014009789266977344001875787684281485389372490401242709340836484070459274890823294938304170696679269615170432896626513086550016120050411794015064919839385679533397813534980509393593012690876051466924587473255370907705384096837539201031683226354816964154869720683490137460066238752601189952812216067293883173598288343738093144582197602508866027726486708285705911662123743368797444239280166476156979981828306808523110290453999237961253260646560567425774495149330285178346375919575603036431313930654474046718837011635745479058589055951229520208681379876315249567689557138251414168058852838594331604091561886338990005568744687710659749699580878689293355608312084175972332131656848090506755766581670037222661859960725695360356399660013482223980773176236114774758931973387262229256426038277792432369061226882382250359037486444503062809578240862862334769483279111345584571646296784782672411266449778715671619918520472464022978399132448196019813007415223189425246929863126117412585362994226091034321052785837802983674785310237697470617544476684545268032474574284123215803511240071514405463231630469826196547378293619390955186259855212638119522846507810897154078372754183885811424719364576921949646824350068876579032210791641021131920630733608839649462176499897417861015856267768691931182039332923004777396758404408101058061490665025352442920366950965737565579295993434743105014801137196283595650517307072305753392537882118467715935402561622556346903484861807204197074943579823558720947272780562150121633107652627568217122424455581933819865763943843606201823031155661068612796389108707758140625457955977607784518889768164365895835165157243764471408892405990796916673993962308391218968902957296521000029309182566898209208945162519417333450570063600926170169113983411002667135613587737038014009789266977344001875787684281485389372490401242709340836484070459274890823294938304170696679269615170432896626513086550016120050411794015064919839385679533397813534980509393593012690876051466924587473255370907705384096837539201031683226354816964154869720683490137460066238752601189952812216067293883173598288343738093144582197602508866027726486708285705911662123743368797444239280166476156979981828306808523110290453999237961253260646560567425774495149330285178346375919575603036431313930654474046718837011635745479058589055951229520208681379876315249567689557138251414168058852838594331604091561886338990005568744687710659749699580878689293355608312084175972332131656848090506755766581670037222661859960725695360356399660013482223980773176236114774758931973387262229256426038277792432369061226882382250359037486444503062809578240862862334769483279111345584571646296784782672411266449778715671619918520472464022978399132448196019813007415223189425246929863126117412585362994226091034321052785837802983674785310237697470617544476684545268032474574284123215803511240071514405463231630469826196547378293619390955186259855212638119522846507810897154078372754183885811424719364576921949646824350068876579032210791641021131920630733608839649462176499897417861015856267768691931182039332923004777396758404408101058061490665025352442920366950965737565579295993434743105014801137196283595650517307072305753392537882118467715935402561622556346903484861807204197074943579823558720947272780562150121633107652627568217122424455581933819865763943843606201823031155661068612796389108707758140625457955977607784518889768164365895835165157243764471408892405990796916673993962308391218968902957296521000029309182566898209208945162519417333450570063600926170169113983411002667135613587737038014009789266977344001875787684281485389372490401242709340836484070459274890823294938304170696679269615170432896626513086550016120050411794015064919839385679533397813534980509393593012690876051466924587473255370907705384096837539201031683226354816964154869720683490137460066238752601189952812216067293883173598288343738093144582197602508866027726486708285705911662123743368797444239280166476156979981828306808523110290453999237961253260646560567425774495149330285178346375919575603036431313930654474046718837011635745479058589055951229520208681379876315249567689557138251414168058852838594331604091561886338990005568744687710659749699580878689293355608312084175972332131656848090506755766581670037222661859960725695360356399660013482223980773176236114774758931973387262229256426038277792432369061226882382250359037486444503062809578240862862334769483279111345584571646296784782672411266449778715671619918520472464022978399132448196019813007415223189425246929863126117412585362994226091034321052785837802983674785310237697470617544476684545268032474574284123215803511240071514405463231630469826196547378293619390955186259855212638119522846507810897154078372754183885811424719364576921949646824350068876579032210791641021131920630733608839649462176499897417861015856267768691931182039332923004777396758404408101058061490665025352442920366950965737565579295993434743105014801137196283595650517307072305753392537882118467715935402561622556346903484861807204197074943579823558720947272780562150121633107652627568217122424455581933819865763943843606201823031155661068612796389108707758140625457955977607784518889768164365895835165157243764471408892405990796916673993962308391218968902957296521

 

Celebrating Ten Years of Identity-Based Encryption (IBE)

Over the past 10 years, IBE has become the one of the fastest deployed encryption technologies as measured by the commercial adoption of Voltage SecureMail™ and the use of IBE as a general purpose key management solution used across the Voltage Security product line. Since its commercial launch 8 years ago, Voltage SecureMail has become one of the most widely adopted secure email products in the world with over one billion secure business emails sent annually and over 50 million worldwide users; those numbers are expected to double by 2014.

IBE was first introduced on Tuesday August 21 at the 2001 Crypto conference in a seminal paper by Dan Boneh, Stanford University, and Matt Franklin, University of California Davis. The paper, entitled “Identity-Based Encryption from the Weil Pairing,” set forth a simple but powerful approach for encrypting information with identity-based keys. This cryptography breakthrough became the founding technology for Voltage Security that was incorporated in 2002 by three Stanford students working with Professor Boneh: Matt Pauker, Rishi Kacker and Guido Appenzeller. In July 2003, the new company launched Voltage SecureMail, an email encryption product using IBE to secure messages without the difficulties and expenses of traditional certificates.

Key metrics in the 10 year history of IBE:

• 50 million Voltage SecureMail users worldwide.
• Approximately one billion IBE secured business emails will be sent in 2011.
• By 2014, it is estimated there will be 100 million Voltage SecureMail licensed users and over two billion secure emails will be sent that year.
• All the messages protected by IBE in 2011, if printed out, would circle the globe seven times.
• Nearly a third of the world’s 20 biggest public companies (per the Forbes Global 2000) have standardized on Voltage SecureMail.

