Boomerang Spectra of Two Classes of Power Functions via Their Differential Spectra

TL;DR

This paper analyzes the boomerang spectra of specific power functions over finite fields, providing explicit spectra for Niho and Gold functions, which enhances understanding of their cryptographic properties against boomerang attacks.

Contribution

It determines the boomerang spectra of two classes of power functions over finite fields, linking them to their differential spectra and revealing their cryptographic characteristics.

Findings

The boomerang spectrum of the power function x^{2^{m+1}-1} over GF(2^{2m}) is explicitly determined.

The boomerang spectrum of the Gold function x^{2^t+1} over GF(2^n) is two-valued.

These spectra provide insights into the cryptographic strength of the functions against boomerang attacks.

Abstract

In EUROCRYPT 2018, Cid $et\;al.$ introduced a new concept on the cryptographic property of S-boxes to evaluate the subtleties of boomerang-style attacks. This concept was named as boomerang connectivity table (BCT for short) . For a power function, the distribution of BCT can be directly determined by its boomerang spectrum. In this paper, we investigate the boomerang spectra of two classes power functions over even characteristic finite fields via their differential spectra. The boomerang spectrum of the power function $ {x^{{2^{m+1}} - 1}} $ over $ {\mathbb{F}_{{2^{2m}}}} $ is determined, where $2^{m+1}-1$ is a kind of Niho exponent. The boomerang spectrum of the Gold function $G(x)=x^{2^t+1}$ over $ {\mathbb{F}_{{2^n}}} $ is also determined. It is shown that the Gold function has two-valued boomerang spectrum.