On the Linear Span of Binary Sequences Obtained from Finite Geometries

Abstract

A class of periodic binary sequences that are obtained from the incidence vectors of hyperplanes in finite geometries is defined, and a general method to determine their linear spans (the length of the shortest linear recursion over GF(2) satisfied by the sequence) is described. In particular, we show that the projective and affine hyperplane sequences of odd order both have full linear span. Another application involves the parity sequence of order n, which has period p" -1 and h e a r span uL(s) where These linear recursive sequences suffer from one drawback: only relatively few terms of the sequence are needed to solve for the generating recursion; i.e., their linear span (the length of the shortest linear recursion over G F ( 2 ) satisfied by the sequence) is short relative to their period.Such easy predictability makes binary n-sequences unsuitable for some applications requiring pseudorandom bits.

Multi-Valued Sequences Generated by Power Residue Symbols over Odd Characteristic Fields

“…According to previous works on geometric sequences [4], [9], [10], a binary sequence is generated with a primitive element ω, trace function Tr (·), and some binarizing function f (·) as Eq. (11), where the trace function maps an extension field element to a prime field element:…”

“…Based on this fact, the proposed approach applies the k-th power residue symbol for mapping the trace sequence S to the multi-valued sequence T , where T consists of integers between 0 to k − 1 and k needs to be a prime factor of p − 1. Most of the previous works in this area are basically focused on binary sequences [4], [5]. On the other hand, this paper adapts power residue symbol for the mapping from S to T , and it is shown that the obtained multi-valued sequences T has a period of n = k(p m − 1)/(p − 1) and a typical periodic autocorrelation.…”

Partial period crosscorrelations of geometric sequences

“…For each of these three cases we need to know the number of times each value of N i,j,τ (s, t, u, v) occurs. This allows us to decompose the sum in equation (1) according to the values of N i,j,τ (s, t, u, v). Fix i, j, τ , with 0 ≤ i, j < D, 0 ≤ τ ≤ q n − 2.…”

“…This large class of pseudorandom sequences includes m-sequences [3], GMW sequences [4,13], Bent sequences [11,14], cascaded GMW sequences [6], the Chan-Games stream cipher [1] and many others. They are readily generated using shift register hardware, may have enormous linear span [1,6,12], and optimally low periodic cross-correlation values [6], so geometric sequences are natural candidates for use in secure spread spectrum applications.…”

Applied cryptography: Protocols, algorithms, and source code in C

“…A construction that involved computing LFSRs over a field of odd characteristic [310] is insecure [842].…”