The Decisional Diffie-Hellman Problem and the Uniform Boundedness Theorem∗

In this paper, we propose an algorithm to solve the Decisional Diffie-Hellman problem over finite fields, whose time complexity depends on the effective bound in the Uniform Boundedness Theorem (UBT). 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 an efficient nonuniform algorithm to solve DDH on finite fields and a nonuniform algorithm to solve elliptic curve discrete logarithm problem faster than the known algorithms. In the other words, if the Decisional Diffie-Hellman problem over finite fields turns out to be nonuniformly hard, then the effective bound in UBT should be very small.