N-variable Boolean Functions Research Articles

Compact Circuits for Efficient Möbius Transform

The Möbius transform is a linear circuit used to compute the evaluations of a Boolean function over all points on its input domain. The operation is very useful in finding the solution of a system of polynomial equations over GF(2) for obvious reasons. However the operation, although linear, needs exponential number of logic operations (around n · 2n−1 bit xors) for an n-variable Boolean function. As such, the only known hardware circuit to efficiently compute the Möbius Transform requires silicon area that is exponential in n. For Boolean functions whose algebraic degree is bound by some parameter d, recursive definitions of the Möbius Transform exist that requires only O(nd+1) space in software. However converting the mathematical definition of this space-efficient algorithm into a hardware architecture is a non-trivial task, primarily because the recursion calls notionally lead to a depth-first search in a transition graph that requires context switches at each recursion call for which straightforward mapping to hardware is difficult. In this paper we look to overcome these very challenges in an engineering sense. We propose a space efficient sequential hardware circuit for the Möbius Transform that requires only polynomial circuit area (i.e. O(nd+1)) provided the algebraic degree of the Boolean function is limited to d. We show how this circuit can be used as a component to efficiently solve polynomial equations of degree at most d by using fast exhaustive search. We propose three different circuit architectures for this, each of which uses the Möbius Transform circuit as a core component. We show that asymptotically, all the solutions of a system of m polynomials in n unknowns and algebraic degree d over GF(2) can be found using a circuit of silicon area proportional to m · nd+1 and circuit depth proportional to 2 · log2(n − d).In the second part of the paper we introduce a fourth hardware solver that additionally aims to achieve energy efficiency. The main idea is to reduce the solution space to a small enough value by parallel application of Möbius Transform circuits over the first few equations of the system. This is done so that one can check individually whether the vectors of this reduced solution space satisfy each of the remaining equations of the system using lower power consumption. The new circuit has area also bound by m · nd+1 and has circuit depth proportional to d · log2 n. We also show that further optimizations with respect to energy consumption may be obtained by using depth-bound Möbius circuits that exponentially decrease run time at the cost of additional logic area and depth.

A unified construction of weightwise perfectly balanced Boolean functions

At Eurocrypt 2016, Méaux et al. presented FLIP, a new family of stream ciphers that aimed to enhance the efficiency of homomorphic encryption frameworks. Motivated by FLIP, recent research has focused on the study of Boolean functions with good cryptographic properties when restricted to subsets of the space F2n. If an n-variable Boolean function has the property of balancedness when restricted to each set of vectors with fixed Hamming weight between 1 and n−1, it is a weightwise perfectly balanced (WPB) Boolean function. In the literature, a few algebraic constructions of WPB functions are known, in which there are some constructions that use iterative method based on functions with low degrees of 1, 2, or 4. In this paper, we generalize the iterative method and contribute a unified construction of WPB functions based on functions with algebraic degrees that can be any power of 2. For any given positive integer d not larger than m, we first provide a class of 2m-variable Boolean functions with a degree of 2d−1. Utilizing these functions, we then present a construction of 2m-variable WPB functions gm;d. In particular, gm;d includes four former classes of WPB functions as special cases when d=1,2,3,m. When d takes other integer values, gm;d has never appeared before. In addition, we prove the algebraic degree of the constructed WPB functions and compare the weightwise nonlinearity of WPB functions known so far in 8 and 16 variables.

Combined weight and density bounds on the polynomial threshold function representation of Boolean functions

In an earlier report it was shown that an arbitrary n-variable Boolean function f can be represented as a polynomial threshold function (PTF) with 0.75×2n or less number of monomials. In this report, we derive an upper bound on the absolute value of the (integer) weights of a PTF that represents f and still obeys the aforementioned density bound. To our knowledge this provides the best combined bound on the PTF density (number of monomials) and PTF weight (sum of the coefficient magnitudes) of general Boolean functions. For the special case of bent functions, it is found that any n-variable bent function can be represented with integer coefficients less than or equal to 2n with density no more than 0.75×2n, and for the case of m-sparse Boolean functions that are almost constant except for small (m≪2n) number of variable assignments, it is shown that they can be represented with small weight PTFs with density at most m+2n−1. In addition, tight PTF weight bounds with conformance to the density bound of 0.75×2n are numerically obtained for the general Boolean functions up to 6 variables.