 World’s Biggest Companies Standardize on Voltage SecureMail

Nearly 30% of the world’s 20 biggest public companies (as listed by Forbes Global 2000) have standardized on Voltage SecureMail powered by IBE including four of the world’s largest global financial institutions; one of the world’s largest retailers, one of the largest U.S. managed healthcare providers and several large regional healthcare providers.

 

 

 

Notable Voltage SecureMail customers from the last year include:

• One of the largest Wall Street banks with over 230,000 employees standardizes on Voltage SecureMail
• A major Wall Street bank and Fortune 100 financial services provider with global operations chooses Voltage SecureMail for its 100,000 employees around the world.
• A major credit card brand with over 60,000 employees standardizes on Voltage SecureMail
• An award-winning regional health care organization replaces a non-functioning email security solution from one of the largest technology companies in the world with a policy-based encryption solution from Voltage SecureMail

• A Fortune 50 global financial services company deploys Voltage SecureMail to over 320,000 internal and several million external users across 86 countries, replacing an aging PKI-based encryption technology.

In addition, over 1000 enterprise companies have standardized on Voltage SecureMail, and thousands of mid-size to smaller business use the cloud-based Voltage SecureMail Cloud™ solution to protect private and confidential information.

More information at www.voltage.com

Another good question about data breaches

I was just asked one of the best questions ever about data breaches. It happened roughly like this.

"You guys seem to do a lot of analysis of data breaches. I just have one question for you about it."

"OK."

"I see why you're doing it, but why isn't anyone else also doing it, too?"

That's a very good question.

Looking at the available data isn't really very hard, and there are lots of companies who you'd think would also be interested in understanding it. I have no idea why they're not.

Visual group theory

Nathan Carter's book Visual Group Theory tries to help you understand group theory through a set of pictures. Some of these made sense to me while others didn't. Here's the illustration of the structure of dihedral groups from a presentation that Carter gave before the book was available. That seems to show their structure in a very useful way.

 

Carter's book runs 297 pages, many of which are illustrations like this one. It's a book that I would have found very useful back when I was taking classes in group theory. It would have definitely helped to make some concepts clearer. If you're interested in group theory, this is probably a book that you'll want to have a copy of.

Carter has also developed software that lets you create visualizations of groups. This is Group Explorer, and it's something that I wish I had time to actually try. Maybe I'll do that on Voltage's next Jack Bauer Day.

The spectrum of a ring

Because the machinery that Voltage's IBE algorithms use comes from algebraic geometry, the engineers who work on our core cryptography need to understand at least a little of it. So I wasn't too surprised this morning when I was asked what the connection is between the spectrum of a ring and the spectrum of a matrix. Here’s the totally non-rigorous answer that I came up with.

Suppose that we have an n x n matrix A over the complex numbers ℂ. The spectrum of A is the set of eigenvalues of A, or the solutions to

f(x) = det(A – x I) = 0

The spectrum of a ring is the set of prime ideals of the ring.

These two ideas sound like they’re totally unrelated at first, but there’s actually a connection. Here’s an example that shows this.

The Cayley-Hamilton theorem tells us that the matrix A satisfies f(x). So if we write

f(x) = a₀ + a₁ x + a₂ x² + … + a_n-1 x^n-1 + xⁿ

then we also have that

f(A) = a₀ I + a₁ A + a₂ A² + … + a_n-1 A^n-1 + Aⁿ = 0

To make things simpler, let's suppose that A has n distinct eigenvalues, and let's write

f(x) = (x - λ₁) (x - λ₂) ... (x - λ_n)

Now consider the ring ℂ[A] where

ℂ[A] = {z₀ + z₁ A + z₂ A² + …, z_i∈ℂ}

Because we have that

f(A) = a₀ I + a₁ A + a₂ A² + … + a_n-1 A^n-1 + Aⁿ= 0

we can reduce elements of ℂ[A] modulo f(A) to get rid of powers of A higher than n - 1. That's just like working in the quotient ring

ℂ[A] / (f(A))

But we can also think of this quotient as

ℂ[A] / ((A - λ₁ I) (A - λ₂ I) ... (A - λ_n I))

From there, the correspondence between the eigenvalues λ_i and the prime ideals (A - λ_i I) seems fairly natural. Or at least it's where most people seem to say, "Ah, I see the connection now." So if you're going to generalize this idea to other rings, calling the set of prime ideals the spectrum of the ring actually seems to make sense.

Brown's identity-based decryption

Dan Brown recently wrote a paper that described what he calls "identity-based decryption." Here’s how he describes this in this paper’s abstract:

Identity-based decryption is an alternative to identity-based encryption, in which Alice encrypts a symmetric key for Bob under a trusted authority’s public key. Alice sends Bob the resulting ciphertext, which Bob can send to the trusted authority. The trusted authority provides Bob the symmetric key only upon verifying Bob’s identity.

I’m not quite sure that this is really a new idea. It’s very similar to what existing implementations of identity-based encryption currently let you do.

Products like Voltage's SecureMail that use IBE to encrypt email let you do the IBE either with or without client software. If you have client software installed, they work like you’d expect:

1. Alice encrypts a message with a symmetric key
2. Alice encrypts the symmetric key with Bob’s IBE public key
3. Bob gets his IBE private key from a key server
4. Bob decrypts the symmetric key with his IBE private key
5. Bob decrypts the message with the symmetric key

But if Bob’s in an environment where he can’t install client software or his IT department won’t let him install any client software, a slightly different approach is used. In Voltage's SecureMail, we call this the Voltage Zero Download Messanger. Here’s how it works:

1. Alice encrypts a message with a symmetric key
2. Alice encrypts the symmetric key with Bob’s IBE public key
3. Bob sends the encrypted message to a secure server
4. The secure server gets Bob’s IBE private key
5. The secure server decrypts the symmetric key with Bob’s IBE private key
6. The secure server decrypts the message with the symmetric key
7. The secure server sends the decrypted message to Bob

That’s extremely close to Brown’s IBD. It just does an additional step or two for Bob.

