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Discrete Logarithms in Finite Fields
, 1996
"... Given a finite field F q of order q, and g a primitive element of F q , the discrete logarithm base g of an arbitrary, nonzero y 2 F q is that integer x, 0 x q \Gamma 2, such that g x = y in F q . The security of many realworld cryptographic schemes depends on the difficulty of computing discr ..."
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Given a finite field F q of order q, and g a primitive element of F q , the discrete logarithm base g of an arbitrary, nonzero y 2 F q is that integer x, 0 x q \Gamma 2, such that g x = y in F q . The security of many realworld cryptographic schemes depends on the difficulty of computing discrete logarithms in large finite fields. This thesis is a survey of the discrete logarithm problem in finite fields, including: some cryptographic applications (password authentication, the DiffieHellman key exchange, and the ElGamal publickey cryptosystem and digital signature scheme); Niederreiter's proof of explicit formulas for the discrete logarithm; and algorithms for computing discrete logarithms (especially Shank's algorithm, Pollard's aemethod, the PohligHellman algorithm, Coppersmith's algorithm in fields of order 2 n , and the Gaussian integers method for fields of prime order).