(A) study of provably secure pseudo-random generator and hard-core predicate

Abstract

Provable security is an important area in modern cryptography. A cryptographic scheme is said to be provably secure if defeating it can be shown to be essentially as difficult as solving a well-known and supposedly difficult problem. Recently, the conjugacy problem in braid groups has been turned out to be cryptographically useful. In this thesis, we study based on the conjugacy problem two fundamental areas of provably secure cryptographic primitives: one-way functions and pseudorandom generators. The difficulty of inverting a one-way function induces a notion of hard-core predicate. Our results are twofold. On the one hand, we view the conjugacy problem as a one-way function. To apply this problem to provably secure schemes, we describe formally the conjugacy problem as a collection of one-way functions in terms of the theory of computation. For this one-way function collection we present a hard-core predicate showing that it inherits the difficulty of solving the conjugacy problem. On the other hand, we view the decisional Ko-Lee problem, a variant of the conjugacy problem, as a computationally infeasible problem. To apply this to provably secure schemes, we describe formally the decisional Ko-Lee assumption in terms of the theory of computation. From this, we construct two practical pseudorandom schemes: a pseudorandom generator and a pseudorandom synthesizer. And then we show that they are provably as secure as the decisional Ko-Lee assumption.