On the Non-linearity and Sparsity of Boolean Functions Related to the Discrete Logarithm in Finite Fields of Characteristic Two

• Published in: Coding and Cryptography pp. 135 - 143
• Main Authors: Brandstätter, Nina, Lange, Tanja, Winterhof, Arne
• Format: Book Chapter Conference Proceeding
• Language: English
• Published: Berlin, Heidelberg Springer Berlin Heidelberg 2006; Springer
• Series: Lecture Notes in Computer Science
• Subjects: Applied sciences; Boolean functions; Computer science; control theory; systems; Cryptography; degree; Discrete logarithm; Exact sciences and technology; Information, signal and communications theory; maximal Fourier coefficient; Memory and file management (including protection and security); Memory organisation. Data processing; non-linearity; Signal and communications theory; Software; sparsity; Telecommunications and information theory; Discrete function; Finite field; Discrete logarithm; Coding; Linearity; One way function; Public key cryptography; Boolean function; Sparse representation
• ISBN: 3540354816; 9783540354819
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/11779360_11

In public-key cryptography the discrete logarithm has gained increasing interest as a one-way function. This paper deals with the particularly interesting case of the discrete logarithm in finite fields of characteristic two. We obtain bounds on the maximal Fourier coefficient, i.e., on the non-linearity, on the degree and the sparsity of Boolean functions interpolating the discrete logarithm in finite fields of characteristic two. These bounds complement earlier results for finite fields of odd characteristic. The proofs of the results for odd characteristic involve quadratic character sums and are not directly extendable to characteristic two. Here we use a compensation for dealing with the quadratic character.