## Cryptology ePrint Archive: Report 2019/012

A Proof of the Beierle-Kranz-Leander’s Conjecture related to Lightweight Multiplication in $F_{2^n}$

Sihem Mesnager and Kwang Ho Kim and Dujin Jo and Junyop Choe and Munhyon Han and Dok Nam Lee

Abstract: Lightweight cryptography is an important tool for building strong security solutions for pervasive devices with limited resources. Due to the stringent cost constraints inherent in extremely large applications, 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 ${F}_{2^n}$. 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.

Category / Keywords: implementation / Lightweight cryptography; constant multiplication; Hamming weight; XOR-count; cycle normal form.