S-boxes generated using Affine Transformation giving Maximum Avalanche Effect Chandrasekharappa T.G.S., Prema K.V. and Kumara Shama

Department of Electronics and Communication Engineering Manipal Institute of Technology Manipal - 576104 INDIA tgscmpl@gmail.com

Abstract: The Advanced Encryption Standard (AES) was published by National Institute of Standards and Technology (NIST) in November 2001, to replace DES (Data Encryption Standard) and Triple DES. The S-box (Substitution box) used in AES is designed to be resistant to known cryptanalytic attacks [1][2]. The property of the S-box is that the output cannot be described as a simple mathematical function of the input. The S-box is designed to provide good avalanche effect. The Avalanche Criteria of S-box depends on the generator matrix A used in affine transformation to construct the S-box. This paper investigates the construction of Sboxes by affine transformation which satisfy maximum Avalanche Criteria.

Keywords: Affine transformation, AES algorithm, Irreducible polynomial, Avalanche Criteria, S-box. 1. Introduction: The S-box, constructed in AES algorithm uses the Affine transformation

y Ax C mod m( x)

(1).

where A is an 8 x 8 matrix with entries in GF(2) and C is a column matrix in GF(2), m(x) is an irreducible polynomial in GF(29). The entries used in A matrix are [f8h; 7ch, 3eh, 1fh, 8fh, c7h, e1h, f1h]T and C = [63h]T (2) To be useful as S-box generator, matrix A should be non-singular. We can generate approximately 263 such non-singular matrices with each irreducible polynomials. The polynomials that result in non-singular matrices are bound by [01; 02; 04; 08; 10; 20; 40; 80]T on lower end and [feh, 7fh, bfh, dfh, efh, f7h, fbh, fdh]T on higher end. As per Avalanche criteria a one bit change in input should result in at least 50% changes in the output bits. A cryptographic function which satisfies above condition is said to be satisfying Strict Avalanche Criteria if and only if a one bit change in input should result in exactly 50% change in the output bits [4][6]. In this work we construct different S-boxes using Affine transformation and different irreducible polynomials for GF(28) and test them for Avalanche Criteria property.

ISSN : 0975-3397

Vol. 3 No. 9 september 2011

3185

Chandrasekharappa T.G.S et al. / International Journal on Computer Science and Engineering (IJCSE)

The rest of the paper is organized as follows. In second section, the S-box construction using Affine transformation is briefed. Section three deals with calculation of Avalanche Criteria for the S-boxes. Section four classifies the polynomials into different difference distribution vectors. Section five gives the possible S-boxes which results in Avalanche Criteria of more than 50% . In section six the experimental results are discussed. Section seven concludes the paper.

2. S-box generation using Affine Transformation: The steps involved in generating a S-box for AES algorithm, using an Affine transformation are as follows: Step 1. S-box is a 16_16 matrix. Initialize the _rst row with [00]; [01]; [02]:::[0f], second row with [10]; [11]; [12]:::[1f] and so on with last row as [f0]; [f1]; [f2]:::[ff]. Step 2. Map each byte into its multiplicative inverse with any one of the irreducible polynomials m(x), with [00] mapped to itself. Step 3. Using Affine transformation in equation(1) , construct S-box with polynomials given in equation(2). The corresponding S-box is shown in Table1.It is possible to construct different S-boxes using different A (A must be a non-singular matrix), C and irreducible polynomials m(x).

3. Avalanche Criteria calculation for the S-boxes generated by affine transformation: For a given function f, the Avalanche Criteria is given by

Si ( f )

xGF ( 28

f ( x) f ( x ei

(3)

where ei’s the vector having...