Card-Based Cryptographic Protocols with Malicious Players Using Private Operations

This paper shows new card-based cryptographic protocols using private operations that are secure against malicious players. Physical cards are used in card-based cryptographic protocols instead of computers. Operations that a player executes in a place where the other players cannot see are called private operations. Using several private operations, calculations of two variable Boolean functions and copy operations were realized with the minimum number of cards. Though private operations are very powerful in card-based cryptographic protocols, there is a problem that it is very hard to prevent malicious actions during private operations. Though most card-based protocols are discussed in the semi-honest model, there might be cases when the semi-honest model is not enough. Thus, this paper shows new protocols that are secure against malicious players. We show logical XOR, logical AND, n-variable Boolean function, and copy protocols. We can execute any logical computations with a combination of these protocols. We use envelopes as an additional tool that can be easily prepared and used by people.

Classification of the Codewords of Weights 16 and 18 of the Reed-Muller Code RM(n-3,n)

Reed-Muller codes are error-correcting codes used in many areas related to coding theory, such as electrical engineering and computer science. The binary rth-order Reed-Muller code RM(r, n) can be viewed as the set of all n-variable Boolean functions of algebraic degree at most r. Despite the intense work on these codes, many problems are known to be hard (notably, determining their covering radius) and remain open to this day. Fourteen years ago, Carlet and Mesnager improved in [IEEE Transactions on Information Theory, “Improving the Upper Bounds on the Covering Radii of Binary Reed-Muller Codes”, 53(1), 2007] the upper bound on the covering radius of the Reed-Muller code of order 2, and they deduced improved upper bounds on the covering radii of the Reed-Muller codes of higher orders. Until 2021, these upper bounds remain the best ones in the literature. The Reed-Muller code RM(n -3, n), which corresponds to the dual of the Reed-Muller code RM(2, n), has attracted much attention. One of the main reasons is that it is precisely the code that has been considered to get the upper bounds derived by Carlet and Mesnager. Those upper bounds have been obtained thanks to the characterization of the codewords of the Reed-Muller code, whose Hamming weights are strictly less than 2.5 times the minimum distance 2n-r due to Kasami, Tokura, and Azumi. Despite their impressive work in the seventieth, a more refined study and profound description of those codewords of RM(n -3, n) whose Hamming weight equals 16, and especially 18, seem necessary, as it could help us significantly in improving the covering radius of Reed-Muller codes. In this paper, we push further the known results on the Reed-Muller codes by focusing on the Reed-Muller code RM(n -3, n). We provide a classification of the codewords of weight 16 and 18 of the Reed-Muller code RM(n -3, n). Our algebraic descriptions allow us to count the number of such codewords and to enumerate all of them explicitly.

Minimum Round Card-Based Cryptographic Protocols Using Private Operations

This paper shows new card-based cryptographic protocols with the minimum number of rounds, using private operations under the semi-honest model. Physical cards are used in card-based cryptographic protocols instead of computers to achieve secure multiparty computation. Operations that a player executes in a place where the other players cannot see are called private operations. Using three private operations—private random bisection cuts, private reverse cuts, and private reveals—the calculations of two variable Boolean functions and copy operations were realized with the minimum number of cards. Though the number of cards has been discussed, the efficiency of these protocols has not been discussed. This paper defines the number of rounds to evaluate the efficiency of the protocols, using private operations. Most of the meaningful calculations using private operations need at least two rounds. This paper presents a new two-round committed-input, committed-output logical XOR protocol, using four cards. Then, we show new two-round committed-input, committed-output logical AND and copy protocols, using six cards. Even if private reveal operations are not used, logical XOR, logical AND, and copy operations can be executed with the minimum number of rounds. Protocols for general n-variable Boolean functions and protocols that preserve an input are also shown. Lastly, protocols with asymmetric cards are shown.

On the transparency order relationships between one Boolean function and its decomposition functions

The concept of transparency order (denoted by TO) is an important criterion of (n,m)-functions to resist against differential power analysis (DPA). In this paper, we give several transparency order relationships of some Boolean functions. We give the lower bound on the transparency order for Boolean function, and obtain the transparency order relationship between one Boolean function and its decomposition Boolean functions. Furthermore, we deduce one relationship among TO(f⊕g), TO(f), TO(g) and TO(fg) for any n-variable Boolean functions f,g. Finally, we study the transparency order for the sum function and for the product function between two variable-disjoint Boolean functions, and calculate the transparency order distributions of 4-variable and 5-variable balanced Boolean functions, respectively.