And by using IBE to do this instead of IBD, you get some important advantages. The biggest of these is probably the fact that you don’t need to securely archive any private keys. This makes an IBE system very simpler to buy and operate, and that gives the technology a big advantage when it’s compared to other alternatives.

So IBD looks like an interesting idea, but I doubt that it would ever get the commercial acceptance that IBE has seen. The last numbers that I saw said that there are somewhere between 40 and 50 million users of IBE worldwide, and I’d guess that most of those users use it because some CISO liked the fact that systems that use it are much cheaper to buy and operate than the alternatives.

Another Jack Bauer Day

We recently had another one of Voltage's Jack Bauer Days. These are days when everyone in engineering gets to work on whatever they want to for 24 hours. I always seem to miss these. I actually missed this one too, so I decided to do whatever I wanted to do from airplane seats and airports.

My first thought was to try to understand abelian varieties, but then I realized that I had a few hours hours instead of a few months, so I decided to try something a bit easier: implementing the Tate pairing and the ate pairing in Lisp. This turned out to actually be fairly easy.  

The ate pairing is useful in many applications of pairing-based cryptography because it can be much faster than the Tate pairing. That's why we're moving to in all of our shipping products. But my Jack Bauer Day implementation didn't seem to have that particular advantage. And now there seems to be a shortage of parentheses at Voltage. Maybe we'll get some more in our next weekly office supply shipment. 

How not to implement format-preserving encryption

There are lots of good reasons to be interested in format-preserving encryption. Most of them involve getting encryption to work in IT environments that have lots of older technologies in them. Lots of these don't handle encrypted data gracefully, and you never quite know which part of your IT environment will choke on even a slight change in data format until you try it.

But if you're going to use FPE, it's probably best to just use either a shipping product that implements this for you or to use someone else's encryption toolkit that does this. It's probably not a good idea to implement it yourself.

But that's just what one of our competitors is recommending. Here's what they said on their web site when someone asked about for FPE. (I've tried to make this anonymous, because I know that this vendor is actually very careful about security, and that this questionable advice probably doesn't really reflect the company's point of view.)

What you could try is:

• Re-format your plain text from the restricted key space format to binary format plain text.
• AES encrypt the binary format plain text to produce binary format cipher text.
• Re-format your cipher text from binary format cipher text to resticted key space format cipher text.

where "restricted key space format" is the restrictions placed on the text based on the format you are trying to preserve. For instance, if your plain text was only ASCII numbers '0' to '9', before encrypting the numbers, you would need to convert the number to the range 0x00 to 0xFF.

Unless you really know what you're doing, following that advice is very likely to lead to an approach that isn't as secure as you think it is. The way that Voltage does FPE uses a technique that's provably secure. Our approach and the proof that it's secure is actually due to Phil Rogaway, which is about as good as you can get when it comes to symmetric encryption. Other approaches may not be as sound.

(As an aside, I actually have seen fairly non-technical QSAs arguing with Phil over this on various Internet forums. It's sort of like seeing an undergraduate English major telling Einstein that his theory of general relativity is totally wrong. But that's probably getting too far off topic. ) 

In any event, it's still true that creating your own cryptographic algorithms is probably not a good idea. It's easy enough to buy products or to license a toolkit that will do FPE in a secure way. That's probably a much better approach than trying to do it yourself.

One way in which working for an encryption company is different

Working for a vendor of encryption technology is a bit different than working for many other technology companies. I was recently reminded of this when I heard some of the engineers at Voltage describing how their current job is different from ones that they've had in the past.

Our engineers seem to be well above average in many ways, and this often made them the "go to" person for math questions in their previous jobs. At Voltage, however, because we deal with public key cryptography a lot, we have lots of people who understand this technology, and that means that they also understand way more math than your typical engineer does. This means that some of the people who previously found themselves as the "go to" person for math questions now find that they don't get those sort of questions at all. One of the people who work on our core cryptography tends to fill that role instead.

I was a bit surprised to hear that, probably because I've worked around people who deal with numbers for most of my life so that I often (incorrectly) assume that the rest of the world is also that way.

Forbes: The Epsilon Hack Attack: Time For “SOX For Consumers”?

In today's Forbes CIO Blog, Voltage co-founder, Matt Pauker, shares his vision for how corporations can reduce the risks associated with sensitive data being in the hands of 3rd party cloud/service providers - insist on a mandatory data encryption clause in all service provider contracts.

Secret message

 

How EHRs will affect privacy

I just read CDW's "Elevated Heart Rates: EHR and IT Security Report" (PDF - giving personal information may be required). It looks like lots of people are concerned about the privacy implications of electronic health records. Here's how CDW's survey found that people thought that EHRs would affect the privacy of personal information and  health data:

Significantly negative: 9 percent

Somewhat negative: 40 percent

No effect: 24 percent

Somwhat positive: 20 percent

Significantly positive: 7 percent

So 49 percent, or almost half, think that the use of EHRs will have a negative effect on their privacy.

The other 51 percent are wrong.

We really don't quite know how to protect sensitive information in a cost-effective way yet, and this applies to EHRs as well as it does to other types of sensitive information.

National governments do a reasonable (but not perfect) job of protecting classified information, but they way that they do it is very expensive and doesn't work well when you're actually worried about things like costs and people being able to do their jobs efficiently.

Or you could just encrypt your sensitive information, but using that approach relies on having strong key management to support the use of encryption, and how to do interoperable key management securely and in a cost-effective way is still an unsolved problem. (This is why a big fraction of Voltage's R&D focuses on key management and lots of our products are really designed to make key management easier - we want to be the first to solve this problem.)

But in the absence of a good way to protect EHRs, we're definitely going to see them compromised. That's whey I get worried every time I hear people in Congress talking about EHRs as a good way to reduce the cost of health care in the US. They might actually allow some cost savings, but it would almost certainly also allow the disclosure of sensitive information on an enormous scale. So because the technology to adequately protect EHRs really isn't there yet, it probably isn't time to move to them yet. Let's work out how to address the privacy concerns first.

