Reducing the Number of Homogeneous Linear Equations in Finding Annihilators

Given a Boolean function f on n-variables, we find a reduced set of homogeneous linear equations by solving which one can decide whether there exist annihilators at degree d or not. Using our method the size of the associated matrix becomes $\nu_f \times (\sum_{i=0}^{d} \binom{n}{i} -- \mu_f)$, where, νf = |{x | wt(x) > d, f(x) = 1}| and μf = |{x | wt(x) ≤d, f(x) = 1}| and the time required to construct the matrix is same as the size of the matrix. This is a preprocessing step before the exact solution strategy (to decide on the existence of the annihilators) that requires to solve the set of homogeneous linear equations (basically to calculate the rank) and this can be improved when the number of variables and the number of equations are minimized. As the linear transformation on the input variables of the Boolean function keeps the degree of the annihilators invariant, our preprocessing step can be more efficiently applied if one can find an affine transformation over f(x) to get h(x) = f(Bx+b) such that μh = |{x | h(x) = 1, wt(x) ≤d}| is maximized (and in turn νh is minimized too). We present an efficient heuristic towards this. Our study also shows for what kind of Boolean functions the asymptotic reduction in the size of the matrix is possible and when the reduction is not asymptotic but constant.

[1]  Subhamoy Maitra,et al.  Basic Theory in Construction of Boolean Functions with Maximum Possible Annihilator Immunity , 2006, Des. Codes Cryptogr..

[2]  Dong Hoon Lee,et al.  Resistance of S-Boxes against Algebraic Attacks , 2004, FSE.

[3]  Subhamoy Maitra,et al.  Cryptographically Significant Boolean Functions: Construction and Analysis in Terms of Algebraic Immunity , 2005, FSE.

[4]  Claude Carlet Improving the algebraic immunity of resilient and nonlinear functions and constructing bent functions , 2004, IACR Cryptol. ePrint Arch..

[5]  Anne Canteaut,et al.  Open Problems Related to Algebraic Attacks on Stream Ciphers , 2005, WCC.

[6]  Frederik Armknecht,et al.  Improving Fast Algebraic Attacks , 2004, FSE.

[7]  Josef Pieprzyk,et al.  Cryptanalysis of Block Ciphers with Overdefined Systems of Equations , 2002, ASIACRYPT.

[8]  Josef Pieprzyk,et al.  Algebraic Attacks on SOBER-t32 and SOBER-t16 without Stuttering , 2004, FSE.

[9]  Bart Preneel,et al.  Evaluating the Resistance of Stream Ciphers with Linear Feedback Against Fast Algebraic Attacks , 2006, ACISP.

[10]  Frederik Armknecht,et al.  Efficient Computation of Algebraic Immunity for Algebraic and Fast Algebraic Attacks , 2006, EUROCRYPT.

[11]  Willi Meier,et al.  Fast Algebraic Attacks on Stream Ciphers with Linear Feedback , 2003, CRYPTO.

[12]  Jean-Pierre Tillich,et al.  Computing the Algebraic Immunity Efficiently , 2006, FSE.

[13]  Deepak Kumar Dalai,et al.  Towards an Efficient Algorithm to find Annihilators by Solving a Set of Homogeneous Linear Equations , 2006 .

[14]  Guang Gong,et al.  Upper Bounds on Algebraic Immunity of Boolean Power Functions , 2006, FSE.

[15]  Subhamoy Maitra,et al.  Results on Algebraic Immunity for Cryptographically Significant Boolean Functions , 2004, INDOCRYPT.

[16]  Claude Carlet,et al.  Algebraic Attacks and Decomposition of Boolean Functions , 2004, EUROCRYPT.

[17]  Bart Preneel,et al.  On the Algebraic Immunity of Symmetric Boolean Functions , 2005, INDOCRYPT.

[18]  Nicolas Courtois Fast Algebraic Attacks on Stream Ciphers with Linear Feedback , 2003, CRYPTO.

[19]  Bart Preneel,et al.  Probabilistic Algebraic Attacks , 2005, IMACC.

[20]  Nicolas Courtois,et al.  On Exact Algebraic [Non-]Immunity of S-Boxes Based on Power Functions , 2006, ACISP.

[21]  Lynn Margaret Batten Algebraic Attacks Over GF(q) , 2004, INDOCRYPT.

[22]  Dong Hoon Lee,et al.  Algebraic Attacks on Summation Generators , 2004, FSE.

[23]  J. Faugère,et al.  Algebraic Immunities of functions over finite fields , 2005 .