Inferring Sequences Produced by Nonlinear Pseudorandom Number Generators Using Coppersmith's Methods

International audienceNumber-theoretic pseudorandom generators work by iterating an algebraic map F (public or private) over a residue ring ℤ N on a secret random initial seed value v 0 ∈ ℤ N to compute values for n ∈ ℕ. They output some consecutive bits of the state value v n at each iteration and their efficiency and security are thus strongly related to the number of output bits. In 2005, Blackburn, Gomez-Perez, Gutierrez and Shparlinski proposed a deep analysis on the security of such generators. In this paper, we revisit the security of number-theoretic generators by proposing better attacks based on Coppersmith's techniques for finding small roots on polynomial equations. Using intricate constructions, we are able to significantly improve the security bounds obtained by Blackburn et al.