A Lattice-Based Method for Recovering the Unknown Parameters of Truncated Multiple Recursive Generators with Constant

Multiple recursive generators with constant, as the high-order extension of linear congruence generators, form an important class of pseudorandom number generators that are widely used in cryptography. The predictability of truncated sequences output by multiple recursive generators with constant that predicts the whole sequences by the truncated high-order bits of the sequences, is a crucial problem in cryptography. This paper studies the predictability of truncated multiple recursive generators with constant. Given a few truncated digits of high-order bits output by a multiple recursive generator with constant, we first convert the multiple recursive generator with constant to multiple recursive generator and then adopt the method we proposed recently to recover the modulus, the coefficients, and the differences of initial state. In particular, we give an estimation of the number of truncated digits required for recovering the differences of initial state by using the expected norm of target vector. We prove by exponential sums that the number of truncated digits required for uniquely determining both the initial state and the constant is finite and give an upper bound. Extensive experiments confirm the correctness of our method.