Given a primitive element g of a finite field GF(q), the discrete logarithm of a nonzero element u GF(q) is that integer k, 1 k q - 1, for which u = g k . The well-known problem of computing discrete logarithms in finite fields has acquired additional importance in recent years due to its applicability in cryptography. Several cryptographic systems would become insecure if an efficient discrete logarithm algorithm were discovered. This paper surveys and analyzes known algorithms in this area, with special attention devoted to algorithms for the fields GF(2 n ). It appears that in order to be safe from attacks using these algorithms, the value of n for which GF(2 n ) is used in a cryptosystem has to be very large and carefully chosen. Due in large part to recent discoveries, discrete logarithms in fields GF(2 n ) are much easier to compute than in fields GF(p) with p prime. Hence the fields GF(2 n ) ought to be avoided in all cryptographic applications. On the other hand, ...

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In this manuscript we consider the special type of dual basis for finite fields, GF (2 m ), where the variants of m are presented in the following contents. Here we introduce our field representing method for its efficient arithmetic(of field multiplication and field inversion). It revealed a very effective role for both software and VLSI implementations, but the aspect of hardware design for its structure is out of this manuscript and so, here, we deal only the case of its software implementation (the efficiency of hardware implementation is appeared in another article submitted to IEEE Transactions on Computers). A brief description of this advantageous characteristics is that (1) the field multiplication can be constructed only by k( m 2 ) rotations and the same amount of vector XOR processes, (2) there is needed no additional work load as basis changing(from standard to the dual basis or from the dual basis to standard basis as the conventional dual based arithmetic does), (3...

This paper proposes a public key cryptosystem based on the symmetric group Sn, and validates its theoretical foundation. The proposed system benefits from the algebraic properties of Sn such as non commutative, high computational speed and high flexibility in selecting keys which make the Discrete Logarithm Problem (DLP) resistant to attacks by algorithms such as Pohlig-Hellman. Against these properties, the only disadvantage of the scheme is its relative large memory and bandwidth requirements. Due to the similarities in the algebraic structures, many other cryptosystems can be translated to their symmetric group analogs, and the proposed cryptosystem is in fact the Generalized El-Gamal cryptosystem which is based on Sn instead of GF(p). Key words: