A discrete logarithm problem over composite modulus

Abstract

AbstractRecently, many applications of integer theory to cryptographic techniques have been observed. The discrete logarithm problem is one such case. Usually, the discrete logarithm problem is the determination of the logarithm for the given arbitrary element with a prime number as the modulus. However, the discrete logarithm problem can also be considered with a composite number as the modulus. It is anticipated that the discrete logarithm problem with a composite number as the modulus is a difficult problem if the prime factors of the composite number, which is used as the modulus, are unknown. Then the problem can be applied to the cryptography.In the general discrete logarithm problem with a composite number as the modulus, it is not always true that an arbitrary element has a logarithm. From such a viewpoint, this paper shows that the exponent of an arbitrary element belonging to the irreducible residue class with a composite number as the modulus has a logarithm. Then the necessary condition in the application to the cryptographic technique is presented.Finally, as an application example of the technique proposed in this paper, a cryptographic technique based on the discrete logarithm problem with the composite number as the modulus is shown.