Codes, bent functions and permutations suitable for DES-like cryptosystems.

*(English)*Zbl 0938.94011Almost bent (AB) and almost perfect nonlinear (APN) functions from \(\{0,1\}^m\) to itself are of importance in several topics in information theory, such as with sequences, correlation-immune and resilient functions, permutations for block ciphers, and for resistance against linear and differential cryptanalysis.

After presenting the basic properties of AB functions, the authors develop the coding-theoretic point of view. To this end, they consider a function \(F\) from \(\text{GF} (2^m)\) to itself with \(F(0)=0\), and relate its properties to the properties of the binary code \(C_F\) with parity check matrix \[ \left[ \begin{smallmatrix} 1&\alpha &\alpha^2 &\ldots &\alpha^{n-1}\\ F(1) &F(\alpha) &F(\alpha^2) &\ldots &F(\alpha^{n-1}) \end{smallmatrix} \right], \] where \(n= 2^m-1\) and \(\alpha\) is primitive in \(\text{GF} (2^m)\).

Properties of the code \(C_F\) are related to \(F\) being AB or APN. Results in coding theory due to Kasami give rise to the only known class of AB functions.

The subject of the paper has recently attracted quite some attention is research. Below are some recent references:

H. Dobbertin, “One-to-one highly nonlinear power functions on \(\text{GF} (2^n)\)”, Appl. Algebra Eng. Commun. Comput. 9, 139–152 (1998; Zbl 0924.94026); “Almost perfect nonlinear power functions on \(\text{GF} (2^n)\): The Niho case”, Inf. Comput. 151, 57–72 (1999; Zbl 1072.94513); “Almost perfect nonlinear power functions on \(\text{GF} (2^n)\): The Welch case”, IEEE Trans. Inf. Theory. 45, 1271–1275 (1999; Zbl 0957.94021); H. D. L. Hollmann and Q. Xiang, “A proof of the Welch and Niho conjectures on cross-correlations of binary \(m\)-sequences”, Finite Fields Appl. 7, No. 2, 253–286 (2001; Zbl 1027.94006).

After presenting the basic properties of AB functions, the authors develop the coding-theoretic point of view. To this end, they consider a function \(F\) from \(\text{GF} (2^m)\) to itself with \(F(0)=0\), and relate its properties to the properties of the binary code \(C_F\) with parity check matrix \[ \left[ \begin{smallmatrix} 1&\alpha &\alpha^2 &\ldots &\alpha^{n-1}\\ F(1) &F(\alpha) &F(\alpha^2) &\ldots &F(\alpha^{n-1}) \end{smallmatrix} \right], \] where \(n= 2^m-1\) and \(\alpha\) is primitive in \(\text{GF} (2^m)\).

Properties of the code \(C_F\) are related to \(F\) being AB or APN. Results in coding theory due to Kasami give rise to the only known class of AB functions.

The subject of the paper has recently attracted quite some attention is research. Below are some recent references:

H. Dobbertin, “One-to-one highly nonlinear power functions on \(\text{GF} (2^n)\)”, Appl. Algebra Eng. Commun. Comput. 9, 139–152 (1998; Zbl 0924.94026); “Almost perfect nonlinear power functions on \(\text{GF} (2^n)\): The Niho case”, Inf. Comput. 151, 57–72 (1999; Zbl 1072.94513); “Almost perfect nonlinear power functions on \(\text{GF} (2^n)\): The Welch case”, IEEE Trans. Inf. Theory. 45, 1271–1275 (1999; Zbl 0957.94021); H. D. L. Hollmann and Q. Xiang, “A proof of the Welch and Niho conjectures on cross-correlations of binary \(m\)-sequences”, Finite Fields Appl. 7, No. 2, 253–286 (2001; Zbl 1027.94006).

Reviewer: L. M. G. M. Tolhuizen (Eindhoven)