E. Filiol and C. Fontaine. Highly nonlinear balanced Boolean functions with a good correlation-immunity. In Advances in Cryptology - EUROCRYPT'98, Springer-Verlag, 1998.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to fix one or two parameters and try to optimize others. The most general model is when researchers fix the parameters n (number of variables) and m (order of correlation immunity) and try to optimize some other criptographically important parameters. Here we can call the works [14] 2] [6], 4] 7] 8] 10] The present paper continues the investigations in this direction and gives new results. In Section 2 we give preliminary concepts, notions and some simple lemmas. In Section 3 we establish a new trade off between resiliency and nonlinearity, namely, we prove that the ....

....2 n 2 Gamma1 [13] Thus, nlmax(n; Gamma1) 2 n Gamma1 Gamma 2 n 2 Gamma1 ; 1) This value can be achieved only for even n. The functions with such nonlinearity are called bent functions. Thus, for even n we have nlmax(n; Gamma1) 2 n Gamma1 Gamma 2 n 2 Gamma1 . It is known [11, 12, 6] that for odd n, n 7, nlmax(n; Gamma1) 2 n Gamma1 Gamma 2 (n Gamma1) 2 , and for odd n, n 15, the inequality nlmax(n; Gamma1) 2 n Gamma1 Gamma 2 (n Gamma1) 2 holds. Bent functions are nonbalanced always, so, for balanced (0 resilient) n variable function f we have nl(f) 2 ....

E. Filiol, C. Fontaine, Highly Nonlinear Balanced Boolean Functions with a Good Correlation Immunity, Advanced in Cryptology, Eurocrypt '98, Helsinki, Finland, Lecture Notes in Computer Sciences, Vol. 1403, 1998, pp. 475--488.

....16] Thus, it is very clear that a lot of interest have been generated in this direction. Construction of resilient (balanced correlation immune) functions have direct application as combining functions in certain models of stream ciphers and there are lot of results available in this direction [2, 23, 5, 10, 21, 22, 27, 16]. However, construction results related to unbalanced correlation immune functions has not yet received a lot of attention (though there are some results available in recent papers [27, 16] In this paper we provide a construction of unbalanced correlation immune Boolean functions on even number ....

E. Filiol and C. Fontaine. Highly nonlinear balanced Boolean functions with a good correlation-immunity. In Advances in Cryptology - EUROCRYPT'98. Springer-Verlag, 1998.

....We review some relevant notation, de nitions and known results in the considered area. Since the function f(x) f : F n 2 7 F m 2 , can be regarded as composed of m Boolean functions f = f 1 ; f m ) we rst introduce some concepts for a Boolean function f : F n 2 7 F 2 [3, 4, 9]. A Boolean function f(x) can be expressed in algebraic normal form (ANF) i.e. there are unique constants a 0 ; a 1 ; a 12 ; a 12 n 2 F 2 such that f(x 1 ; x n ) a 0 a 1 x 1 a n x n a 12 x 1 x 2 a 13 x 1 x 3 a 12 n x 1 x 2 x n ; ....

....Boolean function which is balanced is called an m th order resilient (m resilient) function. This paper will be directed towards the study of trade o s between resiliency and nonlinearity. In the special case of Boolean functions (as assumed above) a lot of work has been done, see for example [4, 9, 11, 15, 19, 20, 26]. Now we generalize the notion above to functions F : F n 2 7 F m 2 . Let F : F n 2 7 F m 2 be a function de ned by F (x) f 1 (x) f m (x) where f 1 ; f m are Boolean functions mapping F n 2 7 F 2 . We start with a formal de nition of a resilient function. ....

E. Filiol, and C. Fontaine, Highly nonlinear balanced Boolean functions with a good correlation-immunity Advances in Cryptology - EUROCRYPT'98, Lecture Notes in Computer Science, vol. 1403, pp. 475488, Springer-Verlag, 1998.

....approach: to x one or two parameters and try to optimize others. The most general model is when researchers x the parameters n (number of variables) and m (order of correlation immunity) and try to optimize some other cryptographically important parameters. Here we can call the works [13] 2] [5], 4] 6] 7] 8] 9] 16] The present paper continues the investigations in this direction and gives new results. In Section 2 we give preliminary concepts and notions. In Section 3 we give a brief review of the investigations on the problem of maximal nonlinearity for n variable ....

