Abstract The algebraic degree of a vectorial Boolean function is one of the main parameters driving the cost of its hardware implementation. Thus, finding decompositions of functions into sequences of functions of lower algebraic degrees has been explored to reduce the cost of implementations. In this paper, we consider such decompositions of permutations over $$\mathbb {F}_{2^n}$$ F 2 n . We prove the existence of a decomposition of the inverse using quadratic and linear power permutations for all permutations when $$2^n-1$$ 2 n - 1 is a prime, and we prove the non-existence of such decompositions for power permutations of differential uniformity strictly lower than 16 when 4|n. We also prove that any permutation admits a decomposition into quadratic power permutations and affine permutations of the form $$ax+b$$ a x + b if $$4 \not \mid n$$ 4 ∤ n . Furthermore, we prove that any permutation admits a decomposition into cubic power permutations and affine permutations. Finally, we present a decomposition of the PRESENT S-Box using the power permutation $$x^7$$ x 7 and affine permutations.

The Difference Distribution Table (DDT) and the differential uniformity play a major role for the design of substitution boxes in block ciphers, since they indicate the function’s resistance against differential cryptanalysis. This concept was extended recently to$c$-DDT and$c$-differential uniformity, which have the potential of extending differential cryptanalysis. Recently, a new theoretical tool, the Boomerang Connectivity Table (BCT) and the corresponding boomerang uniformity were introduced to quantify the resistance of a block cipher against boomerang-style attacks. Here we concentrate on two classes (introduced recently) of permutation polynomials over finite fields of even characteristic. For one of these, which is an involution used to construct a 4-uniform permutation, we explicitly determine the$c$-DDT entries and BCT entries. For the second type of function, which is a differentially 4-uniform function, we give bounds for its$c$-differential and boomerang uniformities.

The Strict Avalanche Criterion (SAC) is a property of vectorial Boolean functions that is used in the construction of strong S-boxes. We show in this paper how to generalize the concept of SAC to address possible c-differential attacks, in the realm of finite fields. We define the concepts of c-Strict Avalanche Criterion (c-SAC) and c-Strict Avalanche Criterion of order m (c-SAC(m)), and generalize results of (Li and Cusick, 2005). We also show computationally how the new definition is not equivalent to the existing concepts of c-bent1-ness (Stanica et al., 2020), nor (for n = m) PcN-ness (Ellingsen et al., 2020)

In a prior paper (Ellingsenet al., 2020), two of us, along with P. Ellingsen, P. Felke, and A. Tkachenko, defined a new (output) multiplicative differential and the corresponding$c$-differential uniformity, which has the potential of extending differential cryptanalysis. Here, we continue the work by looking at some APN functions through the mentioned concept and showing that their$c$-differential uniformity increases significantly in some cases.

Nowadays, the necessity of electronic information increases rapidly. As a consequence, often, that information needs to be shared among mutually distrustful parties. In this area, private set intersection (PSI) and its variants play an important role when the participants wish to do secret operations on their input sets. Unlike the most modern public key cryptosystems relying on number theoretic problems, lattice‐based cryptographic constructions provide security in the presence of a quantum computer. Consequently, developing PSI and its variants using lattice based cryptosystem becomes an interesting direction for research. This study presents the first size‐hiding post quantum PSI cardinality (PSI‐CA) protocol whose complexity is linear in the size of the sets of the participants. The authors use space‐efficient probabilistic data structure (Bloom filter) as its building block. Further, they extend the authors’ PSI‐CA to its authorised version, i.e. authorised PSI‐CA. Security for both of them is achieved in the standard model based on the hardness of the decisional learning with errors problem.

In this paper we define a new (output) multiplicative differential, and the corresponding c-differential uniformity. With this new concept, even for characteristic 2, there are perfect c-nonlinear (PcN) functions. We first characterize the c-differential uniformity of a function in terms of its Walsh transform. We further look at some of the known perfect nonlinear (PN) functions and show that only one remains a PcN function, under a different condition on the parameters. In fact, the p-ary Gold PN function increases its c-differential uniformity significantly, under some conditions on the parameters. We then precisely characterize the c-differential uniformity of the inverse function (in any dimension and characteristic), relevant for the Rijndael (and Advanced Encryption Standard) block cipher.