Sign up for the 2011 Key Management Summit

 
 

The 2011 Key Management Summit is almost here. This year it's being held in Pacific Grove, California, right down the street from the Monterey Bay Aquarium, the Pebble Beach golf course and 17-mile Drive. Previous events were sponsored by the IEEE and were collocated with the IEEE MSST conference. But because key management has moved away from its roots in storage, this year's event isn't being held with MSST. It's not even an IEEE event this year. That means that the IEEE won't cover any losses that the event might suffer, so key management vendors Voltage, Thales and SafeNet volunteered to handle that responsibility. It's still an industry event, just one that vendors are picking up the tab for this time.

But there are more reasons to go to this event besides a nice location. It's a great chance to learn about key management from people working in academia, the government and industry. Each of these types of participants usually have very different points of view, and you can learn lots of interesting things from each of them.

The academic point of view might not be very helpful for solving today's key management problems, but it will probably a good indication of what's coming in five to 10 years.

The government speakers probably have some interesting things to say also. Some of the biggest key management systems in the world are run and used by government agencies, and the people behind these projects often have lots of interesting insights that are hard to find elsewhere.

And when it comes to knowing who's actually buying what, there's no better of point of view than that of the vendors who are making and selling key management solutions today.

From the most recent program for this event, it certainly looks like there are couple of excellent opportunities to hear things that will be of interest to anyone working in the information security field. These two talks look particularly interesting:

Dorothy Denning, Naval Postgraduate School, "The History of Key Management"

Dan Boneh, Stanford University, "Social Keys: New Directions in Public Key Management"

Back in the dot-com era, Dorothy Denning was one of the most vocal supporters of the Clinton administration's key escrow plans, which would have required all users of strong cryptography to use a version of the technology that the government could get the keys for and decrypt. With a court order, of course. She's seems to have changed her mind in the past 10 years or so, but it will still be interesting to hear her description of what really happened in the political battles over key escrow.

Dan Boneh is probably very well known to people in touch with the academic cryptography research community, although he might be less well known by people in the business world. In addition to being the inventor of the first practical and secure identity-based encryption scheme, he won the 2005 RSA Award for the field of mathematics for his work in public-key cryptography. He's one of the world's leading researchers in the field and can definitely give you a good idea of where academic research in data security is headed.

But it's not just academics that will be at this year's event. Here's a list of the other talks that are confirmed as of today:

Tony Steiber, Wells Fargo, "Crisis and Opportunity of Cryptographic Key Management"

Chris Kostick, Ernst & Young, "Auditing an Enterprise Key Management Project"

Elaine Barker, NIST, "Key Management Framework"

Ramon Krikken, Burton Group, "So we're managing a bunch of keys… now what?"

Bob Griffin, RSA, "The OASIS KMIP Standard: Interoperability for the Cryptographic Ecosystem"

Rami Shalom, SafeNet, "Universal Key Management in an Age of Encryption Fragmentation"

Bob Griffin, RSA, "Where Are My Keys?"

Jon Geater, Thales, "Key Management Control Strategies in the Cloud Information System"

Boris Schumperli, Cryptomathic, "A New Approach to Key Management in the Cloud"

A panel discussion on cloud key management, led by Ramon Krikken, Burton Group.

And even though it's not an official part of the event, the best part might actually be what you learn from talking to people working in the field over lunch or dinner. That's when you'll often learn all sort of things that you wouldn't hear in a more formal setting. Past events have also had very interesting discussions between the major key management vendors about which of their products were selling and which ones weren't. And at one past event, one leading key management vendor even learned from one of their biggest customers that they weren't happy with certain features of the vendor's products. The sort of stuff that you want to hear but that's hard to learn in other ways. For only $325, that's a pretty good deal.

But it's even better than it already sounds. (Try to imagine that being said by either Billy Mayes or Anthony Sullivan, the famous infomercial pitchmen.) 

That $325 even includes a room at the Asilomar Conference Grounds and your meals while you’re at the meeting. I’m not part of the program committee for this year’s event because people didn’t want to see too much participation from a single vendor so I haven’t see the budget for this event, but I’m very surprised that they were able to do this for only $325. In many cases, just the cost of the meeting rooms and insurance that hotels make you get for events like this can put a floor of around $200 to $250 on what you can charge to break even, so the fact that they were able to get the meeting facilities plus a room and meals for only $325 is quite impressive.

And although the web page for the KMS doesn’t mention it, I’ve been told that attendees will also get a KMS 2011 t-shirt.

So there will be lots of interesting discussions and a cost that’s probably well below what you’d expect. Milton Friedman would probably suggest that you take this opportunity to buy something at a low price instead of selling something at a low price, and you can do that by signing up for this event here.

The 20th anniversary of the RSA Conference

Just in case you haven't heard, this year's RSA Conference marks the 20th anniversary of this event. This year's event start in just a few days. I'm not sure what's being planned for this year, but there will almost certainly be an event worth attending. The best part will probably be the party that Voltage puts on. That was definitely one of the high points of last year's event.

And that's coming from a person who just worries about arcane issues in cryptography and information security and sometimes doesn't actually talk to another person for days at a time.

So if you haven't received an invitation to this year's party yet, be sure to contact your Voltage sales rep and ask for one.

The value of voltage.com

Sure, Voltage sells several millions of dollars of enterprise software each year, but what is the domain name "voltage.com" itself worth? I recently came across a web site (that I can't seem to find again after cutting and pasting these values) that had the following estimates for the value of this domain name: 70 bars of gold, 16 camels, 42,115 bottles of beer or 4,761 bottles of vodka.

Really.

I have absolutely no idea how they came up with those values. They might even be just random values meant to entertain instead of being based on some sort of useful information, but I'm now sort of curious exactly how much software we sell each year as measured by those values. Just not curious enough to actually figure out what a reasonable conversion from dollars to bars of gold, camels, bottles of beer or bottles of vodka is.

The best part of stumbling across this bit of Internet trivial might have been learning that our competitors all have much less valuable domain names, even when measured in camels or other unconventional units. Maybe I'll mention that to them when I stop by their booths to talk to them at the RSA Conference that's starting in just a few weeks.

