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

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.

[1]  Claude Carlet,et al.  Covering Sequences of Boolean Functions and Their Cryptographic Significance , 2002, Des. Codes Cryptogr..

[2]  Arne Winterhof,et al.  Approximation of the discrete logarithm in finite fields of even characteristic by real polynomials , 2006 .

[3]  Igor E. Shparlinski,et al.  Cryptographic applications of analytic number theory - complexity lower bounds and pseudorandomness , 2003, Progress in computer science and applied logic.

[4]  Wolfgang M. Schmidt,et al.  Equations over Finite Fields: An Elementary Approach , 1976 .

[5]  C. Carlet.  On Cryptographic Complexity of Boolean Functions , 2002 .

[6]  I. Shparlinski.  Cryptographic Applications of Analytic Number Theory , 2003 .

[7]  Todd Cochrane.  On a trigonometric inequality of vinogradov , 1987 .

[8]  Arne Winterhof,et al.  Incomplete Additive Character Sums and Applications , 2001 .

[9]  Claude Carlet,et al.  On the degree, nonlinearity, algebraic thickness, and nonnormality of Boolean functions, with developments on symmetric functions , 2004, IEEE Transactions on Information Theory.

[10]  Palash Sarkar,et al.  Spectral Domain Analysis of Correlation Immune and Resilient Boolean Functions , 2000, IACR Cryptol. ePrint Arch..

[11]  Alfred Menezes,et al.  Handbook of Applied Cryptography , 2018 .

[12]  Igor E. Shparlinski,et al.  Number Theoretic Methods in Cryptography , 1999 .

[13]  Cunsheng Ding,et al.  Highly nonlinear mappings , 2004, J. Complex..

[14]  Tanja Lange,et al.  Interpolation of the Discrete Logarithm in Fq by Boolean Functions and by Polynomials in Several Variables Modulo a Divisor of Q-1 , 2003, Discret. Appl. Math..

[15]  Tanja Lange,et al.  Incomplete character sums over finite fields and their application to the interpolation of the discrete logarithm by Boolean functions , 2002 .

[16]  Yuliang Zheng,et al.  Connections among nonlinearity, avalanche and correlation immunity , 2003, Theor. Comput. Sci..

[17]  Pantelimon Stanica.  Nonlinearity, local and global avalanche characteristics of balanced Boolean functions , 2002, Discret. Math..