....nonlinearity of a Boolean function does not exceed 2 n 1 2 n 2 1 [12] Thus, nlmax(n; 1) 2 n 1 2 n 2 1 , This value can be achieved only for even n. The functions with such nonlinearity are called bent functions. Thus, for even n we have nlmax(n; 1) 2 n 1 2 n 2 1 . It is known [10, 11, 5] that for odd n, n 7, nlmax(n; 1) 2 n 1 2 (n 1) 2 , and for odd n, n 15, the inequality nlmax(n; 1) 2 n 1 2 (n 1) 2 holds. Bent functions are nonbalanced always, so, for balanced (0 resilient) n variable function f we have nl(f) 2 n 1 2 n 2 1 , and nlmax(n; m) 2 n 1 2 ....

E. Filiol, C. Fontaine, Highly Nonlinear Balanced Boolean Functions with a Good Correlation Immunity, Advanced in Cryptology, Eurocrypt '98, Helsinki, Finland, Lecture Notes in Computer Sciences, Vol. 1403, 1998, pp. 475-488.

....correlation immunity m, the following holds: m d n. Further, if the function is balanced then m d n 1. Currently, the exact nature of trade o among order of correlation immunity, nonlinearity and algebraic degree has also been investigated [15, 20, 21, 3, 14] Earlier, a series of papers [2, 17, 4, 6, 11, 13, 15] have approached the construction problem by xing the number of variables and the order of correlation immunity (and possibly the algebraic degree) and then trying to design balanced Boolean functions with as high nonlinearity as possible. However, the existence of the current papers [15, 20, 21, ....

E. Filiol and C. Fontaine. Highly nonlinear balanced Boolean functions with a good correlationimmunity. In Advances in Cryptology - EUROCRYPT'98. Springer-Verlag, 1998. 11

....function, of degree d and order of correlation immunity m, the following holds: m d n. Further, if the function is balanced then m d n 1. However, the exact nature of trade o between order of correlation immunity and nonlinearity has not been previously investigated. A series of papers [1, 22, 3, 5, 12, 16, 20] have approached the construction problem in the following fashion. Fix the number of variables and the order of correlation immunity (and possibly the degree) and try to design balanced functions with as high nonlinearity as possible. Many interesting ideas have been used and successively better ....

E. Filiol and C. Fontaine. Highly nonlinear balanced Boolean functions with a good correlationimmunity. In Advances in Cryptology - EUROCRYPT'98. Springer-Verlag, 1998.

....cipher systems. Some of the important properties for a Boolean function to be used in stream cipher systems are balancedness, correlation immunity (CI) algebraic degree and nonlinearity. Construction of Boolean functions possessing a good combination of these properties have been proposed in [4, 10, 11, 13]. However, it is important to study the exact nature of the relationship between the above mentioned properties. This topic has received a lot of attention in recent times as evidenced by the papers [1, 10, 13, 15] The most recent paper to consider these relationships is by Carlet [1] where use ....

E. Filiol, C. Fontaine. Highly nonlinear balanced Boolean functions with a good correlation immunity. In Proceedings of Eurocrypt'99, LNCS 1592 , pages 475-488. 1998.

.... function) A Boolean function must be chosen to resist certain cryptographic attacks (in particular, correlation attacks [15] linear attacks) The different criteria for a Boolean function (balancedness, correlation immunity, nonlinearity, algebraic degree) are studied extensively in many works [14, 5, 3, 13, 7, 2, 4, 16, 8, 10]. It was shown that some criteria can not be satisfied similtaniously, so the problem is to find a necessary tradeoff between them. In this work we study the problem of the constructing of Boolean functions 4 that satisfied to criteria of balancedness and correlation immunity, have good ....

....there are no known methods which can construct such an optimized function with better or even equal nonlinearity . Here using our method (4) we construct a Boolean function with the same parameters and better nonlinearity. Denote by h n a maximal nonlinear function on V n . It is known [11, 12, 1, 8] that for odd n, n 15, the inequality N(h n ) 2 n Gamma1 Gamma 2 (n Gamma1) 2 holds. We start with the function h 19 , N(h 19 ) 2 18 Gamma2 9 . If W (h 19 ) is even then changing the value of h 19 at an arbitrary vector we obtain the function h 0 19 (if W (h 19 ) is odd then we ....

E. Filiol, C. Fontaine, Highly Nonlinear Balanced Boolean Functions with a Good Correlation Immunity, Advanced in Cryptology, Eurocrypt '98, Helsinki, Finland, Lecture Notes in Computer Sciences, Vol. 1403, 1998, pp. 475--488.