For a function $f$ from $\mathbb {F}_{2}^{n}$ to $\mathbb {F}_{2}^{n}$ , the planarity of $f$ is usually measured by its differential uniformity and differential spectrum. In this paper, we propose the concept of vanishing flats, which supplies a combinatorial viewpoint on the planarity. First, the number of vanishing flats of $f$ can be regarded as a measure of the distance between $f$ and the set of almost perfect nonlinear functions. In some cases, the number of vanishing flats serves as an “intermediate” concept between differential uniformity and differential spectrum, which contains more information than differential uniformity, however less than the differential spectrum. Secondly, the set of vanishing flats forms a combinatorial configuration called partial quadruple system, since it conveys a detailed structural information about $f$ . We initiate this study by considering the number of vanishing flats and the partial quadruple systems associated with monomials and Dembowski-Ostrom polynomials. In addition, we present an application of vanishing flats to the partition of a vector space into disjoint equidimensional affine spaces. We conclude the paper with several further questions and challenges.

Boolean functions are usually studied under the assumption that each input bit is considered independent and identically distributed. However, in the case of some stream ciphers, a keystream bit is generated by using a nonlinear Boolean function with inputs from a restricted domain. At Eurocrypt 2016, one such stream cipher (FLIP) has been proposed, where a Boolean function on n variables was exploited with inputs of weight n/2 only. Recently, Carlet et al. studied several properties of such functions and obtained certain bounds on linear approximations of direct sum in the restricted domain. In this paper, we observe that for a direct sum like f = f1+ f2, the inputs to each sub-function f1, f2do not follow a uniform distribution in the restricted domain. In this regard, we study the properties of the Boolean functions by considering a general probability distribution on the inputs. We further obtain several bounds related to the biases of direct sums. Finally, we obtain a lower bound on the bias of the nonlinear filter function of FLIP. Our results provide a general framework to study security parameters of ciphers over restricted domain.

In this paper, we introduce generalized hyperbent functions from F2nto ℤ2k, and investigate decompositions of generalized (hyper)bent functions. We show that generalized (hyper)bent functions f from F2nto ℤ2kconsist of components which are generalized (hyper)bent functions from F2ntoZ2k'for some k' <; k. For even n, most notably we show that the g-hyperbentness of f is equivalent to the hyperbentness of the components of f with some conditions on the Walsh-Hadamard coefficients. For odd n, we show that the Boolean functions associated to a generalized bent function form an affine space of semibent functions. This complements a recent result for even n, where the associated Boolean functions are bent.

Two (so-called 𝒞, D) classes of permutation-based bent Boolean functions were introduced by Carlet [4] two decades ago, but without specifying some explicit construction methods for their construction (apart from the subclass 𝒟 0 ). In this article, we look in more detail at the 𝒞 class, and derive some existence and nonexistence results concerning the bent functions in the 𝒞 class for many of the known classes of permutations over 𝔽 2 n . Most importantly, the existence results induce generic methods of constructing bent functions in class 𝒞 which possibly do not belong to the completed Maiorana-McFarland class. The question whether the specific permutations and related subspaces we identify in this article indeed give bent functions outside the completed Maiorana-McFarland class remains open.

In this paper, we introduce a sequence of discrete Fourier transforms and define new versions of bent functions, which we shall call (weak and strong) octa/hexadeca and, in general, 2k-bent functions. We investigate relationships between these classes and completely characterize the octabent and hexadecabent functions in terms of bent functions. We further find relative difference sets based upon these functions.