More on cryptographic key strength

Just in case the discussion of cryptographic key strength here isn't enough for you, here's a You Tube video that also talks about it:

Antifragility and IBE

There's an interesting interview with Nassim Taleb, the author of The Black Swan, on The Economist's web site. In this interview Taleb talks about how some systems actually benefit from shocks.

When I saw the video of this interview, I was reminded how the invention of IBE may be exactly the sort of shock that he's talking about. IBE lets you encrypt sensitive data, and it does this in a way that's very easy for users and very inexpensive for administrators. This means that it makes encrypting lots of data feasible that wouldn't be feasible otherwise, and it's probably no coincidence that the dramatic increase in the use of encrypted email over the past few years coincides with the introduction of IBE and widespread use of IBE.

Taleb didn't use IBE as an example of a case where a big change can actually end up having a big benefit, but that's probably because he's not a big user of enterprise software. Among people like that, particularly the ones who worry about information security, IBE seems to be much more widely known.

These days, when our sales guys are tied up and I end up manning the Voltage booth at trade shows, I'm rarely asked to explain what IBE is and how it works. Five years ago, that wasn't the case. These days it seems that the more common questions are from existing customers, and they're often about how they can use their existing key management system that they got when they bought Voltage SecureMail for other applications.  That's definitely a big change.

Happy Voltage Day!

Today is apparently Voltage Day, the first of many to come. Here's what my calendar shows:

 

It actually looks like every day will be a Voltage day this year. At least the weekdays.

 
 

 

Planning for our next Jack Bauer Day

A few times per year, Voltage has Jack Bauer Days, where the people in engineering can do whatever they want to for 24 hours. Although taking the day off to stay home and watch episodes of 24 on Netflix is certainly allowed, most people stay up all night working on some sort of programming project that they've always wanted to try. While using the sucks/rocks web site to compare HTTPS and S-HTTP, I also used the site to compare the popularity of common programming languages. Here's what I got:

 

Based on this, I think that I'll try using LISP to implement the Tate pairing for our next Jack Bauer Day. (That's if I manage to be around for it. I've actually missed every one of them so far because I've been out of town at some standards meeting. Maybe I'll have better luck this time.)

Just in case anyone asks why I'm implementing the Tate pairing in LISP, I have a response prepared that justifies it as an important project. After all, my justification goes, there are still people out there who use EMACS. They'd probaby love to have a way to implement identity-based encryption in that environment, but the lack of an implementation of the Tate pairing in LISP is probably the biggest single obstacle to letting them do this. So if I do the hard part for them, they might become big users of identity-based encryption. Or at least that's the best reason that I've come up with so far.

A sign of our success

Because we've grown too big for our current building, Voltage will be moving soon. But not before I had a chance to notice that Google Maps actually labels Voltage's location as "Voltage Security" if you zoom in on our current address.

Seeing this reminded me of the following exchange from The Jerk, in which an excited Navin R. Johnston (played by Steve Martin) tells his less-than-excited boss Harry Hartounian (played by Jackie Mason) how excited he is that the new phone book has arrived:

NAVIN: The new phone book's here! The new phone book's here!

HARRY: Boy, I wish I could get that excited about nothing.

NAVIN: Nothing? Are you kidding? Page 73 - Johnson, Navin R.! I'm somebody now! Millions of people look at this book everyday! This is the kind of spontaneous publicity - your name in print - that makes people. I'm in print! Things are going to start happening to me now.

So maybe because we're in Google Maps, things are going to start happening for us. (Although if you look at the number of businesses using our technology and our current rate of growth, you'd probably think that that's actually happened already.)

Another Jack Bauer Day

We recently had another Jack Bauer Day at Voltage. These are days when everyone in engineering can do pretty much anything that you want to for 24 hours.

It's truly amazing to see what our engineers can pull off in 24 hours. One of them completely overhauled a feature of our existing products to make it much more user-friendly. Another one single-handedly added a very complicated feature that a big customer was asking for. Yet another managed to implement an identity-based encryption algorithm in JavaScript. Others did more research-oriented things, like implementing a homomorphic encryption scheme.

After the event, each of the teams gave a short presentation on what they had accomplished. It was hard to fit all of these into the two-hour slot that we had, but it was a very interesting two hours.

Every day can be Voltage Day

I just learned that Despair.com, the people who are probably best known for their parodies of corporate motivational posters, sells calendars that you can customize. In addition to picking which demotivational poster appears for each month, you can also customize each day with up to 120 characters so that you'll never miss important events like National S'mores Day or Hug Your Cat Day. A better use might be to label every day as Send an Encrypted Email Using VoltageSecureMail Day. At only 68 character (counting spaces), that's well within the allowed limit for a customization string.

(When I ordered my custom calendar, I didn't actually have every day as Send an Encrypted Email Using Voltage SecureMail Day - I had April 1 marked as Send an Encrypted Email Using XXX Day, where XXX is actually the name of one of our competitors.)

A webinar on EMV

it looks like our marketing guys are having another webinar. This one's Understanding EMV – Future Directions for the U.S. Market, and here's a quick description of what it will cover:

Merchants, issuers and acquirers need to understand the elements that comprise EMV, how EMV adoption may impact PCI compliance initiatives and what additional security needs to be implemented to ensure comprehensive data protection from Point-of-Sale (POS) and Card Not Present (CNP) to back-end Merchant IT systems.

In this webcast you will learn:

Why EMV is gaining more awareness in the United States as the standard for securing interactions from the card to the payment terminal

What threats EMV is designed to mitigate and how EMV affects PCI compliance

Why organizations need to address the complete security of cardholder data across their back-end IT systems

This webinar will be October 12, 2010 at 11 am Pacific/2 pm Eastern and will last 60 minutes.

If that sounds interesting, you can register for this webinar here.

Even more users

In a previous post I mentioned how a recent press release from our marketing people mentioned that we now have over 4.5 million licensed users of SecureMail, but due to the way that we license the software, it's not clear exactly how many total users that represents.