An improved hybrid genetic algorithm to construct balanced Boolean function with optimal cryptographic properties

Boolean functions are used as nonlinear filter functions and combiner functions in several stream ciphers. The security of these stream ciphers largely depends upon cryptographic properties of Boolean functions. Finding a balanced Boolean function with optimal cryptographic properties is an open research problem in the cryptographic community. Since the number of n-variable Boolean functions is $$2^{2^n}$$ , it is not computationally feasible to search the entire space of such functions for cryptographically significant functions when $$n \ge 6$$ . In general, the construction of Boolean functions with optimal or near optimal cryptographically significant properties is formulated as combinatorial optimization problems. In this paper, we apply the Genetic algorithm with the integration of a local search procedure called hybrid GA (HGA) searching for the Boolean functions with high nonlinearity and low autocorrelation. The performance of the Genetic algorithm depends upon the tuning parameters such as crossover mechanism, mutation probability, selection criteria, and the choice of fitness/cost function. To achieve an optimal trade-off among two cryptographic properties, the choice of the cost function plays an important role for finding an optimal solution. So our main focus is to construct balanced Boolean functions using HGA with a new cost function and compare it with existing cost functions. Our experimental results shows that the HGA is more efficient than GA and the new cost function is more efficient than existing cost functions for reaching the optimal solutions.

The number of affine equivalent classes and extended affine equivalent classes of vectorial Boolean functions

Affine equivalent classes and extended affine equivalent (EA-equivalent for short) classes of vectorial Boolean functions have important applications in cryptography, logic circuit, sequences for communications, etc. Recently, Y. Zhang et al., computed the number of affine equivalent classes of n-variable Boolean functions when 1≤n≤10 by group isomorphism (Zhang et al., 2016). However, the case for affine equivalent vectorial Boolean function remains a challenging open problem. Furthermore, little result for EA-equivalent vectorial Boolean function is known except a trivial lower bound mentioned in Mullen and Panario (2013, P246). In this paper, we focus on the challenging problem of calculating the number of affine equivalent classes and EA-equivalent classes of vectorial Boolean functions. First, for EA-equivalence, we prove that the trivial lower bound proposed in Mullen and Panario (2013) has at least 3 effective figures if n is not too small. We then show that the lower bound also holds for affine equivalent classes. Furthermore, we give an explicit formula and calculate the exact number of affine equivalent classes of (n,m)-functions with 1≤m,n≤11 by GAP. The results in this paper are helpful for the theory and applications of the classifications of vectorial Boolean functions.

Tests with Stuck-At and Shift Faults on Circuit Inputs

We investigate left-shifts of variables in Boolean functions and stuck-at faults. Some bounds of Shannon functions are established, producing the minimum test lengths for detection and diagnosis of any n-variable Boolean function with respect to a given fault source. (The Shannon function of the test length is the maximum of the minimum test lengths over all n-variable Boolean functions.) A left-shift in an n-variable Boolean function simultaneously increments the indexes of all the variables by a natural number and substitutes some constants for variables with indexes greater than n. The Shannon function of the relevant test is shown to be exactly of length 2; the order of growth of the Shannon function of the length of the diagnosis test is $$ \Theta \left(\sqrt{2^n}\right). $$ With the variables left-shifted by k places, 1 ≤ k ≤ n, the Shannon function $$ {L}^{diagn}\left({U}_{n,k}^{shifts},n\right) $$ of the length of the complete diagnosis test has the following bounds: $$ \min \left({2}^k-1,{2}^{n-k}\right)\le {L}^{diagn}\left({U}_{n,k}^{shifts},n\right)\le \min \left({2}^k,{2}^{n-k}+1\right). $$ The article also considers local stuck-at faults of multiplicity k on the inputs of k-input, one-output circuits realizing a Boolean function. An asymptotic expression is derived for the log Shannon function of the diagnosis test length $$ {\log}_2{L}^{diagn}\left({U}_k^{lc},n\right)\sim k\ \mathrm{as}\ n\to \infty, k=k(n)\to \infty, \kern1em 1\le k\le n/2,{\log}_2n=o(k). $$ An asymptotic lower bound of the Shannon function of the diagnosis test length with stuck-at faults on circuit inputs is shown to be Ldiagn(Uc, n) ≥ 2 ⋅ 2[n/2] ⋅ (1 + o(1)) as n → ∞ .

Quantum Secure Multiparty Computation with Symmetric Boolean Functions**Supported in part by the National Key R&D Program of China (Grant No. 2017YFB0802400), the National Natural Science Foundation of China (Grant Nos. 61373171 and 11801564), the Program for Excellent Young Talents in University of Anhui Province, China (Grant No. gxyqZD2019060), the Basic Research Project of Natural Science of Shaanxi Province, China (Grant

We propose a class of n-variable Boolean functions which can be used to implement quantum secure multiparty computation. We also give an implementation of a special quantum secure multiparty computation protocol. An advantage of our protocol is that only 1 qubit is needed to compute the n-tuple pairwise and function, which is more efficient comparing with previous protocols. We demonstrate our protocol on the IBM quantum cloud platform, with a probability of correct output as high as 94.63%. Therefore, our protocol presents a promising generalization in realization of various secure multipartite quantum tasks.

