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Unformatted text preview: 8]. A third system, Luccio-Mazzone , is insecure . A signature scheme based on birational permutations  was broken the day after it was presented . 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 . Similar schemes are in . Gustavus Simmons suggested J-algebras as a basis for public-key algorithms [1455,145]. This idea was abandoned after efficient methods for factoring polynomials were invented . Special polynomial semigroups have also been studied [1619,962], but so far nothing has come of it. Harald Niederreiter proposed a public-key algorithm based on shift-register sequences . Another is based on Lyndon words  and another on propositional calculus . And a recent public-key algorithm gets its security from the matrix cover problem . Tatsuaki Okamoto and Kazuo Ohta compare a number of digital signature schemes in . Prospects for creating radically new and different public-key cryptography algorithms seem dim. In 1988 Whitfield Diffie noted that most public-key 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 public-key cryptography is worrisome. A breakthrough in either the problem of factoring or of calculating discrete logarithms could render whole classes of public-key algorithms insecure. Dif...
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- Fall '10