An upper bound for the linearity of Exponential Welch Costas functions

Abstract The maximum correlation between a function and affine functions is often called the linearity of the function. In this paper, we determine an upper bound for the linearity of Exponential Welch Costas functions using Fourier analysis on Z n . Exponential Welch Costas functions are bijections on Z p − 1 , where p is an odd prime, defined using an exponential function of Z p . Their linearity properties were recently studied by Drakakis, Requena, and McGuire (2010) [1] who conjectured that the linearity of an Exponential Welch Costas function on Z p − 1 is bounded from above by O ( p 0.5 + ϵ ) , where ϵ is a small constant. We prove that the linearity is upper bounded by 2 π p ln p + 4 p , which is asymptotically strictly less than what was previously conjectured.

[1]  Gary McGuire,et al.  On the Security of Blockwise Secure Modes of Operation Beyond the Birthday Bound , 2010 .

[2]  Xiutao Feng,et al.  Linear Approximations of Addition Modulo 2n-1 , 2010, IACR Cryptol. ePrint Arch..

[3]  James L. Massey,et al.  SAFER K-64: One Year Later , 1994, FSE.

[4]  Verónica Requena,et al.  On the Nonlinearity of Exponential Welch Costas Functions , 2010, IEEE Transactions on Information Theory.

[5]  P. Vijay Kumar,et al.  Generalized Bent Functions and Their Properties , 1985, J. Comb. Theory, Ser. A.

[6]  Jacques Stern,et al.  Linear Cryptanalysis of Non Binary Ciphers , 2007, Selected Areas in Cryptography.

[7]  Mitsuru Matsui,et al.  Linear Cryptanalysis Method for DES Cipher , 1994, EUROCRYPT.

[8]  On the exponential sum , 1972 .

[9]  Mitsuru Matsui,et al.  A New Method for Known Plaintext Attack of FEAL Cipher , 1992, EUROCRYPT.

[10]  Rudolf Lide,et al.  Finite fields , 1983 .

[11]  James L. Massey,et al.  SAFER K-64: A Byte-Oriented Block-Ciphering Algorithm , 1993, FSE.

[12]  Xuejia Lai,et al.  A Proposal for a New Block Encryption Standard , 1991, EUROCRYPT.

[13]  O. S. Rothaus,et al.  On "Bent" Functions , 1976, J. Comb. Theory, Ser. A.

[14]  Gary McGuire,et al.  APN permutations on Zn and Costas arrays , 2009, Discret. Appl. Math..

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