Improved lower bound for algebraic immunity of a subclass of MM bent Boolean functions

It is known that algebraic immunity of an n-variable Boolean function is upper bounded by []. In literature, it was shown algebraic immunity of a subclass of Maiorana-McFarland (MM) bent Boolean functions is lower bounded by 2 and upper bounded by . In this paper, this lower bound is further improved. Given a Boolean function of -variables with algebraic immunity k ≥ 4, the improved lower bound helps a cryptographer to find a MM bent function of this class whose algebraic immunity is at least k − 2.

Quantum computing cryptography: Finding cryptographic Boolean functions with quantum annealing by a 2000 qubit D-wave quantum computer

As the building block in symmetric cryptography, designing Boolean functions satisfying multiple properties is an important problem in sequence ciphers, block ciphers, and hash functions. However, the search of n-variable Boolean functions fulfilling global cryptographic constraints is computationally hard due to the super-exponential size O(22n) of the space. Here, we introduce a codification of the cryptographically relevant constraints in the ground state of an Ising Hamiltonian, allowing us to naturally encode it in a quantum annealer, which seems to provide a quantum speedup. Additionally, we benchmark small n cases in a D-Wave machine, showing its capacity of devising cryptographic Boolean functions with certain relevant properties. We have complemented it with local search and chain repair to improve the D-Wave quantum annealer performance related to the low connectivity. This work shows how to codify super-exponential cryptographic problems into quantum annealers and paves the way for reaching quantum supremacy.

Generic constructions of $$\mathbb {Z}$$-bent functions

$$\mathbb {Z}$$-bent functions, mappings from $$\mathbb {F}_2^n$$ to a subset of $$\mathbb {Z}$$, were introduced by Dobbertin and Leander (Des Codes Cryptogr 49:3–22, 2008) as an attempt to capture the origin of standard bent functions and in particular to understand better a recursive construction framework of bent functions. Nevertheless, many questions have been left open in Dobbertin and Leander (2008) such as efficient construction methods of $$\mathbb {Z}$$-bent functions of different levels, where these levels specify precisely a subset of $$\mathbb {Z}$$ containing both the image values of f and its normalized Fourier coefficients. In this article, using different design techniques, we provide several generic construction methods of $$\mathbb {Z}$$-bent functions of arbitrary levels, thereby solving an open problem posed in Dobbertin and Leander (2008). On the other hand, apart from an independent theoretical interest in these objects, our rigor treatment of the so-called gluing technique reveals that this approach is equivalent to a classical concept of concatenation. More precisely, gluing four suitable n-variables $$\mathbb {Z}$$-bent functions of level one to obtain an $$(n+2)$$-variable bent function directly corresponds to a concatenation of four suitable n-variable Boolean functions. Nevertheless, the recursive framework based on $$\mathbb {Z}$$-bent functions remains to be investigated further in this context.

On the q-bentness of Boolean functions

For each non-constant q in the set of n-variable Boolean functions, the q-transform of a Boolean function f is related to the Hamming distances from f to the functions obtainable from q by nonsingular linear change of basis. Klapper conjectured that no Boolean function exists with its q-transform coefficients equal to $$\pm \, 2^{n/2}$$ (such function is called q-bent) when q is non-affine balanced. In our early work, we only gave partial results to confirm this conjecture for small n. Here we prove thoroughly that the conjecture is true for all n by investigating the nonexistence of the partial difference sets in abelian groups with special parameters. We also introduce a new family of functions called $$(\delta ,q)$$ -bent functions, which give a measurement of q-bentness.

Characterization of bipartite Cayley graphs in terms of Boolean functions

A Boolean function on n-variables is a function f from Vn into field GF(2) and we denote the set of all n-variable Boolean functions by Fn. For every f ∈Fn, the Walsh transform of f is defined as Wf (a) = Σx∈GF(2n)f(x)(−1)a.x. For the Boolean function f, the list [Wf (0),…, (2n −1)] is called the Walsh spectrum of f. The aim of this paper is to study the Walsh spectrum of Cayley graphs in terms of Boolean functions. We also characterize some Boolean functions whose related bipartite Cayley graphs have four distinct eigenvalues.

