A Proof of the Beierle-Kranz-Leander Conjecture related to Lightweight Multiplication in $\mathds{F}_{2^n}$

TL;DR

This paper proves a conjecture related to optimizing lightweight finite field multiplication in cryptography, which is crucial for resource-constrained devices, by applying linear algebra techniques to confirm the conjecture's correctness.

Contribution

The paper provides a formal proof of the Beierle-Kranz-Leander conjecture on XOR-count optimization in finite field multiplication, advancing cryptographic implementation efficiency.

Findings

Confirmed the conjecture's correctness using linear algebra.

Established optimal basis choice for lightweight multiplication.

Provides a reference for cryptography algorithm development in constrained devices.

Abstract

Lightweight cryptography is a key tool for building strong security solutions for pervasive devices with limited resources. Due to the stringent cost constraints inherent in extremely large applications (ranging from RFIDs and smart cards to mobile devices), the efficient implementation of cryptographic hardware and software algorithms is of utmost importance to realize the vision of generalized computing. In CRYPTO 2016, Beierle, Kranz and Leander have considered lightweight multiplication in $\mathds{F}_{2^n}$. Specifically, they have considered the fundamental question of optimizing finite field multiplications with one fixed element and investigated which field representation, that is which choice of basis, allows for an optimal implementation. They have left open a conjecture related to two XOR-count. Using the theory of linear algebra, we prove in the present paper that their conjecture is correct. Consequently, this proved conjecture can be used as a reference for further developing and implementing cryptography algorithms in lightweight devices.