Autocorrelations of Maximum Period FCSR Sequences

Abstract

Let a be a maximum period feedback with carry shift register sequence (l-sequence) with connection integer q = p e and period T = p e−1 (p − 1). It is shown that the expected value of its autocorrelations is 0, and its variance is O(q ln 4 q). Thus when q is sufficiently large, with high probability, the autocorrelations are low. Furthermore, it is shown that when e ≥ 2, for any integer i, 1 ≤ i ≤ e/2, when the shift is a multiple of T /2p i , the absolute value of the autocorrelations of a is T /p 2i−1 , and the sign relies on the parity of the multiple. Introduction.Pseudorandom sequences are important in many areas of communications and computing such as cryptography, spread spectrum communications, error correcting codes, and quasi-Monte Carlo integration. In the study of pseudorandom sequences, we are often interested in the correlation properties of the sequences. These properties not only are important measures of randomness [4] but also have practical applications in spread spectrum communication systems, radar systems, cryptanalysis, and so on.Feedback with carry shift register (FCSR) sequences, especially the l-sequences, have many fine pseudorandom properties analogous to those of m-sequences. Let q = p e , with p an odd prime integer and e ≥ 1, and let 2 be a primitive root modulo q. The class of binary sequences known as l-sequences can be described in several ways [8], [9]. An l-sequence is the output sequence from a maximum period FCSR with connection number q. It is the 2-adic expansions of a rational number r/q, where gcd(r, q) = 1, and it is the sequence a n = (A · 2 −n (mod q))(mod 2), where gcd(A, q) = 1.Up to now, research has been done on the distribution properties and linear complexity of l-sequences [5], [6], [7], [8], [9], [13], [14]. The lattice test on such sequences has also been done by Couture and L'Ecuyer [1], [2], [3] and L'Ecuyer [10], [11]. Their ordinary autocorrelations have not been studied, although their arithmetic autocorrelations have been shown to be zero [5].The autocorrelation function of a binary periodic sequence a = (a 0 , a 1 , a 2 , . . . ) with period T is defined as C a (τ ) =

A note on the crosscorrelation of maximal length FCSR sequences

“…In [6], the authors have investigated the expected value and the variance of autocorrelations of an l-sequence. They derived the following main result.…”

“…As for usual correlation properties of l-sequences, it is relatively difficult to research. Instead of directly evaluating autocorrelations of an l-sequence, [6] has investigated the expected value and the variance of them. But up to now almost no paper, in the literature, has given any theoretical result on the usual crosscorrelation of l-sequences as far as we know.…”

Linearity properties of binary FCSR sequences

“…It is widely believed that l-sequences have very good pseudorandom properties, and research has been done on distribution properties, linear complexities and correlation properties of them, see [2][3][4][5][6]. But an l-sequence should never be used by itself as a keystream for its low 2-adic complexity [7].…”

Algebraic Shift Register Sequences

“…The ordinary autocorrelations of -sequences seems to be quite difficult to compute, although Xu and Qi have been able to do so for some special shifts [192]. Instead we consider the arithmetic version of the autocorrelation.…”