Soo Hak Sung

New Block Cipher: ARIA

In this paper, we propose a 128-bit block cipher ARIA which is an involution substitution and permutation encryption network(SPN). We use the same S-boxes as Rijndael to eliminate defects which are caused by a totally involution structure. In the diffusion layer of ARIA, a 16× 16 binary matrix of the maximum branch number 8 is used to avoid some attacks well applied to the reduced round of Rijndael. ARIA uses only basic operations, S-box substitutions and XOR’s together with an involution structure so that it can be efficiently implemented on various platforms.

Almost sure convergence theorems of weighted sums of random variables

Let be random variables and a triangular array of constants. In this paper we find the various conditions on {ani } and {Xn } under which converges to zero almost surely. It is shown that for the identically distributed random variables the conditions a.s. A generalization of Marcinkiewiczs law of large numbers is given. It is shown that for the i.i.d. case the conditions and EX = 0 imply that a.s

On the Correlation Immune Functions and Their Nonlinearity

In this paper, we consider the relationship between the nonlinearity and correlation immunity of functions suggested in [1], [3]. For the analysis of such functions, we present a simple method of generating the same set of functions, which makes us possible to construct correlation immune functions with controllable correlation immunity and nonlinearity.

Strong laws for weighted sums of i.i.d. random variables

Strong laws of large numbers are established for the weighted sums of i.i.d. random variables which have higher-order moment condition. One of the results of Bai and Cheng (2000, Statist. Probab. Lett. 46, 105-112) is extended.

On the Security of Rijndael-Like Structures against Differential and Linear Cryptanalysis

Rijndael-like structure is a special case of SPN structure. The linear transformation of Rijndael-like structures consists of linear transformations of two types, the one is byte permutation ? and the other is linear transformation ? = (?1, ?2, ?3, ?4), where each of ?i separately operates on each of the four columns of a state. Furthermore, ? and ? have some interestingprop erties. In this paper, we present a new method for upper boundingthe maximum differential probability and the maximum linear hull probability for Rijndael-like structures. By applyingour method to Rijndael, we obtain that the maximum differential probability and the maximum linear hull probability for 4 rounds of Rijndael are bounded by 1.06 × 2-96.

Moment Inequalities and Complete Moment Convergence

Let and be sequences of random variables. For any and , bounds for and are obtained. From these results, we establish general methods for obtaining the complete moment convergence. The results of Chow (1988), Zhu (2007), and Wu and Zhu (2009) are generalized and extended from independent (or dependent) random variables to random variables satisfying some mild conditions. Some applications to dependent random variables are discussed.

Global avalanche characteristics and propagation criterion of balanced Boolean functions

Abstract The GAC (Global Avalanche Characteristics) were introduced by Zhang and Zheng (1995) as a measure of cryptographic strength of Boolean functions. Two indicators σ ƒ and Δ ƒ related to GAC are introduced. Son et al. (1998) gave a lower bound on σ ƒ for a balanced Boolean function. In this paper, we provide an improved lower bound. Moreover, we provide bounds on nonlinearity for a balanced Boolean function satisfying the propagation criterion with respect to t vectors.

Global avalanche characteristics and nonlinearity of balanced Boolean functions

Abstract The GAC (global avalanche characteristic) was introduced to measure cryptographic behavior, such as, propagation characteristic, in a global manner. The two indicators σf and Δf related to GAC are introduced. However, an important problem to compute lower bounds of two indicators for balanced Boolean functions was not solved yet. In this paper, we derive lower bounds on the two indicators for balanced Boolean functions: σf ⩾ 22n + 2n + 3 and Δf ⩾ 8. Moreover, we obtain an upper bound on nonlinearity for balanced Boolean functions: N f ⩽ 2 n−1 − 1 2 2 − n 2 √σ f , which improves the previously known results.

WEAK LAWS OF LARGE NUMBERS FOR ARRAYS UNDER A CONDITION OF UNIFORM INTEGRABILITY

For an array of dependent random variables satisfying a new notion of uniform integrability, weak laws of large numbers are obtained. Our results extend and sharpen the known results in the literature.

COMPLETE CONVERGENCE FOR ARRAYS OF ROWWISE INDEPENDENT RANDOM VARIABLES

Abstract. Under some conditions on an array of rowwise inde-pendent random variables, Hu et al.(1998) obtained a completeconvergence result for law of large numbers with rate fa n ;n ‚ 1 g which is bounded away from zero. We investigate the general situ-ation for rate fa n ;n ‚ 1 g under similar conditions. 1. IntroductionThe concept of complete convergence of a sequence of random vari-ables was introduced by Hsu and Robbins [5] as follows. A sequence fU n ;n ‚ 1 g of random variables converges completely to the constant µ ifX 1n =1 P ( jU n i µj > † ) 0. We refer to [3] for a survey on results on complete conver-gence related to strong laws.Recently, Hu et al. [6] and Hu and Volodin [8] proved the follow-ing complete convergence theorem for arrays of rowwise independentrandom variables.Theorem 1. Let fX ni ; 1 • i • k n ;n ‚ 1 g be an array of rowwiseindependent random variables and fa n ;n ‚ 1 g a sequence of positive Received April 19, 2002.2000 Mathematics Subject Classification: 60F15, 60G50, 60B12.Key words and phrases: Arrays, rowwise independence, sums of independent ran-dom variables, complete convergence, Rademacher type