Solving X^2^2k+2^k+1+(X+1)^2^2k+2^k+1=b over \GF2^4k

2023/05/22 by Kwang Ho Kim, Sihem Mesnager, Kim, Kwang Ho +3
Computer Science · Social Sciences · #Coding theory and cryptography #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Islamic Finance and Communication #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2305.12645

openalex publication_date 2023/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let F(X)=X^2^2k+2^k+1 be the power function over the finite field \GF2^4k which is known as the Bracken-Leander function. In \citeBCC10,BL10,CV20,Fu22,XY17, it was proved that the number of solutions in \GFq⁴ to the equation F(X)+F(X+1)=b is in \0,2,4\ for any b∈ \GFq⁴ and the number of the b giving i solutions have been determined for every i. However, no paper provided a direct and complete method to solve such an equation, and this problem remained open. This article presents a direct technique to derive an explicit solution to that equation. The main result in \citeBCC10,BL10,Fu22,XY17, determining differential spectrum of F(X)=X^2^2k+2^k+1 over \GF2^4k, is re-derived simply from our results.

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