2018/11/27 by Sihem Mesnager, Mesnager, Sihem, Kwang Ho Kim +3
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #Cryptographic Implementations and Security #FOS: Computer and information sciences #Information Theory (cs.IT) #cs.IT #graph theory and CDMA systems #math.IT
paper · pdf · doi:10.48550/arxiv.1811.11280
arxiv created 2018/11/27 · openalex publication_date 2018/11/27 · arxiv updated 2018/11/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
To determine the dimension of null space of any given linearized polynomial is one of vital problems in finite field theory, with concern to design of modern symmetric cryptosystems. But, the known general theory for this task is much far from giving the exact dimension when applied to a specific linearized polynomial. The first contribution of this paper is to give a better general method to get more precise upper bound on the root number of any given linearized polynomial. We anticipate this result would be applied as a useful tool in many research branches of finite field and cryptography. Really we apply this result to get tighter estimations of the lower bounds on the second order nonlinearities of general cubic Boolean functions, which has been being an active research problem during the past decade, with many examples showing great improvements. Furthermore, this paper shows that by studying the distribution of radicals of derivatives of a given Boolean functions one can get a better lower bound of the second-order nonlinearity, through an example of the monomial Boolean function gμ=Tr(μx^22r+2r+1) over any finite field \GFn.