Abstract
The propagation criterion is one of the main cryptographic criteria on Boolean functions used in block ciphers. Quadratic Boolean functions satisfying the propagation criterion of high degree were given by Preneel et al., but their algebraic degree is too small for a cryptograhic use. Then designing Boolean functions of high algebraic degree and high degree of propagation has been the goal of several papers. In this paper, we investigate the work of Kurosawa and Satoh in order to optimize the algebraic degree and the degree of propagation, and the work of Honda, Satoh, Iwata, and Kurosawa, by giving in particular a construction of Boolean functions satisfying PC(3) and having a very large algebraic degree. We also show that among symmetric functions, only the quadratic ones satisfy the propagation criterion of degree greater than 1. A particular case of this result is that symmetric bent functions must be quadratic – a result that needed a whole paper to be proved before.
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References
A. Bernasconi, On Boolean functions satisfying odd order propagation criteria, 3rd International Workshop on Boolean Problems, IWSBP’98, (1998), 117–124.
A. Bonnecaze, P. Solé, and A.R. Calderbank, Quaternary quadratic residue codes and unimodular lattices, IEEE Transactions on Information Theory 41 (1995), 366–377.
P. Camion, C. Carlet, P. Charpin, and N. Sendrier, On correlation-immune functions, in Advances in Cryptology Proc. of Crypto ‘81, LNCS576 (1991), 86–100.
C. Carlet, On the propagation criterion of degree l and order k in Advances in Cryptology Proc. of EUROCRYPT’98, LNCS1403 (1998), 462–474.
C. Carlet, On cryptographic propagation criteria for Boolean functions, Special Issue on Cryptology of Information and Computation 150 (1999), 32–56.
C. Carlet and P. Guillot, A new representation of Boolean functions, AAECC, (1999), 94–103.
J. Daemen, R. Govaerts, and J. Vandewalle, A practical approach to the design of high speed self-synchronizing stream ciphers, Singapore ICCS/ISITA ‘82 Conference Proceedings, IEEE, (1992), pp. 279–283.
J. F. Dillon. Elementary Hadamard Difference sets, Ph. D. Thesis, Univ. of Maryland, 1974.
R. Forré, The strict avalanche criterion: spectral properties of Boolean functions and an extended definition, in Advances in Cryptology Proc. of CRYPTO’88, LNCS403 (1989), 450–468.
J. von zur Gathen and J. Roche, Polynomials with two values Combinatorica 17 (3) (1997), 345–362.
T. Honda, T. Satoh, T. Iwata, and K. Kurosawa, Balanced Boolean functions satisfying PC(2) and very large degree, in Proceedings of SAC’97, (1997), 64–72.
K. Kurosawa and T. Satoh, Design of SAC/PC(l) of order k Boolean functions and three other cryptographic criteria, in Advances in Cryptology, Proc. of EURO-CRYPT ‘87, LNCS1223 (1997), 434–449.
W. Meier and O. Staffelbach, Nonlinearity criteria for cryptographic functions, in Advances in Cryptology Proc. of EUROCRYPT’89, LNCS434 (1990), 549–562.
V.S. Pless and W.C. Huffman (Eds.), The Handbook of Coding Theory, North-Holland, New York, 1998.
V.S. Pless and Z. Qian, Cyclic codes and quadratic residue codes over 7L4 IEEE Transactions on Information Theory 42(5) (1996), 1594–1600.
B. Preneel, W. Van Leekwijck, L. Van Linden, R. Govaerts, and J. Vandewalle, Propagation characteristics of Boolean functions, in Advances in Cryptology Proc. of EUROCRYPT’90, LNCS473 (1991), 161–173.
B. Preneel, R. Govaerts, and J. Vandewalle, Boolean functions satisfying higher-order propagation criterion, in Advances in Cryptology Proc. of Eurocrypt’91, LNCS547 (1991), 141–152.
O.S. Rothaus, On bent functions Journal of Combinatorial Theory (A) 20 (1976), 300–305.
P. Savicky, On the bent functions that are symmetric European J. of Combinatorics 15 (1994), 407–410.
T. Siegenthaler, Correlation-immunity of nonlinear combining functions for cryptographic applications IEEE Transactions on Information Theory 30(5) (1984), 776–780.
A.F. Webster and S.E. Tavares, On the design of S-box, in Advances in Cryptology Proc. of CRYPTO’85, LNCS218 (1986), 523–534.
Y. Zheng and X. M. Zhang, On relationships among avalanche, nonlinearity, and correlation-immunity, in Advances in Cryptology Proc. of ASIACRYPT’00, LNCS 1976 (2000), 470–482.
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Gouget, A. (2004). On the Propagation Criterion of Boolean Functions. In: Feng, K., Niederreiter, H., Xing, C. (eds) Coding, Cryptography and Combinatorics. Progress in Computer Science and Applied Logic, vol 23. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7865-4_9
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DOI: https://doi.org/10.1007/978-3-0348-7865-4_9
Publisher Name: Birkhäuser, Basel
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