Global avalanche characteristics and nonlinearity of balanced Boolean functions

Son, Jung Je; Lim, Jung In; Chee, Seongtaek; Sung, Soo Hak

doi:10.1016/s0020-0190(98)00009-x

Cited by 23 publications

(6 citation statements)

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“…In addition we would like cipher functions to satisfy other cryptographic properties also, such as correlation immunity [20] and the strict avalanche criterion (SAC) [25]. Previous work on the design of balanced functions includes [6,7,[17][18][19]21]. The existing body of research concentrates on specific constructions, supported by algebraic proofs that the resulting Boolean functions will be both balanced and satisfy one or more other properties.…”

Section: Introductionmentioning

confidence: 99%

Heuristic design of cryptographically strong balanced Boolean functions

Millan

Clark

Dawson

1998

Lecture Notes in Computer Science

108

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Abstract. Advances in the design of Boolean functions using heuristic techniques are reported. A genetic algorithm capable of generating highly nonlinear balanced Boolean functions is presented. Hill climbing techniques are adapted to locate balanced, highly nonlinear Boolean functions that also almost satisfy correlation immunity. The definitions for some cryptographic properties are generalised, providing a measure suitable for use as a fitness function in a genetic algorithm seeking balanced Boolean functions that satisfy both correlation immunity and the strict avalanche criterion. Results are presented demonstrating the effectiveness of the methods. IntroductionIt is well known that the resistance of a product cipher to modern cryptanalytic attacks such as linear and differential cryptanalysis [10,1] depends critically upon the nonlinearity of the Boolean functions comprising the round function. Typically these functions must be balanced, so there is considerable interest in the design of highly nonlinear balanced Boolean functions. In addition we would like cipher functions to satisfy other cryptographic properties also, such as correlation immunity [20] and the strict avalanche criterion (SAC) [25]. Previous work on the design of balanced functions includes [6,7,[17][18][19]21]. The existing body of research concentrates on specific constructions, supported by algebraic proofs that the resulting Boolean functions will be both balanced and satisfy one or more other properties. In contrast, some recent publications [13,12] address the issue of applying combinatorial optimisation methods to the design of Boolean functions. The methods of gradient descent (or hill climbing (HC)) and the use of a genetic algorithm have proven useful in the quasi-random generation of highly nonlinear Boolean functions. In this paper we present a modification of the genetic algorithm (GA) presented in [12] so that it is confined to balanced Boolean functions. When combined with the two-step hill climbing algorithm [13] a very effective means of generating highly nonlinear balanced Boolean functions is obtained. The results

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Section: Introductionmentioning

confidence: 99%

Heuristic design of cryptographically strong balanced Boolean functions

Millan

Clark

Dawson

1998

Lecture Notes in Computer Science

108

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“…In order to measure the global properties of Boolean functions, Zhang and Zheng introduced another criterion: the global avalanche characteristics of one Boolean function(GAC) [4], and they gave the lower and upper bounds on the two indicators: the sum-of-squares indicator σ f (2 2n ≤ σ f ≤ 2 3n ) and the absolute indicator △ f (0 ≤ △ f ≤ 2 n ). Son et al [5] derived the balanced Boolean function f (x) ∈ B n satisfying σ f ≥ 2 2n + 2 n+3 (n ≥ 3) and △ f ≥ 8. Sung et al [6] provided an improved lower bound.…”

Section: Introductionmentioning

confidence: 99%

“…(7) and Lemma 2, we have f 1 (x) and f 2 (x) are at most three values: 0, ±L(f ), and are perfectly uncorrelated. [5] shown: the smaller σ f , the better the GAC of a function f (x) ∈ B n . Combing Theorem 2, we know, if two decomposition functions f 1 , f 2 satisfy the two following conditions:…”

mentioning

confidence: 99%

New bounds on the sum-of-squares indicator

Zhou¹,

Dong²,

Zhang³

et al. 2012

7th International Conference on Communications and Networking in China

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The two indicators the sum-of-squares indicator and the absoluate indicator between two different Boolean functions f and g are introduced by Yu Zhou, Min Xie, Guozhen Xiao, On the global avalanche characteristics of two Boolean functions and the higher order nonlinearity. Information Sciences. 180(2010) 256-265, to measure of cryptographic behavior in a global manner. In this paper, we derive a new bound on the sumof-squares indicator. Moreover, we obtain some bounds on the sum-of-squares indicator of a Boolean function f by using two decomposition functions f1, f2. Finally, we give a method to construct balanced Boolean functions with n(n >= 6) variables by the disjoint spectra functions, where n is an even integer, satisfying strict avalanche criterion, high nonlinearity and lower GAC.

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“…Son et al have published lower bounds on σ f for balanced functions [15]. They show that σ f ≥ 2 2n + 2 n+3 (and also give upper bounds on nonlinearity of balanced functions in terms of σ f ).…”

Section: A Experiments With Sum-of-squares As the Cost Functionmentioning

confidence: 99%

“…The tradeoffs between these criteria are improperly understood and have been the subject of much research, e.g. [1], [8], [10], [13], [15], [16], [17]. The more criteria that have to be taken into account, the more difficult the problem.…”

Section: Introductionmentioning

confidence: 99%

Searching for cost functions

Clark¹,

Jacob²,

Stepney³

Proceedings of the 2004 Congress on Evolutionary Computation (IEEE Cat. No.04TH8753)

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Abstract-Boolean function design is at the heart of cryptography, and is the subject of a great deal of theoretical research. We have use a simulated annealing approach to find functions with particular desirable cryptographic properties; for functions of a small numbers of variables, results with properties as good as (and sometimes better than) the best so far have been achieved. The success of this approach is very sensitive to the cost function chosen; here we investigate this property, and describe a metasearch approach to finding the most effective cost function for this class of problems.

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Global avalanche characteristics and nonlinearity of balanced Boolean functions

Cited by 23 publications

References 1 publication

Heuristic design of cryptographically strong balanced Boolean functions

Heuristic design of cryptographically strong balanced Boolean functions

New bounds on the sum-of-squares indicator

Searching for cost functions

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