On the Existence of Secure Keystream Generators

Journal of Cryptology 14 (2001) 1-15.

Andrew Klapper, 779A Anderson Hall, Dept. of Computer Science, University of Kentucky, Lexington, KY, 40506-0046, klapper at cs.uky.edu.

Abstract Designers of stream ciphers have generally used ad hoc methods to build systems that are secure against known attacks. There is often a sense that this is the best that can be done, that any system will eventually fall to a practical attack. In this paper we show that there are families of keystream generators that resist all possible attacks of a very general type in which a small number of known bits of a keystream are used to synthesize a generator of the keystream (called a synthesizing algorithm). Such attacks are exemplified by the Berlekamp-Massey attack. We first formalize the notions of a family of finite keystream generators and of a synthesizing algorithm. We then show that for any function h(n) that is in O(2^{n/d}) for every d>0, there is a secure family B of periodic sequences in the sense that any efficient synthesizing algorithm outputs a generator of size h(log(per(B))) given the required number of bits of a sequence B in B of large enough period. This result is tight in the sense it fails for any faster growing function h(n). We also consider several variations on this scenario.

Index Terms -- Binary sequences, nonlinear feedback registers, security, cryptography, stream ciphers.