By Huaxiong Wang, Yongqing Li, Shengyuan Zhang

Li Y., Ling S., Niederreiter H., Wang H., Xing C. (eds.) Coding and Cryptology.. lawsuits of the overseas Workshop, Wuyi Mountain, Fujian, China, 11-15 June 2007 (WS, 2008)(ISBN 9812832238)(O)(288s)

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Algebraic attacks and decomposition of Boolean functions. In Advances in Cryptology - EUROCRYPT 2004, number 3027 in Lecture Notes in Computer Science, pp. 474–491. Springer Verlag, 2004. 31. F. Rodier. Asymptotic nonlinearity of Boolean functions. Designs, Codes and Cryptography, no 40:1 2006, pp 59-70. 32. S. Rønjom and T. Helleseth. A new attack on the filter generator. IEEE Transactions on Information theory 53 (5), pp. 1752-1758, 2007. 33. C. E. Shannon. Communication theory of secrecy systems.

A. Kabatiansky and V. I. Levenshtein, Bounds for packings on the sphere and in the space, Problems of Information Transmission. 14(1), 3–25, (1978). 11. V. M. Sidelnikov, Extremal polynomials used in bounds of code volume, Problemy Peredachi Informatsii. 16(3), 17–30, (1980). 12. C. Bachoc, Linear programming bounds for codes in Grassmannian spaces, IEEE Trans. Inform. Theory. 52, 2111–2126, (2006). fr. Because of the recent algebraic attacks, a high algebraic immunity is now an absolutely necessary (but not sufficient) property for Boolean functions used in stream ciphers.

It (n) is upper bounded by 21+n/2 n/2 , but this upper bound is approximately in Ω 2 √ n n/2 √ 2 2π , and is therefore asymptotically huge. 2 is not automatically verified, we still need to exhibit a related infinite class of functions. In ref. 8 is asserted that the function whose support equals the union of the set of vectors of weight n2 − 4 and of the set of vectors of weights at least n2 except those of weight n2 + 4 has optimum algebraic immunity. This result is probably false (see ref. 5) – it is true up to some (even) value of n.