The Computational Difficulty of Solving Cryptographic Primitive Problems Related to the Discrete Logarithm Problem

To the authors’ knowledge, there are not many cryptosystems proven to be as difficult as or more difficult than the discrete logarithm problem. Concerning problems related to the discrete logarithm problem, there are problems called the double discrete logarithm problem and the e-th root of the discrete logarithm problem. These two problems are likely to be difficult and they have been utilized in cryptographic protocols such as verifiable secret sharing scheme and group signature scheme. However, their exact complexity has not been clarified, yet. Related to the e-th root of the discrete logarithm problem, we can consider a square root of the discrete logarithm problem. Again, the exact complexity of this problem has not been clarified, yet. The security of cryptosystems using these underlying problems deeply depends on the difficulty of these underlying problems. Hence it is important to clarify their difficulty. In this paper we prove reductions among these fundamental problems and show that under certain conditions, these problems are as difficult as or more difficult than the discrete logarithm problem modulo a prime. key words: discrete logarithm problem, double discrete logarithm problem, square root of discrete logarithm problem, e-th root of discrete logarithm problem