Open Problems on With-Carry Sequence Generators
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Pseudorandom sequences are used in a wide range of applications in computing and communications, including cryptography. It is common to use linear feedback shift registers (LFSRs) to generate such sequences, either directly or as components in more complex structures. Much of the analysis of such sequences is done using the algebra of polynomials and power series over finite fields. The subjects of this chapter are feedback with carry shift registers (FCSRs) and algebraic feedback shift registers (AFSRs, generalizations of both LFSRs and FCSRs), sequence generators that are analogous to LFSRs, but whose state update involves arithmetic with a carry. Their analysis is based on algebraic structures with carry, such as the integers and the N-adic numbers. After a brief review of the basics on LFSRs, FCSRs, and AFSRs, we describe several open problems. These include: given part of a sequence, how to find an optimal generator of the sequence; how to construct sequences that cannot be generated by short LFSRs, FCSRs, or AFSRs; and the analysis of various statistical properties related to these generators.