Prime Finite Field Arithmetic

Abstract

The modular exponentiation operation is a common operation for scrambling; it is used in several cryptosystems. For example, the Diffie-Hellman key exchange scheme requires modular exponentiation [64]. Furthermore, the ElGamal signature scheme [80] and the Digital Signature Standard (DSS) of the National Institute for Standards and Technology [90] also require the computation of modular exponentiation. However, we note that the exponentiation process in a cryptosystem based on the discrete logarithm problem is slightly different: The base (M) and the modulus (n) are known in advance. This allows some precomputation since powers of the base can be precomputed and saved [35]. In the exponentiation process for the RSA algorithm, we know the exponent (e) and the modulus (n) in advance but not the base (M); thus, such optimizations are not likely to be applicable.