NRS Marseille France rodierimlunivmrsfr 1 Introduction Boolean functions on the space are not only important in the theory of errorcorrecting codes but also in cryptography where they occur in private key systems In both cases the properties of syst ID: 26016 Download Pdf

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NRS Marseille France rodierimlunivmrsfr 1 Introduction Boolean functions on the space are not only important in the theory of errorcorrecting codes but also in cryptography where they occur in private key systems In both cases the properties of syst

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Onthenonlinearityofbooleanfunctions Fran cois Rodier Institut de Math ematiques de Luminy – C.N.R.S. Marseille – France rodier@iml.univ-mrs.fr 1 Introduction Boolean functions on the space are not only important in the theory of error-correcting codes, but also in cryptography, where they occur in private key systems. In both cases, the properties of systems depend on the nonlinearity of a boolean function. The nonlinearity is linked to the covering radius of Reed-Muller codes cf. for instance P. Langevin [9]). It is also an important cryptographic parameter cf. the article by

F. (habaud and S. Vaudenay [+] and the thesis of (aroline Fontaine [,] or the recent article by (. (arlet [-]). It is useful to have at one.s disposal boolean functions with highest nonlinearity. These functions have been studied in the case where is even, and have been called “bent0 functions cf. 1illon [2]). For these, the degree of nonlinearity is well known, we know how to construct several series of them, but we do not know yet their number, nor their classi3cation cf. works by (arlet, in particular the article of (. (arlet and P. 4uillot, [5]). In the case where is odd, the situation is

quite di7erent. 8e do not know the value of the maximal nonlinearity but for some value of , and we have only a conjecture for the other values. The problem of the research of the maximum of the degree of nonlinearity comes down to minimize the Fourier transform of boolean functions. It is a problem analogous to Fourier series on the real torus, where one wants to minimize the transform of these functions on which take values - for a 3nite set and 0 elsewhere), or one wants to minimize the values of polynomials with coe=cients - random polynomials) on the set of complex numbers of module -. In

this article, we have been inspired by the works of Salem and Zygmund [-5] and by Kahane [A] on random polynomials, and we have transposed them on boolean functions. In this way, we 3nd an evaluation of the mean of the maximum of the absolute values the Fourier transforms of boolean functions, which is not very far from the theoretical minimum value m/ . This gives an evaluation of the mean of the degrees of nonlinearity of these functions. 8e 3nd in particular the fact that the boolean functions have in majority a high nonlinearity, a result found recently by (. (arlet [-]. Moreover, by

transposing a work of 1. Bewman and C. Byrnes [--] on the norms in of polynomials, we studied also a weaker conjecture about the moments of order 2 of the Fourier transform of boolean functions.

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This paper is an extended abstract of the paper [-2] which contains complete proofs of these results. 2 Preliminaries 2.1 Boolean functions Let be a positive integer and D5 Deﬁnition 2.1 A boolean function with variables is a map from the space into E boolean function is linear if it is a linear form on the vector space . It is a=ne if it is equal to a linear function up to a

constant. 2.2 Nonlinearity Deﬁnition 2.2 We call nonlinearity of a boolean function the distance from to the set of aﬃne functions with variables: nl ) D min aﬃne f,h where is the Hamming distance. Gne can show that the degree of nonlinearity is equal to nl )D5 where ) D sup -) )+ and denote the usual scalar product in . 8e call ) the spectral amplitude of the boolean function 2.3 Covering radius of Reed-Muller code of the rst order This spectral amplitude is linked to the covering radius of Reed-Muller code. Indeed a Reed-Muller code of order - on is the vector space of

a=ne boolean functions on . The covering radius of the code is the highest degree of nonlinearity of boolean functions on

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2.4 Known results, conjecture The covering radius of Reed-Muller code of the 3rst order is well known. For an even dimension , bent functions reach the lower bound 5 m/ of spectral amplitude. For odd ,5 m/ 5 has been a long time the only known lower bound of ). In -9A+, Patterson and 8iedemann [-+] have shown that one can do better for D -5. They have exhibited a boolean function such that )D 5, +5 15 They have conjectured that inf m/ -) 2.5 Case of torus

on R Fourier series on the torus that is on the group of complex numbers of module equal to -) present an analogous problem. Let us replace the functions -) for , which are characters of by characters of the torus isx for The conjecture can then be rewritten lim inf s,n isx D- where s,n -. So, it claims that there exists a sequence of polynomials )D =0 s,n with s,n -, and a sequence of positive numbers which tend to zero such that for all D-, | -H Several authors such as C. I. Littlewood [-0], and P. Ird os [K] have asked for the same problem. The latter has conjectured that on the contrary

there exists δ> - such that for all integer and complex number of module -, one has | . C-P. Kahane [A]) solved the problem for complex coe=cients s,n of module -. Le has proved that in this case, lim inf D -. But nothing has been done for the initial problem. Moreover, Kahane used to solve this problem exponentials of the form πin /a , which are exponentials of quadratic forms in , but in our case they do not give any complete result for odd dimensions . 8ith that he works out a polynomial which solves almost the problem. Then he adjusts this polynomial by using an argument of

probabilities. 3 The space of boolean functions with an in nity of variables To study asymptotically the boolean functions, we will need the notion of boolean functions with an in3nity of variables.