The robust symmetrical number system (RSNS) is a number theoretic transform based on N ≥ 2 sequences that can extract the maximum amount of information from symmetrical folding waveforms. The sequences, based on coprime moduli, exhibit an integer gray code property making the RSNS well suited for many applications that benefit from an inherent error detection and correction capability, such as analog-to-digital converters, direction finding arrays, and radar waveform design. To use the RSNS, it is necessary to know the greatest length of combined sequences without ambiguities, called the dynamic range M̂, for which only a few closed-form expressions currently exist. In this paper, an efficient algorithm for computing M and its position within the combined set of sequences is presented and shown to be independent of the size of the moduli. The algorithm is used to generate the equations for several groups of additional moduli arrangements. Closed-form expressions for M are conjectured and proved using the obtained congruence equations that define the ambiguity locations.

In this paper, we consider the spectra of Boolean functions with respect to the action of unitary transforms obtained by taking tensor products of the Hadamard kernel, denoted byH, and the nega-Hadamard kernel, denoted byN. The set of all such transforms is denoted by {H,N}n. A Boolean function is said to be bent4if its spectrum with respect to at least one unitary transform in {H,N}nis flat. We obtain a relationship between bent, semibent, and bent4functions, which is a generalization of the relationship between bent and negabent Boolean functions proved by Parker and Pott [cf., LNCS 4893 (2007), 9-23]. As a corollary to this result, we prove that the maximum possible algebraic degree of a bent4function onnvariables is [n/2] and, hence, solve an open problem posed by Riera and Parker [cf., IEEE-TIT 52:9 (2006), 4142-4159].

Parkerconsidered a new type of discrete Fourier transform, called nega-Hadamard transform. We prove several results regarding its behavior on combinations of Boolean functions and use this theory to derive several results on negabentness (that is, flat nega-spectrum) of concatenations, and partially symmetric functions. We derive the upper bound$\lceil {{ n}\over { 2}} \rceil $for the algebraic degree of a negabent function on$n$variables. Further, a characterization of bent–negabent functions is obtained within a subclass of the Maiorana–McFarland set. We develop a technique to construct bent–negabent Boolean functions by using complete mapping polynomials. Using this technique, we demonstrate that for each$\ell \geq 2$, there exist bent–negabent functions on$n = 12\ell $variables with algebraic degree$ {{ n}\over { 4}}+1 = 3\ell + 1$. It is also demonstrated that there exist bent–negabent functions on eight variables with algebraic degrees 2, 3, and 4. Simple proofs of several previously known facts are obtained as immediate consequences of our work.

We show that there is significant benefit to using a reconfigurable computer to enumerate bent Boolean functions for cryptographic applications. Bent functions are rare, and the only known way to generate all bent functions is by a sieve technique in which many prospective functions are tested. The speed-up achieved depends on the number of variables n; for n = 8, we show that the reconfigurable computer achieves better than a 60,000× speed-up over a conventional computer. Further, we introduce the transeunt triangle as a means to reduce the number of functions that must be considered. For n = 6, this reduction is better than 500,000,000 to 1. Previously, the transeunt triangle had been used only in the design of exclusive OR logic circuits; it converts a truth table to the algebraic normal form. However, this fact has never been proven rigorously, and that shortcoming is removed in this paper. Our proof provides a practical benefit; it yields a new realization of the transeunt triangle that has less complexity and delay. Finally, we show computational results from a reconfigurable computer.

Under mild conditions on n, p, we give a lower bound on the number of n-variable balanced symmetric polynomials over finite fields GF(p), where p is a prime number. The existence of nonlinear balanced symmetric polynomials is an immediate corollary of this bound. Furthermore, we prove that X(2t, 2t+1lscr-1) are balanced and conjecture that these are the only balanced symmetric polynomials over GF(2), where X(d, n) = Sigma1lesi1<i2<hellip

The design of Boolean functions with properties of cryptographic significance is a hard task. In this paper, we adopt an unorthodox approach to the design of such functions. Our search space is the set of functions that possess the required properties. It is “Boolean‐ness” that is evolved.

The design of Boolean functions with properties of cryptographic significance is a hard task. In this paper, we adopt an unorthodox approach to the design of such functions. Our search space is the set of functions that possess the required properties. It is 'Booleanness' that is evolved.