EUROCRYPT 1985: Advances in Cryptology — EUROCRYPT’ 85 pp 135-141

# Generalized Multiplexed Sequences

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 219)

## Abstract

Let LSR 1, LSR2, ..., LSRk and LSR be k+1 linear feedback shift registers with characteristic polynomials f1(x), f2(x), ..., fk(x) and g(x) over $$\mathbb{F}_2$$ and output sequences a 1, a 2, ..., a k and b respectively, where a i=(ai0, ai1, ...), i=1, 2, ..., k, b=(b0, b1, ...). Let $$\mathbb{F}_2^k = \left\{ {\left( {c_1 , c_2 , ..., c_k } \right)\left| {c_k \in \mathbb{F}_2 } \right.} \right\}$$ be the k-dimensional space over $$\mathbb{F}_2$$ and γ be an injective map from $$\mathbb{F}_2^k$$ into the set {0, 1, 2, ..., n−1}, 2k⩽n, of course. Constructing k-dimensional vector sequence A=(A0, A1, ...) where At=(a1t, a2t, ..., akt), t=0, 1, 2, 3, ... and applying γ to each term of the sequence A, we get the sequence γ(A)=(γ(A0), γ(A1), ...) where γ(At) ∈ {0, 1, ..., n−1}, for all t. Using γ(A) to scramble the output sequence b of LSR, we get the sequence u=(u0, u1, ...) where $$u_t = b_{t + \gamma \left( {A_t } \right)}$$, for all t. we call γ a scrambling function and u the Generalized Multiplexed Sequence (generalizing Jenning’s Multiplexed Sequence, see ref, ), in brief, GMS. In the present paper, the period, characteristic polynomial, minimum polynomial and translation equivalence properties of the GMS are studied under certain assumptions. Let Ω be the algebraic closure of $$\mathbb{F}_2$$. Throughout this paper, andy algebraic extension of $$\mathbb{F}_2$$ are assumed to be contained in Ω. Let f(x) and g(x) be polynomials over $$\mathbb{F}_2$$ without multiple roots. Let f*g be the monic polynomial whose roots are all the distinct elements of the set S={α·β | α, β∈Ω, f(α)=0, g(β)=0}. It is well known that f*g is polynomial over $$\mathbb{F}_2$$. Let G(f) denote the vector space consisting of all output sequences of LSR with characteristic polynomial f(x).

## References

1. 
S.M. Jennings, Multiplexed Sequences: Some Properties of the Minimum Polynomial. Lecture Notes in Computer Science, No.149, Springer-Verlag, 1983, 189–206.Google Scholar
2. 
N. Zierler and W.H. Mills, Products of Linear Recurring Sequences, J. of Algebra 27(1973), 147–157.Google Scholar