Unformatted text preview: 8]. A third system, LuccioMazzone [993], is insecure [717]. A signature scheme based on birational permutations [1425] was broken the day after it was presented [381]. Tatsuaki Okamoto has several signature schemes: one is provably as secure as the Discrete Logarithm Problem, and another is provably as secure as the Discrete Logarithm Problem and the Factoring Problem [1206]. Similar schemes are in [709]. Gustavus Simmons suggested Jalgebras as a basis for publickey algorithms [1455,145]. This idea was abandoned after efficient methods for factoring polynomials were invented [951]. Special polynomial semigroups have also been studied [1619,962], but so far nothing has come of it. Harald Niederreiter proposed a publickey algorithm based on shiftregister sequences [1166]. Another is based on Lyndon words [1476] and another on propositional calculus [817]. And a recent publickey algorithm gets its security from the matrix cover problem [82]. Tatsuaki Okamoto and Kazuo Ohta compare a number of digital signature schemes in [1212]. Prospects for creating radically new and different publickey cryptography algorithms seem dim. In 1988 Whitfield Diffie noted that most publickey algorithms are based on one of three hard problems [492, 494]: 1. Knapsack: Given a set of unique numbers, find a subset whose sum is N. 2. Discrete logarithm: If p is a prime and g and m are integers, find x such that gx a M (mod p). 3. Factoring: If N is the product of two primes, either a) factor N, b) given integers M and C, find d such that Md a C (mod N), c) given integers e and C, find M such that Me a C (mod N), or d) given an integer x, decide whether there exists an integer y such that x a y2 (mod N). According to Diffie [492,494], the Discrete Logarithm Problem was suggested by J. Gill, the Factoring Problem by Knuth, and the knapsack problem by Diffie himself. This narrowness in the mathematical foundations of publickey cryptography is worrisome. A breakthrough in either the problem of factoring or of calculating discrete logarithms could render whole classes of publickey algorithms insecure. Dif...
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This note was uploaded on 10/18/2010 for the course MATH CS 301 taught by Professor Aliulger during the Fall '10 term at Koç University.
 Fall '10
 ALIULGER
 Cryptography

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