Abstract:We s h ow t hat some widely accepted criteria for cryptographic functions, including t he strict avalanche c r i t erion SAC and t he propagation criterion, have v arious limitations in capturing properties of vital importance to cryptographic algorithms, and propose a new criterion called GAC t o m easure the global avalanche c haracteristics of cryptographic functions. We also introduce two i n dicators related to t he n ew criterion, one forecasts the sum-of-squares while the o t her the absolute avalanche c haracteristics of a function. Lower and u p per bounds on the t wo i n dicators are derived, and t wo methods are presented to construct cryptographic functions that a c hieve n early optimal global avalanche c haracteristics.

A Boolean function is said to be correlation immune if its output leaks no information about its input values. Such functions have many applications in computer security practices including the construction of key stream generators from a set of shift registers. Finding methods for easy construction of correlation immune functions has been an active research area since the introduction of the notion by Siegenthaler. In this paper we study balanced correlation immune functions using the theory of Hadamard matrices. First we present a simple method for directly constructing balanced correlation immune functions of any order. Then we prove that our method generates exactly the same set of functions as that obtained using a method by Camion, Carlet, Charpin and Sendrier. Advantages of our method over Camion et al's include (1) it allows us to calculate the nonlinearity, which is a crucial criterion for cryptographically strong functions, of the functions obtained, and (2) it enables us to discuss the propagation characteristics of the functions. Two examples are given to illustrate our construction method. Finally, we investigate methods for obtaining new correlation immune functions from known correlation immune functions. These methods provide us with a new avenue towards understanding correlation immune functions.

Three of the most important criteria for cryptographidly strong Boolean functions are the balancedness, the nonlinearity and the propagation criterion. This paper studies systematic methods for constructing Boolean fnnctiom satiafying Borne or aH of the three criteria We show that concatenating, splitting, moand multiplying aequaas can yield balanced Boolean funtiom with a very high nonlinearity. In particular, we ahow that balanced Boolean functione obtained by modifying and multiplying oequences achieve a nonlinearity highthan that attainable by any previously h o r n com'€mction method. We ale0 preeeat methods for constructing highly nonlinear b a l a n d Boolean functions satisfying the propagation criterion with respect to off but one or t h e vectors. A technique is developed to tramform the vectors w4ere the propagation criterion is not SstSOd in such a way that the & tions constructed satisfy the propagation criterion of high degree while preservhg the balancednew and nonlinearity of the functions. The alge braic degrem of functions oonetrnded are also & a d , together with eXeUXlph a U S t d * the &OW COMtruCtiOM, I < 1 Preliminaries Let f be a function on V,._The (1,-1)-sequence defined by ((-l)f@O), (-l)f(Q1),. . ., (-l)f(aan-ll) is called the sequence of f, and the (0, 1)-sequence defined by (f (a g) , f(al), ..., f (a p-l) > is called the truth table of f , where ai, 0 5 i 2n-1, denotes the vector in V,, whose integer representation is i. A (0,l)sequence ((1,-1)-sequence) is said baJonccd if it contains an equal number of zeros and ones (ones and minus ones). A function is balanced if its sequence is balanced. Supported in part by the Australian Research Council under the reference numbers ** Supported in part by the Australian Reseatch Cbuncil under the reference n u m k *** Supported in part by the Australian W c h Council under the reference number ~ A49130102, A9030136, A49131885 and A49232172. A49130102.

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Three of the most important criteria for cryptographically strong Boolean functions are the balancedness, the nonlinearity and the propagation criterion. The main contribution of this paper is to reveal a number of interesting properties of balancedness and nonlinearity, and to study systematic methods for constructing Boolean functions satisfying some or all of the three criteria. We show that concatenating, splitting, modifying and multiplying (in the sense of Kronecker) sequences can yield balanced Boolean functions with a very high nonlinearity. In particular, we show that balanced Boolean functions obtained by modifying and multiplying sequences achieve a nonlinearity higher than that attainable by any previously known construction method. We also present methods for constructing balanced Boolean functions that are highly nonlinear and satisfy the strict avalanche criterion (SAC). Furthermore we present methods for constructing highly nonlinear balanced Boolean functions satisfying the propagation criterion with respect to all but one or three vectors. A technique is developed to transform the vectors where the propagation criterion is not satisfied in such a way that the functions constructed satisfy the propagation criterion of high degree while preserving the balancedness and nonlinearity of the functions. The algebraic degrees of functions constructed are also discussed, together with examples illustrating the various constructions.