On the nonlinearity of idempotent quadratic functions and the weight distribution of subcodes of Reed-Muller codes

International audienceThe Walsh transform \hat{Q} of a quadratic function Q : F2^n → F2 satisfies |\hat{Q(b)}| ∈ {0, 2 n+s 2 } for all b ∈ F_{2^n} , where 0 ≤ s ≤ n − 1 is an integer depending on Q. In this article, we investigate two classes of such quadratic Boolean functions which attracted a lot of research interest. For arbitrary integers n we determine the distribution of the parameter s for both of the classes, C1 = {Q(x) = Tr_n(\sum^{(n−1)/2}_{ i=1} a_ix^{2^i +1}) : a_i ∈ F2}, and the larger class C2, defined for even n as C2 = {Q(x) = Tr_n(^{(n/2)−1}_ { i=1} a_ix^{2^i +1}) + Tr_n/2 (a_{n/2} x^{2^n/2 +1}) : a_i ∈ F2}. Our results have two main consequences. We obtain the distribution of the non-linearity for the rotation symmetric quadratic Boolean functions, which have been attracting considerable attention recently. We also present the complete weight distribution of the corresponding subcodes of the second order Reed-Muller codes

Idempotent and p-potent quadratic functions: distribution of nonlinearity and co-dimension

The Walsh transform QˆQ^ of a quadratic function Q:Fpn→FpQ:Fpn→Fp satisfies |Qˆ(b)|∈{0,pn+s2}|Q^(b)|∈{0,pn+s2} for all b∈Fpnb∈Fpn , where 0≤s≤n−10≤s≤n−1 is an integer depending on Q. In this article, we study the following three classes of quadratic functions of wide interest. The class C1C1 is defined for arbitrary n as C1={Q(x)=Trn(∑⌊(n−1)/2⌋i=1aix2i+1):ai∈F2}C1={Q(x)=Trn(∑i=1⌊(n−1)/2⌋aix2i+1):ai∈F2} , and the larger class C2C2 is defined for even n as C2={Q(x)=Trn(∑(n/2)−1i=1aix2i+1)+Trn/2(an/2x2n/2+1):ai∈F2}C2={Q(x)=Trn(∑i=1(n/2)−1aix2i+1)+Trn/2(an/2x2n/2+1):ai∈F2} . For an odd prime p, the subclass DD of all p-ary quadratic functions is defined as D={Q(x)=Trn(∑⌊n/2⌋i=0aixpi+1):ai∈Fp}D={Q(x)=Trn(∑i=0⌊n/2⌋aixpi+1):ai∈Fp} . We determine the generating function for the distribution of the parameter s for C1,C2C1,C2 and DD . As a consequence we completely describe the distribution of the nonlinearity for the rotation symmetric quadratic Boolean functions, and in the case p>2p>2 , the distribution of the co-dimension for the rotation symmetric quadratic p-ary functions, which have been attracting considerable attention recently. Our results also facilitate obtaining closed formulas for the number of such quadratic functions with prescribed s for small values of s, and hence extend earlier results on this topic. We also present the complete weight distribution of the subcodes of the second order Reed–Muller codes corresponding to C1C1 and C2C2 in terms of a generating function

A Lower Bound on the Constant in the Fourier Min-Entropy/Influence Conjecture

We describe a new construction of Boolean functions. A specific instance of our construction provides a 30-variable Boolean function having min-entropy/influence ratio to be $128/45 \approx 2.8444$ which is presently the highest known value of this ratio that is achieved by any Boolean function. Correspondingly, $128/45$ is also presently the best known lower bound on the universal constant of the Fourier min-entropy/influence conjecture

Count of rotational symmetric bent Boolean functions

Counting the Boolean functions having specific cryptographic features is an interesting problem in combinatorics and cryptography. Count of bent functions for more than eight variables is unexplored. In this paper, we propose an upper bound for the count of rotational symmetric bent Boolean functions and characterize its truth table representation from the necessary condition of a rotational symmetric bent Boolean function

A Survey of Metaheuristic Algorithms for the Design of Cryptographic Boolean Functions

Boolean functions are mathematical objects used in diverse domains and have been actively researched for several decades already. One domain where Boolean functions play an important role is cryptography. There, the plethora of settings one should consider and cryptographic properties that need to be fulfilled makes the search for new Boolean functions still a very active domain. There are several options to construct appropriate Boolean functions: algebraic constructions, random search, and metaheuristics. In this work, we concentrate on metaheuristic approaches and examine the related works appearing in the last 25 years. To the best of our knowledge, this is the first survey work on this topic. Additionally, we provide a new taxonomy of related works and discuss the results obtained. Finally, we finish this survey with potential future research directions

A survey of metaheuristic algorithms for the design of cryptographic Boolean functions

Boolean functions are mathematical objects used in diverse domains and have been actively researched for several decades already. One domain where Boolean functions play an important role is cryptography. There, the plethora of settings one should consider and cryptographic properties that need to be fulfilled makes the search for new Boolean functions still a very active domain. There are several options to construct appropriate Boolean functions: algebraic constructions, random search, and metaheuristics. In this work, we concentrate on metaheuristic approaches and examine the related works appearing in the last 25 years. To the best of our knowledge, this is the first survey work on this topic. Additionally, we provide a new taxonomy of related works and discuss the results obtained. Finally, we finish this survey with potential future research directions.</p