I was recently talking to the people who run our cloud computing offering, VSN, about what their experience says that the right multiple should be. Their answer was about 7, which means that those 4.5 million licensed users represent a total of about 31.5 million total users of SecureMail. That's even more than I first thought.

The Rochester Security Summit

It looks like I'll be talking at the Rochester Security Summit next month about what we've learned at Voltage about the patterns that we can find in data breaches. Some of this material will be what I've mentioned on this blog as well as the article on this topic that I wrote for CSO Magazine, but there will also be lots of new material. If you're interested in this subject, you might want to stop by and see this talk.

Unfortunately, Rochester is three hours ahead of California, so a talk that's scheduled at 9 AM in Rochester feels like it's really at 6 AM to me, and for those of use who aren't morning people, this can be extremely unpleasant. Fortunately, I'll be out in New York a few days before the Rochester Security Summit, so the people that I'm visiting earlier in the week will probably be the ones that get to see me suffering the worst of the effects of the jet lag.

That's a lot of users

Our marketing people issued an interesting press release last week. There was some stuff in it about a huge growth rate, lots of consecutive quarters of profitability, and similar things, but what I found the most interesting is that we now have over 4.5 million licensed users of our SecureMail product.

Note that that's 4.5 million licensed users. Our sales guys typically license our email product to an enterprise by the number of internal users, so the actual number of users is actually much greater than that. Perhaps even much greater. So although it's impossible to get an accurate estimate for how many users we really have, it's not hard to believe that there are probably over 20 million users of SecureMail now.

That's a lot of users.

Get your BN curve here

It looks like some researchers at RWTH Aachen University have an on-line tool for creating BN curves. These are elliptic curves that are particularly useful for pairing-based cryptography, like the identity-based encryption that Voltage uses. If you're interested in implementing pairing-based cryptography, this is a very useful resource to have.

It's generally true that there's no such thing as a free lunch, and this even applies to identity-based encryption. From the user's point of view IBE is great because it's simpler to use than alternatives. From an administrator's point of view IBE iss great because it's extremely simple to keep running. These two combine to make its TCO much lower than the TCO of alternatives.

On the other hand, all of these good features don't come for free, but they really involve things that users and administrators don't see. In particular, it can be very difficult to find elliptic curves that are suitable for use in IBE algorithms. Most elliptic curves don't work very well for this and it can be a bit tricky finding ones that do. BN curves happen to be an example of a type of curve that do work well, so having a place where you can get parameters for such curves can be very useful.

The parameters that you can get from this on-line tool aren't optimized to give you very good performance, so they're not what you'd want to use in a shipping commercial product, but if you're just doing development and testing they're very useful.

Practice-oriented Provable-Security

In 2009, Mihir Bellare and Phil Rogaway shared the ACM's prestigous Paris Kanellakis Theory and Practice Award for their creation of the idea of "Practice-Oriented Provable-Security." Here's the citation for the award that explains why they received it:

Historically, cryptographic schemes used in practice were designed in ad hoc ways and subject to failure. Practice-Oriented, Provable-Security (POPS), developed by Bellare and Rogaway in a series of papers in the 1990s, changed this, giving us the means to create high-assurance practical cryptography, meaning schemes that were backed by the theoretical guarantee of provable security while meeting practical needs and expectations.

Today, POPS-based schemes are cornerstones of Internet security, implemented in most communication security protocols and software - these schemes are used every time someone makes a credit card-based Internet purchase. Meanwhile, the models, techniques and approaches that Bellare and Rogaway introduced, including the random oracle model, have become the foundation of a new subfield of cryptography, inspiring a great amount of follow-on work. Their papers are amongst the most cited in cryptography and their work is discussed in dozens of textbooks.

Bellare and Rogaway changed the perception of theory in practice. Prior to their work, practitioners ignored theory or were even antagonistic to it. Today, they not only choose to implement and standardize proven-secure schemes, but make provable security a requirement in some of their calls for algorithms. That this requirement can be met owes much to Bellare and Rogaway's work. 

In other words, Bellare and Rogaway created a framework for cryptographers to use to prove the security of their inventions and this framework is really the single thing that's most responsible for transforming cryptography from an art into a science.

Before POPS, the only way to ensure that a cryptographic scheme was secure was to wait a while to see if anyone could find a weakness with it. With the invention of POPS that's no longer necessary. It might even be a waste of time to wait to see if a weakness can be found because if there's a valid proof because the very existence of the proof tells you that there can't be one.

Many of the technologies that we use at Voltage have proofs of security. This includes both our Identity-based Encryption and Format-Preserving Encryption. The things that we use that don't have proofs of their security are just things that older standards define: techniques standardized before POPS typically don't have proofs of their security, but there's no really alternative to using them.

I'd hope that newer standards won't have this problem. All of the discussions that I've seen recently in various standards groups have required a proof of security before a new crypotgraphic scheme is taken seriously.

SHARE in Boston in August

Phil Smith of Voltage will be talking at the SHARE meeting in Boston next month. His talk will be on Tuesday, August 3 from 9:30 - 10:30 am, and the topic is Enterprise Encryption 101. Here's a quick summary of what he'll be talking about:

We've all seen the seemingly weekly news about yet another data breach: millions of credit card numbers, SSNs, or other personal information exposed. Encryption is the technology that minimizes the cost of such data breaches, by making the "leaked" data useless to the thief. So more and more sites are investigating encryption, some even before a breach occurs. But where do you start with this technology? How do you make a sensible choice among dozens of vendors, between hardware and software? Where and when do you encrypt data, and is that sufficient? What about emerging standards and legislation, such as PCI DSS, Red Flag, GLBA, SB1386, Directive 95/46/EC, et al.? Come hear about implementing encryption from a business perspective -- what you need to worry about and how to approach it. This is not a comparison of encryption technologies per se, but rather a look at the issues surrounding them. While the presenter works for an encryption vendor, this is a general presentation, with minor content at the end that discusses the Voltage SecureData product as an example.

If you can't make it to Phil's talk, you can download the slides for his talk here. That's probably not quite the same as seeing the talk in person, but it's probably much cheaper.

