Nonuniform polynomial time algorithm to solve decisional diffie-hellman problem in finite fields under conjecture

Abstract

In this paper, we show that curves which are defined over a number field of small degree but have a large torsion group over the number field have considerable cryptographic significance. If those curves exist and the heights of torsions are small, they can serve as a bridge for prime shifting, which results a nonuniform polynomial time algorithm to solve DDH on finite fields and a nonuniform subexpontial time algorithm to solve elliptic curve discrete logarithm problem. At this time we are unable to prove the existence of those curves. To the best of our knowledge, this is the first attempt to apply the ideas related to the Uniform Boundedness Theorem(UBT), formerly known as Uniform Boundedness Conjecture, in cryptography.