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, [1]), 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. [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. [2]
N. Zierler and W.H. Mills, Products of Linear Recurring Sequences, J. of Algebra 27(1973), 147–157.Google Scholar