Voltage Security or Lots Creative Guy?

Last weekend I took my sons to a local game store where they run demos of various boardgames. This particular weekend the demo took longer that usual so I had some time to kill and I tried making entertaining anagrams for "Voltage Security."

One of them, "cattle ye vigours," seemed the one that might be deemed "most likely to be said by a pirate." Maybe this September 19 (talk like a pirate day), I'll hear a few people saying something like "Arr, cattle ye vigours matey!"

Another one, "evil cages tryout" seemed to be a reflection on our fairly rigorous hiring process.

When I got to "lots creative guy" I stopped, thinking that that particular anagram was fairly appropriate. We are known for our innovative technologies, after all.

The location of the 2011 Key Management Summit

We're starting to look for good places to hold the 2011 Key Management Summit. It will almost certainly be held somewhere on the west coast of the US, and probably in California. We have several sites that we're looking at now that are good candidates for this, but we haven't yet decided which one we'll actually use. So if you're interested in attending this event and have a preference for a location for it, now's the time to let us know.  

The next Key Management Summit

It looks like Terence Spies, the CTO of Voltage, will be the general chair of the next Key Management Summit. As soon as Terence gets a chance to get organized, we'll have a better idea of where and when the next event will be held.

Explaining the silence here

It looks like another big credit card processor has signed up to use Voltage's encryption technologies. This time it's Elavon, which means that three of the top five processors (Heartland, Fifth Third and Elavon) are now Voltage customers. Four of the top six POS vendors (Hypercom, Exadigm, UIC, XAC) are also.

This might explain why you don't actually see our sales guys in the office much. When they're not around it's much quieter and easier for the rest of us to work, so I certainly hope that their success in the payments industry continues.

Webinar - Managing Third Party Data Privacy

Our marketing guys have yet another webinar planned, this time it will be held on June 16, 2010 from 10 am Pacific/1 pm Eastern and will last for 60 minutes. The topic this time is Managing 3rd Party Data Privacy - Protecting Your Own and Your Partners Information. Like some of our previous webinars, this one's also sponsored by FS-ISAC, the Financial Services - Information Sharing and Analysis Center.

Here's what they plan to talk about:

• How to ensure the protection of partner information and confidently manage third-party access to card holder and personal data, including protecting terabytes of files on mainframe and legacy systems
• Why a comprehensive end-to-end approach to data protection is critical to ensure the privacy of personal and sensitive information
• What market-leading organizations — such as AAA — really require in an enterprise data protection solution

The fact that one of the infrastructure consultants for AAA will be talking about his experiences is probably a good enough reason to check out this webinar. Every industry has it's own interesting set of problems that it faces and it's fascinating to see how they find clever uses of IT to solve these problems. I've never dealt with an organization like AAA, so I have no idea of the particular challenges that they face, but I'm certainly looking forward to hearing about them on this webinar.

PCI tokenization guidance could benefit payment processors

Interesting news article by industry reporter Rob Westervelt who has been following business and technology trends in the payments sector:

	PCI tokenization guidance could benefit payment processors By Robert Westervelt, News Editor 27 May 2010 | SearchSecurity.com

Hyperelliptic curves

Hyperelliptic curves are interesting for many reasons. The reason that we’re particularly interested in them at Voltage is that you can implement pairings using them and it might turn out that pairings on hyperelliptic curves can be more efficient than pairings on elliptic curves.

An elliptic curve is the set of points defined by an equation like

y² = x³ + ax + b

This gives you a structure that’s much like a torus – a shape that has a single hole in it, like a doughnut.

A hyperelliptic curve is defined by an equation like

y² = f(x)

where f is a monic polynomial of degree 2g + 1.

(There's really no reason to restrict the degree of f to being odd, and a polynomial of degree 2g + 2 will also work just as well. This makes some things trickier, so most people just stick to the odd degree case. One complication is that you actually get two different points at infinity instead of just one. More about that in a future post.)

When g = 1 this reduces to an elliptic curve, but the term “hyperelliptic curve” is usually reserved for the case where g is 2 or greater.

The number g defines the genus of the curve, a number which tells you how many holes the structure corresponding to an elliptic curve’s torus has. If the genus is 2, then the curve has a structure much like a doughnut with 2 holes, etc. The graph above is the graph of a hyperelliptic curve of genus 2.

Working with hyperelliptic curves causes some problems that aren’t there for elliptic curves. It’s possible to easily define a way to add points on an elliptic curve using the usual connect-the-dots algorithm, but this can’t be done for hyperelliptic curves. It’s possible, however, to add sets of g points on a hyperelliptic curve instead of single points. On a curve of genus 2, for example, we might add P = {P₁,P₂} to Q = {Q₁,Q₂} to get R = {R₁,R₂}. (Note that I've used brackets here instead of parentheses to make the notation more set-like. The order in which you list the elements isn't important for a set, and the order of the points isn't important here, either.)

The way that we add these sets of points is by thinking them as divisors, so that we're really adding divisors instead of adding points on a curve. The set of divisors on a curve, along with the rule that defines how to add them is called the Jacobian of the curve.

For a curve of genus g, we can reduce any divisor to one no bigger than one of the form

∑_i=1:g(P_i) – g(O)

For a curve of genus g = 2, for example, we can reduce any divisor to one like

P = (P₁) + (P₂) – 2(O)

So that when we calculate something like

P + Q = R

for a hyperelliptic curve we actually calculating

(P₁) + (P₂) – 2(O) + (Q₁) + (Q₂) – 2(O) = (R₁) + (R₂) – 2(O)

How we find R₁ and R₂ from P₁, P₂, Q₁ and Q₂ is very similar to how we add points on an elliptic curve. In the case of an elliptic curve, we fit a line through two points, find the third point where the line intersects the curve, and reflect this point across the x-axis.

In the case of a hyperelliptic curve of genus 2, we fit a cubic polynomial to the points P₁, P₂, Q₁ and Q₂. We then find the two additional points where this polynomial intersects the curve and then reflect each of those points across the x-axis to get R₁ and R₂. Here’s a picture that shows what it looks like when we add the points P (the two red dots) and Q (the two green dots) to get R (the two black dots).

