The Exact Enumeration of $4$-nomial and $5$-nomial Multiples of the Product of Primitive Polynomials over GF(2)

TL;DR

This paper precisely enumerates 4- and 5-nomial multiples of primitive polynomial products over GF(2), aiding cryptographic design and disproving a previous conjecture on their counts.

Contribution

It provides the exact counts of 4- and 5-nomial multiples of primitive polynomial products over GF(2), advancing understanding of their structure and cryptographic implications.

Findings

Exact number of 4-nomial multiples determined

Exact number of 5-nomial multiples determined

Disproves a previous conjecture on these counts

Abstract

Linear feedback shift registers (LFSRs) are used to generate secret keys in stream cipher cryptosystems. There are different kinds of key-stream generators like filter generators, combination generators, clock-controlled generators, etc. For a combination generator, the connection polynomial is the product of the connection polynomials of constituent LFSRs. For better cryptographic properties, the connection polynomials of the constituent LFSRs should be primitive with coprime degrees. The cryptographic systems using LFSRs as their components are vulnerable to correlation attacks. The attack heavily depends on the $t$-nomial multiples of the connection polynomial for small values of $t$. In 2005, Maitra, Gupta, and Venkateswarlu provided a lower bound for the number of $t$-nomial multiples of the product of primitive polynomials over GF(2). The lower bound is exact when $t=3$. In this article, we provide the exact number of $4$-nomial and $5$-nomial multiples of the product of primitive polynomials. This helps us to choose a more suitable connection polynomial to resist the correlation attacks. Next, we disprove a conjecture by Maitra, Gupta, and Venkateswarlu.