....clever way to choose the function g(x) such that the resulting sequence has a large linear span, a long period and good statistical properties. Examples include filter function generators [22, 26, 41, 24, 25, 3, 27, 44, 5, 47, 12, 46, 28, 19, 17, 18, 38, 32, 33] combinatorial function generators [22, 48, 43, 52, 8], and clock controlled generators and shrinking generators[1, 13, 30, 6] Unfortunately, the trace function destroys the structure of Reed Solomon code. It is difficult to get sequences satisfying cryptographic requirements from this approach. If one can fix the linear span, then there is no ....

E. Filiol and C. Fontaine, Highly nonlinear balanced Boolean functions with a good correlation-immunity, Advances in Cryptology-Eurocrypt'98, Lecture Notes in Computer Science, vol. 1403, Springer-Verlag, Berlin, 1998, pp. 474488.

....are introduced while randomness is lessened. In the search for the best possible trade o , to keep good randomness properties forbidding to get combinatorial information on the function input, the function should have the highest possible degree. 2 As a consequence, Boolean functions designed in [11] are more suitable for cryptographic applications than those presented in [22, 32] since these latter have a slightly lower degree. This fact has been con rmed by our tests when considering output sequences produced by nonlinear feedback shift registers. The statistical results are slightly but ....

E. Filiol, C. Fontaine, Highly Nonlinear Balanced Boolean Functions with a Good CorrelationImmunity, Advances in Cryptology - EUROCRYPT'98, Lecture Notes in Computer Sciences 1403, Springer Verlag, 1998.

....recover the initial state of the k target LFSRs independently of the other unknown key bits. Such correlations always exist but functions offering a good cryptographic resistance generally offer a high correlation order k thus imposing on the cryptanalyst to consider simultaneously several LFSRs [6]. In this case it is obvious that the k distinct LFSRs can be seen as a unique LFSR of length L; the length and the feedback polynomial P of this LFSR can be derived from the feedback polynomials of the constituent polynomials [17] In the following we will generalize by speaking of the unique ....

.... affine function u Delta x , 2 F 2 , both seen as Reed Muller codewords [12] dH (f; u Delta x ) 2 n Gamma1 Gamma ( Gamma1) 2 b f (u) It is well known that a combining function must fulfill some criteria to yield a cryptographically secure combination generator (see e.g. [6]) For correlation attacks and fast correlation attacks the main criterion is that f should be far from all affine functions regarding Hamming distance. The existence of a good approximation of f by an affine function makes fast correlation attacks feasible [2, 8, 9] The Hamming distance between ....

E. Filiol, C. Fontaine, Highly Nonlinear Balanced Boolean Functions with a Good Correlation-Immunity, Advances in Cryptology - EUROCRYPT'98, LNCS 1403, Springer Verlag, 1998.

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E. Filiol and C. Fontaine. Highly nonlinear balanced Boolean functions with a good correlation-immunity. In Advances in Cryptology - EUROCRYPT'98, Springer-Verlag, 1998.

No context found.

E. Filiol and C. Fontaine. Highly nonlinear balanced Boolean functions with a good correlation-immunity. In Advances in Cryptology - EUROCRYPT'98, Springer-Verlag, 1998.

No context found.

Eric Filiol and Caroline Fontaine. Highly Nonlinear Balanced Boolean functions with a Good Correlation-Immunity. In Advances in Cryptology EUROCRYPT'98, pages 475--488. Springer Verlag LNCS 1403, 1998.

No context found.

E. Filiol, C. Fontaine, Highly Nonlinear Balanced Boolean Functions with a Good Correlation Immunity, Advanced in Cryptology, Eurocrypt '98, Helsinki, Finland, Lecture Notes in Computer Sciences, Vol. 1403, 1998, pp. 475--488.

No context found.

E. Filiol and C. Fontaine. Highly nonlinear balanced Boolean functions with a good correlation-immunity. In Advances in Cryptology - EUROCRYPT'98. SpringerVerlag, 1998.

No context found.

E. Filiol and C. Fontaine. Highly nonlinear balanced Boolean functions with a good correlation-immunity. In Advances in Cryptology - EUROCRYPT '98. Springer-Verlag, 1998.

No context found.

E. Filiol and C. Fontaine. Highly nonlinear balanced Boolean functions with a good correlation-immunity. In Advances in Cryptology - EUROCRYPT '98. Springer-Verlag, 1998.

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