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3.1 The space 8e recall that . 8e de3ne M as being the algebra of boolean functions on and as being the space of in3nite sequences of elements of which are almost all equal to zero. 8e de3ne M D M as being the algebra of boolean functions on 8e have the restriction mappings FM 8e will endow M with a probability which will be the Laar measure on it with total mass -. For each , the

probability of the event is given by )D where D5 4 Distribution of 8e have by Parseval identityF 8e will show that in fact ) is often close to 4.1 Upper bound of The following result shows that few boolean function have a high spectral amplitude. Theoreme 4.1 If is a boolean function on , and a positive real, one has -H log Corollary 4.1 We have almost surely lim sup m/ 5 log 5 where is in the space Remark 4.1 In particular, for a given , a majority of boolean functions are of spectral amplitude lower than log )D5 +1 log 5) . Carlet and independently Olejar and Stanek proved this result by

using approximations of sums of binomial coeﬃcients [1, 12]. 4.2 ,ower bound of The following theorem shows that the spectral amplitudes of boolean functions are not too small. It is inspired by Salem and Zygmund [-5] who deal with the real torus.

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Theoreme 4.2 If is a boolean function on , for all such that <η< there exists a constant positive and depending only of such that log Corollary 4.2 We have almost surely lim inf m/ log 5 where is in the space 4.3 -.etches of the proofs The proofs come from the following three ingredients. 8e denote by ) the expectation of

a random variable If is a square integrable random variable, if 0 <λ< - and a> -F - and P If )D -) )+ and exp )), one has ≤E exp λS )) Gne gets by elementary computationsF exp ≤E exp -H exp q exp q 5 -tudying Let us go back using an idea of 1. Bewman and C. Byrnes [--]. They have remarked that, in the case of Fourier series on , the norm in of int had a nice expression. It is the same for boolean functions. For a boolean function, let us denote )D 8e remark that 5 m/ (onsequently, the conjecture -) implies a weaker conjectureF

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Conjecture 5.1 If runs over

the boolean functions on , one has lim inf m/ D- 8e have the following simple expression for ). Lemme 5.1 If is a boolean function on =0 with -) )+ 5.1 Distribution of From lemma 5.-, one can compute the expectations ) and ). Osing these one can prove the following proposition. Proposition 5.1 If is a boolean function on , and a positive real number, +H 20 Corollary 5.1 If , one has almost surely lim m/ D+ 5.2 Asymptotic results From lemma 5.-, we have -D =0 with 5.2.1 Convergence of the distribution of the random variable Proposition 5.2 The distribution of converges in law to the

distribution of density πx x/ x> 0) 5.2.2 A conjecture about the distribution of So, the random variables have almost the same distribution. They seem to be almost independent. In view of the central limit theorem, one may therefore conjecture, that the sequence =0 converges in law to the 4aussian law 0 20) with density A0 80

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This conjecture and the conjecture that lim inf m/ D - would follow from a better understanding of ), or of the ...X ) as it is shown in the next paragraph. Let us de3ne )D log exp =0 If we prove that ) D lim ) exists for every taking eventually

in3nite values) plus some technical conditions, we would deduce by a theorem on large deviation [5], the conjecture 5.-, that is for given , for every large enough, there exists such that <. References [-] (. (arlet, On cryptographic complexity of Boolean functions , to appear in the pro- ceedings of FqK. [5] (. (arlet and P. 4uillot, A characterization of binary bent functions , C. (ombin. Theory Ser. E ,K -99K), +5A–++5. [+] F. (habaud, S. Vaudenay, Links between di7erential and linear cryptanalysis, Iu- rocrypt 92, 950 -992), +5K–+K5. [2] C. 1illon, -lementary Hadamard

.iﬀerence sets ,Th` ese de doctorat, Oniversity of Maryland, -9,2. [5] E. 1embo, G. Zeitouni, Large deviations techni1ues and applications Epplications of Mathematics, +A. Springer-Verlag, Bew Rork, -99A. [K] P. Ird os, Some unsolved problems , Michigan Math. C. 2 -95,), 59-–+00. [,] (. Fontaine, Contribution 2 a la recherche de fonctions bool eennes hautement non lin eaires et au mar1uage d’images en vue de la protection des droits d’auteur ,Th` ese, OniversitS e Paris VI, -99A. [A] C-P. Kahane, Some random series of functions , (ambridge Studies in Edvanced Mathematics, 5. (ambridge

Oniversity Press, (ambridge-Bew Rork, -9A5. [9] P. Langevin, Les sommes de caract2 eres et la formule de Poisson dans la th eorie des codes, des s e1uences et des fonctions bool eennes , Labilitation ` a 1iriger les Recherches, OniversitS e de TGOLGB et du VER, -999, http://www.univ-tln.fr/~langevin/ [-0] C. Littlewood, On polynomials , C. London Math. Soc. 2- -9KK), +K,–+,K.

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[--] 1. Bewman and C. Byrnes, The norm of a polynomial with coeﬃcients -, Emer. Math. Monthly 9, -990), no. -, 25–25 [-5] 1. Glejar, and M. Stanek, On cryptographic properties of random Boolean

functions C.O(S2n A, -99A), ,05–,-,. [-+] B. Patterson and 1. 8iedemann, The covering radius of the 5 15 -K) Reed-Muller codeisatleast -K 5,K, IIII Trans. Inform. Theory 59, n + -9A+), +52–+5K. [-2] F. Rodier, Sur la non-lin earit e des fonctions bool eennes , submitted to Ecta Erith- metica 5005), preprintF http://iml.univ-mrs.fr/editions/preprint2002/preprint2002.html [-5] R. Salem, E. Zygmund Some properties of trigonometric series whose terms have random signs , Ecta Math. 9- -952), 525–+0-

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