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Graphical SAC analysis of [S.sub.8] APA S-box.


In [5], L. Cui and Y. Cao present APA S-box for Advanced Encryption Standard. With the APA structure, the algebraic complexity of the improved AES S-box is increased from 9 to 253, and its inverse S-box keeps 255. Furthermore, other good cryptographic characteristics of AES S-box are inherited.

In [3], H. Dilpazir, I. Hussain, T. Shah and H. Mehmood present new [S.sub.8] APA S-boxes by applying the permutations of [S.sub.8] (symmetric group of order 8!) on original APA S-box [5], and also examine algebraic complexity, differential uniformity, bijective property and nonlinearity of proposed [S.sub.8] APA S-boxes. In this letter we analyze the strict avalanche criterion of [S.sub.8] APA S-boxes and observe the strong points of [S.sub.8] APA S-boxes as compared it to APA S-box [5].

This letter is structured as follows. In section 2, we present problem statement. Section 3 present constructions of [S.sub.8] APA S-boxes [3]. Section 4 consists of analysis of Strict Avalanche Criterion (SAC) for [S.sub.8] APA S-boxes and its comparison with APA S-box. Section 4 present conclusions.

Problem Statement

Graphical strict avalanche criterion is a new approach to investigate SAC for S- box. In this letter we analyze [S.sub.8] APA S-box for graphical strict avalanche criterion with 10, 100 and 1000 counts and bring to a close that when we raise the number of counts the analysis of S-box becomes closer to optimal graph. Also we put side by side the graphs of [S.sub.8] APA S-box with APA S-box and detect that with change in count numbers from 10 to 1000, both S-boxes shows dissimilar graphs, with this difference we can evaluate which s-box is good in these analysis.

Construction of [S.sub.8] APA S-box

An Affine-Power-Affine (APA) S-box is defined as follows:

APASbox = A * P * A

Where "A" denotes the affine surjection and "P" denotes the power permutation with "good" cryptographic characteristics in GF (2n).

In [3] [S.sub.8] APA S-boxes are constructed by applying the permutations of [S.sub.8] on the APA S-box [5], and sequentially construct 40320 new S-boxes. The procedure is as follows
APA S-box

[down arrow]

Convert elements of APA S-box into binary form

[down arrow]

Apply permutations of [S.sub.8] on elements

[down arrow]

We get 40320 [S.sub.8] APA S-boxes_

Comparison of Properties of S-box
S-box Non-linearity Differential

Opt. Value 120 4
AES [1] 112 4
Gray S-box [5] 112 4
[S.sub.8] AES 112 4
 S-box [2]
[S.sub.8] Gray 112 4
 S-box [2a]
[S.sub.8] APA 112 4
 S-box [3]

S-box Algebraic Bijective

Opt. Value 255 terms Yes
AES [1] 9 terms Yes
Gray S-box [5] 255 terms Yes
[S.sub.8] AES 9 terms Yes
 S-box [2]
[S.sub.8] Gray 255 terms Yes
 S-box [2a]
[S.sub.8] APA 253 terms Yes
 S-box [3]

Strict Avalanche Criterion

In [4], three graphical methods to analyze strict avalanche criterion (SAC) are given. These methods are given below

1. Analysis of Frequency of various Hamming weight (Avalanche effect);

2. Analysis of Differential values (completeness);

3. 3-Analysis of Hamming Weight According to Bit Position (Strong S-box).




Fig-1 and Fig-2 represents the analysis of frequency of various hamming weight with 10 counts for APA S-box and [S.sub.8] APA S-box respectively and Fig-3 represents comparison between Fig-1 and 2. Nature of the graphs is not similar with optimal graph of [4].




Now when we increase number of counts from 10 to 100, we examine in Fig-4 and Fig-5 that graphs of both boxes are acceptable and at some extent similar with optimal graphs of [4]. Fig-6 shows comparison, which shows that APA S-box is better than [S.sub.8] APA S-box for 100 counts because its graph is more similar to optimal graphs of [4].




Now we study the same analysis for 1000 counts and detect from Fig-7 and Fig-8 that both graphs are almost same as optimal graphs of [4]. Fig-9 shows comparison, which shows that [S.sub.8] APA S-box is better than APA S-box, because its peak is higher than APA S-box and it is more similar to optimal graph.

Analysis of Differential values of [S.sub.8] Gray S-box:

For 10 counts

Fig-10 and Fig-11 shows analysis of differential values with 10 counts for APA and [S.sub.8] APA S-box respectively. Both shows almost same poor performence.



For 100 counts



For 1000 counts



Now we repeate the same analysis for 100 and 1000 counts and detect that both S- boxes have almost same performence in this analysis.

Analysis of Hamming Weight According to Bit Position of [S.sub.8] Gray S-box

For 10 counts




For 100 counts




For 1000 counts




From Fig-16 to Fig-24, we analyze APA S-box and [S.sub.8] APA S-box for analysis of hamming weight according to bit position, and observe that performence is improved when we raise number of counts from 10 to 1000 and graphs of both boxes is almost similar to the optimal graphs of [4].


In this letter we present the SAC analysis of [S.sub.8] APA S-box. The analysis consists of performing AVC test, completeness test, and strong S-Box test [4]. All these analysis shows that [S.sub.8] APA S-box satisfied the strict avalanche criterion similar to APA Sbox, and we can use it in block ciphers for secure communication. One more thing which is important, the number of [S.sub.8] APA S-boxes are 40320, so we can utilize these numbers of boxes in any kind of cryptographic algorithm.


[1] J. Daemen and V. Rijmen, AES proposal: Rijndael AES algorithm submission, .1999.

[2] I. Hussain, T. Shah and H. Mehmood, A New Algorithm to Construct Secure Keys for AES, International Journal of Contemporary Mathematical Sciences, Vol. 5, no. 26, 1263-1270, 2010.

[3] I. Hussain, T. Shah and H. Mahmood, [S.sub.8] Gray S-boxes Algorithm, International Mathematical forum, 2010 (Accepted).

[4] H. Dilpazir, I. Hussain, T. Shah and H. Mehmood, [S.sub.8] Affine Power Affine S boxes, International Journal of Contemporary Mathematical Sciences, 2010 (Accepted).

[5] P. P. Mar, K. M. Latt, New analysis methods on strict avalanche criterion of S-boxes World Academy of Science, Engineering and Technology 48. 2008.

[6] L. Cui and Y. Cao, "A new S-box structure named Affine-Power-Affine," International Journal of Innovative Computing, Information and Control, 3(3), Vol. 3, no. 3, 751-759 2007.

Iqtadar Hussain (1), Tariq Shah (2) and Khuram Shahzad Aslam (3)

(1,2) Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan

(3) Comsats Institute of Information technology, Abbottabad

E-mail: (1), (2), (3)
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Author:Hussain, Iqtadar; Shah, Tariq; Aslam, Khuram Shahzad
Publication:International Journal of Difference Equations
Date:Jun 1, 2011
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