Because doing operations on hyperelliptic curves are defined in terms of divisors, it shouldn't be much of a surprise that pairings, that are also calculated using divisors, work just as well on hyperelliptic curves as they do on elliptic curves. That means hyperelliptic curves might end up be useful in pairing-based cryptography. If there's a good use for them, they'll probably appear in Voltage's products in the future. Right now, however, elliptic curves seem to be good enough.

Another model for usability

After reading the recent post about the usability lessons that software vendors could learn from the MMORPG Progress Quest, an alert reader suggested another good candidate for a very usable product, and that's the Holly Hop Drive, as seen in the episode "Parallel Universe" of the TV Show Red Dwarf.

Here's how the Holly Hop Drive is described in Red Dwarf:

LISTER: (Holding up the Holly Hop Drive) Is this it?

HOLLY: What do you think?

LISTER: It’s just a box with “STOP” and “START” on it!

HOLLY: It’s fairly straightforward. If you want to start it you press “START,” and you can work out the rest of the controls for yourself.

I can't say for sure whether or not the usability of Voltage's SecureMail was modeled on the Holly Hop Drive, but I don't recall it ever being mentioned. They do seem somewhat similar, though. In one case you just need to press "START" to get it working; in the other case you just need to click on "Send Secure."

Is cloud security a distraction?

An alert reader recently pointed me to an on-line comment that said the following about using computers in the classroom:

My school district blocks content but students are very good at finding and sharing proxy sites. They are also not as tech savvy as administrators and school board members seem to think they are. They are no more interested in using the laptops to learn than they are in using books. The laptops simply give them a more exciting way to be off task.

The alert reader then suggested that a similar comment about cloud computing security might be appropriate. Something like "CIOs are no more interested in cloud security than they are in other aspects of security. Cloud security just gives them a more exciting way to waste time."

I don't know how well it's actually understood by CIOs, but one of the biggest security challenges of cloud computing, is also the easiest to solve, and that's the problem of not exposing sensitive data in cloud computing environments. If you just encrypt your sensitive data that goes into the cloud then most of your problems disappear. That's not wasting time. It's solving a real security problem.

The main reason that all of your problems don't totally disappear is that key management is still a problem. There's actually a good discussion of this in the report Using Encryption to Protect Sensitive Data in Cloud Computing Environments that's available from the Burton Group. That's why Voltage is now developing key management technologies that work well in cloud computing. We have some fairly good solutions or this right now and we'll have some even better ones soon.

You'll probably see those discussed at future Voltage webinars.

Usability lessons from Progress Quest

Voltage is known for its innovative encryption technologies, but we're also known for how easy our products are to use. Not too many years ago, it was extremely hard for the average person to encrypt their email. The classic paper "Why Johnny Can't Encrypt" describes exactly how hard this can be for a typical user and anyone interested in the usability of encryption should read it.

With Voltage's SecureMail, on the other hand, a user doesn't have to do anything more than click on the "Send Secure" button instead of the "Send" button. If you're implementing SecureMail at a gateway appliance, they don't even have to do that – it can just happen automatically. Decrypting is just as easy.

Because we worry so much about the usability of our products, I'm very interested in seeing any enterprise security products that might actually be easier to use than SecureMail. If we ever find one of these, we'll probably be able to learn a thing or two from it. That's why I got so excited when I recently learned of an application that may actually be easier to use than SecureMail. In this case, however, it's not enterprise software. It's the game Progress Quest.

Progress Quest is a massively multiplayer online role-playing game (MMORPG). Before I heard of Progress Quest, I had never actually played a MMORPG, but that didn't stop me from being a government expert on the topic. I say that because I was actually the invited speaker at a government workshop on MMORPGs a couple of years ago. Unfortunately, the fact that I had to sign an NDA for this event means that I can't say much more about it.

Here's how the manual for Progress Quest describes the game:

Progress Quest is a next generation computer role-playing game. Gamers who have played modern online role-playing games, or almost any computer role-playing game, or who have at any time installed or upgraded their operating system, will find themselves incredibly comfortable with Progress Quest's very familiar gameplay. Progress Quest follows reverently in the footsteps of recent smash hit online worlds, but is careful to streamline the more tedious aspects of those offerings. Players will still have the satisfaction of building their character from a ninety-pound level 1 teenager, to an incredibly puissant, magically imbued warrior, well able to snuff out the lives of a barnload of bugbears without need of so much as a lunch break. Yet, gone are the tedious micromanagement and other frustrations common to that older generation of RPG's.

You start Progress Quest by picking the class and race of the character that you'll be playing. After that, the game does everything else for you. I even created a Progress Quest character: Elrond Hubbard, a Demicanadian Ur-Paladin with a name that's almost funny. If you're more adventurous you can pick races like Double Wookiee or Enchanted Motorcycle and classes like Fighter/Organist or Battle-Felon. I wasn't.

If you let Progress Quest run, your character will gradually increase in power and gain useful magical treasures. As I write this, Elrond Hubbard is currently Level 60 and has +23 Fine Gilded Plasma Vambraces. I'm not really sure if that's good or bad, but I certainly didn't have to pay any $9.95 monthly fees to get my character to where he is now.

Surprisingly enough, or at least surprisingly enough to surprise to a one-time government expert like me, Progress Quest seems to be fairly popular. The good reviews of it dramatically outnumber the bad reviews. And that's for a game where the player does absolutely nothing.

I'm never surprised to learn that most people really don't want to worry about encryption at all - they're too busy doing their jobs to worry about fighting with software that's hard to use. But I never would have thought that people would actually enjoy a game in which they do absolutely nothing.

In any event, I suppose that the bottom line is that we haven't quite figured out what we can learn from Progress Quest that will help us make SecureMail better, but that doesn't mean that we won't keep trying.

(If anyone wants to quote me about Progress Quest, here's my position on it: "Of all the games available for the PC, this is one of them.")