Quantum algorithms on Walsh transform and Hamming distance for Boolean functions

Walsh spectrum or Walsh transform is an alternative description of Boolean functions. In this paper, we explore quantum algorithms to approximate the absolute value of Walsh transform $$W_f$$Wf at a single point $$z_{0}$$z0 (i.e., $$|W_f(z_{0})|$$|Wf(z0)|) for n-variable Boolean functions with probability at least $$\frac{8}{\pi ^{2}}$$8?2 using the number of $$O(\frac{1}{|W_f(z_{0})|\varepsilon })$$O(1|Wf(z0)|?) queries, promised that the accuracy is $$\varepsilon $$?, while the best known classical algorithm requires $$O(2^{n})$$O(2n) queries. The Hamming distance between Boolean functions is used to study the linearity testing and other important problems. We take advantage of Walsh transform to calculate the Hamming distance between two n-variable Boolean functions f and g using O(1) queries in some cases. Then, we exploit another quantum algorithm which converts computing Hamming distance between two Boolean functions to quantum amplitude estimation (i.e., approximate counting). If $$Ham(f,g)=t\ne 0$$Ham(f,g)=t?0, we can approximately compute Ham(f, g) with probability at least $$\frac{2}{3}$$23 by combining our algorithm and $${Approx-Count(f,\varepsilon )\, algorithm}$$Approx-Count(f,?)algorithm using the expected number of $$\varTheta ( \sqrt{\frac{N}{(\lfloor \varepsilon t\rfloor +1)}}+\frac{\sqrt{t(N-t)}}{\lfloor \varepsilon t\rfloor +1})$$?(N(??t?+1)+t(N-t)??t?+1) queries, promised that the accuracy is $$\varepsilon $$?. Moreover, our algorithm is optimal, while the exact query complexity for the above problem is $$\varTheta (N)$$?(N) and the query complexity with the accuracy $$\varepsilon $$? is $$O(\frac{1}{\varepsilon ^{2}}N/(t+1))$$O(1?2N/(t+1)) in classical algorithm, where $$N=2^{n}$$N=2n. Finally, we present three exact quantum query algorithms for two promise problems on Hamming distance using O(1) queries, while any classical deterministic algorithm solving the problem uses $$\varOmega (2^{n})$$Ω(2n) queries.

The Multiplicative Complexity of 6-variable Boolean Functions.

The multiplicative complexity of a Boolean function is the minimum number of two-input AND gates that are necessary and sufficient to implement the function over the basis (AND, XOR, NOT). Finding the multiplicative complexity of a given function is computationally intractable, even for functions with small number of inputs. Turan et al. [1] showed that n-variable Boolean functions can be implemented with at most n-1 AND gates for n ≤ 5. A counting argument can be used to show that, for n ≥ 7, there exist n-variable Boolean functions with multiplicative complexity of at least n. In this work, we propose a method to find the multiplicative complexity of Boolean functions by analyzing circuits with a particular number of AND gates and utilizing the affine equivalence of functions. We use this method to study the multiplicative complexity of 6-variable Boolean functions, and calculate the multiplicative complexities of all 150357 affine equivalence classes. We show that any 6-variable Boolean function can be implemented using at most 6 AND gates. Additionally, we exhibit specific 6-variable Boolean functions which have multiplicative complexity 6.

The number of boolean functions with multiplicative complexity 2

Multiplicative complexity is a complexity measure defined as the minimum number of AND gates required to implement a given primitive by a circuit over the basis (AND, XOR, NOT). Implementations of cyphers with a small number of AND gates are preferred in protocols for fully homomorphic encryption, multiparty computation and zero-knowledge proofs. Fischer and Peralta (2002) computed the number of n-variable Boolean functions with multiplicative complexity 1. In this paper, we study Boolean functions that can be constructed with two AND gates. By characterising the structure of these functions in terms of affine equivalence relations, we provide a closed-form formula for the number of Boolean functions with multiplicative complexity 2.

The number of boolean functions with multiplicative complexity 2

Multiplicative complexity is a complexity measure defined as the minimum number of AND gates required to implement a given primitive by a circuit over the basis (AND, XOR, NOT). Implementations of cyphers with a small number of AND gates are preferred in protocols for fully homomorphic encryption, multiparty computation and zero-knowledge proofs. Fischer and Peralta (2002) computed the number of n-variable Boolean functions with multiplicative complexity 1. In this paper, we study Boolean functions that can be constructed with two AND gates. By characterising the structure of these functions in terms of affine equivalence relations, we provide a closed-form formula for the number of Boolean functions with multiplicative complexity 2.