Register Synthesis for Algebraic Feedback Shift Registers Based on Non-Primes

In this paper, we describe a solution to the register synthesis problem for a class of sequence generators known as Algebraic Feedback Shift Registers. These registers are based on the algebra of π-adic numbers, where π is an element in a ring R, and produce sequences of elements in R/(π). We give several cases where the register synthesis problem can be solved by an efficient algorithm. Consequently, any keystreams over R/(π) used in stream ciphers must be unable to be generated by a small register in these classes. This paper extends the analyses of feedback with carry shift registers and algebraic feedback shift registers by Goresky, Klapper, and Xu.