ENCYCLOPEDIA OF Espionage, Intelligence, and Security Volume ...

Cryptology, Historyalgorithm underlying the system derives its security fromthe difficulty in factoring very large composite numbers.The RSA algorithm was the mathematical foundation forthe development of a public two-key cryptographic systemcalled Pretty Good Privacy (PGP).Applications of number theory allow the developmentof mathematical algorithms which can make information(data) unintelligible to everyone except for intendedusers. In addition, mathematical algorithms canprovide real physical security to data—allowing only authorizedusers to delete or update data. One of the problemsin developing tools to crack encryption codes involvesfinding ways to factor very large numbers. Advancesin applications of number theory, along with significantimprovements in the power of computers, have madefactoring large numbers less daunting.In general, the larger the key size used in a systemsystems, the longer it will take computers to factor thecomposite numbers used in the keys.Specialized mathematical derivations of number theorysuch as theory and equations dealing with ellipticalcurves are also making an increasing impact on cryptology.Although, in general, larger keys provide increasingsecurity, applications of number theory and elliptical curvesto cryptological algorithms allow the use smaller keyswith any loss of security.Advancements in number theory are also used tocrack important cryptologic systems. Attempting to crackencryoption codes (the encryption procedures) often requiresuse of advanced number theories that allow, forinstance, an unauthorized user to determine the productof the prime numbers used to start the encryption process.Factoring this product is, at best, a time consuming processto determine the underlying prime numbers. An unsophisticatedapproach, for example, might be to simplyattempt or apply all prime numbers. Other more elegantattempts involve algorithms termed quadratic sieves, amethod of factoring integers, developed by Carl Pomerance,that is used to attack smaller numbers, and field sievesalgorithms that are used in attempts to determine largerintegers. Advances in number theory allowed factoring oflarge numbers to move from procedures that, by manualmanipulation, could take billions of years, to proceduresthat—with the use of advanced computing—can be accomplishedin weeks or months. Further advances innumbertheory may lead to the discovery of a polynomialtime factoring algorithm that can accomplish in hourswhat now takes months or years of computer time.Advances in factoring techniques and the expandingavailability of computing hardware (both in terms of speedand low cost) make the security of the algorithms underlyingcryptologic systems increasingly vulnerable.These threats to the security of cryptologic systemsare, in some regard, offset by continuing advances indesign of powerful computers that have the ability togenerate larger keys by multiplying very large primes.Despite the advances in number theory, it remains easierEncyclopedia of Espionage, Intelligence, and Securityto generate larger composite numbers than it is to factorthose numbers.Other improvements related to applications of numbertheory involve the development of ”non-reputable“transactions. Non-reputable means that parties can notlater deny involvement in authorizing certain transactions(e.g., entering into a contract or agreement). Many cryptologistsand communication specialists assert that a globalelectronic economy is dependent on the developmentof verifiable and non-reputable transactions that carry thelegal weight of paper contracts. Legal courts around theworld are increasingly faced with cases based on disputesregarding electronic communications.❚ FURTHER READING:BOOKS:Burn R. P. A Pathway into Number Theory, 2nd. ed. NewYork: Cambridge University Press, 1997 .Niederreiter, Harald. Mathematical Foundations of Codingand Cryptology. Singapore: World Scientific Press,2003 .Wagstaff, Samuel S., Jr., Cryptanalysis of Number TheoreticCyphers Boca Raton, FL: CRC Press, 2002 .SEE ALSOCryptology, HistoryCryptonym❚ JUDSON KNIGHTCryptology, HistoryCryptology is the study of both cryptography, the use ofmessages concealed by codes or ciphers, and cryptanalysis,or the breaking of coded messages. It is nearly as oldas civilization itself, although ciphers and codes prior tothe late medieval period in western Europe tended to beextremely simple by today’s standards. Advances in mathematicsmade possible the development of ever moresophisticated systems. Further improvements in cryptologyaccompanied the creation of modern standing armiesand intelligence services during the nineteenth century.Following the world wars and the creation of the computer,cryptology entered a far more advanced stage,resulting in the creation of codes and ciphers so sophisticatedthat virtually no amount of human genius unaidedby computer technology can break them.Ancient CryptologyEarly examples of cryptology can be found in the work ofMesopotamian, Egyptian, Chinese, and Indian scribes. Inthose four cradles of civilization, which emerged during287