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Secure multiparty computation (MPC) schemes allow two or more parties to conjointly compute a function on their private input sets while revealing nothing but the output. Existing state-of-the-art number-theoretic-based designs face the threat of attacks through quantum algorithms. In this context, we present secure MPC protocols that can withstand quantum attacks. We first present the design and analysis of an information-theoretic secure oblivious linear evaluation (OLE), namely ${\sf qOLE}$ in the quantum domain, and show that our ${\sf qOLE}$ is safe from external attacks. In addition, our scheme satisfies all the security requirements of a secure OLE. We further utilize ${\sf qOLE}$ as a building block to construct a quantum-safe multiparty private set intersection (MPSI) protocol.

This paper makes the first bridge between the classical differential/boomerang uniformity and the newly introduced $c$-differential uniformity. We show that the boomerang uniformity of an odd APN function is given by the maximum of the entries (except for the first row/column) of the function's $(-1)$-Difference Distribution Table. In fact, the boomerang uniformity of an odd permutation APN function equals its $(-1)$-differential uniformity. We next apply this result to easily compute the boomerang uniformity of several odd APN functions. In the second part we give two classes of differentially low-uniform functions obtained by modifying the inverse function. The first class of permutations (CCZ-inequivalent to the inverse) over a finite field $\mathbb{F}_{p^n}$ ($p$, an odd prime) is obtained from the composition of the inverse function with an order-$3$ cycle permutation, with differential uniformity $3$ if $p=3$ and $n$ is odd; $5$ if $p=13$ and $n$ is even; and $4$ otherwise. The second class is a family of binomials and we show that their differential uniformity equals~$4$. We next complete the open case of $p=3$ in the investigation started by G\" olo\u glu and McGuire (2014), for $p\geq 5$, and continued by K\"olsch (2021), for $p=2$, $n\geq 5$, on the characterization of $L_1(X^{p^n-2})+L_2(X)$ (with linearized $L_1,L_2$) being a permutation polynomial. Finally, we extend to odd characteristic a result of Charpin and Kyureghyan (2010) providing an upper bound for the differential uniformity of the function and its switched version via a trace function.

Finding functions, particularly permutations, with good differential properties has received a lot of attention due to their possible applications. For instance, in combinatorial design theory, a correspondence of perfect $c$-nonlinear functions and difference sets in some quasigroups was recently shown [1]. Additionally, in a recent manuscript by Pal and Stanica [20], a very interesting connection between the $c$-differential uniformity and boomerang uniformity when $c=-1$ was pointed out, showing that that they are the same for an odd APN permutations. This makes the construction of functions with low $c$-differential uniformity an intriguing problem. We investigate the $c$-differential uniformity of some classes of permutation polynomials. As a result, we add four more classes of permutation polynomials to the family of functions that only contains a few (non-trivial) perfect $c$-nonlinear functions over finite fields of even characteristic. Moreover, we include a class of permutation polynomials with low $c$-differential uniformity over the field of characteristic~$3$. As a byproduct, our proofs shows the permutation property of these classes. To solve the involved equations over finite fields, we use various techniques, in particular, we find explicitly many Walsh transform coefficients and Weil sums that may be of an independent interest.

The Feistel Boomerang Connectivity Table and the related notion of $F$-Boomerang uniformity (also known as the second-order zero differential uniformity) has been recently introduced by Boukerrou et al.~\cite{Bouk}. These tools shall provide a major impetus in the analysis of the security of the Feistel network-based ciphers. In the same paper, a characterization of almost perfect nonlinear functions (APN) over fields of even characteristic in terms of second-order zero differential uniformity was also given. Here, we find a sufficient condition for an odd or even function over fields of odd characteristic to be an APN function, in terms of second-order zero differential uniformity. Moreover, we compute the second-order zero differential spectra of several APN or other low differential uniform functions, and show that our considered functions also have low second-order zero differential uniformity, though it may vary widely, unlike the case for even characteristic when it is always zero.

In 1953, Carlitz~\cite{Car53} showed that all permutation polynomials over $\F_q$, where $q>2$ is a power of a prime, are generated by the special permutation polynomials $x^{q-2}$ (the inversion) and $ ax+b$ (affine functions, where $0\neq a, b\in \F_q$). Recently, Nikova, Nikov and Rijmen~\cite{NNR19} proposed an algorithm (NNR) to find a decomposition of the inverse function in quadratics, and computationally covered all dimensions $n\leq 16$. Petrides~\cite{P23} found a class of integers for which it is easy to decompose the inverse into quadratics, and improved the NNR algorithm, thereby extending the computation up to $n\leq 32$. Here, we extend Petrides' result, as well as we propose a number theoretical approach, which allows us to cover easily all (surely, odd) exponents up to~$250$, at least., Comment: 18 pages

It was shown by Boukerrou et al.~\cite{Bouk} [IACR Trans. Symmetric Cryptol. 1 2020, 331--362] that the $F$-boomerang uniformity (which is the same as the second-order zero differential uniformity in even characteristic) of perfect nonlinear functions is~$0$ on $\F_{p^n}$ ($p$ prime) and the one of almost perfect nonlinear functions on $\F_{2^n}$ is~$0$. It is natural to inquire what happens with APN or other low differential uniform functions in even and odd characteristics. Here, we explicitly determine the second-order zero differential spectra of several maps with low differential uniformity. In particular, we compute the second-order zero differential spectra for some almost perfect nonlinear (APN) functions, pushing further the study started in Boukerrou et al.~\cite{Bouk} and continued in Li et al. \cite{LYT} [Cryptogr. Commun. 14.3 (2022), 653--662], and it turns out that our considered functions also have low second-order zero differential uniformity.

We investigate pairs of permutations $F,G$ of $\mathbb{F}_{p^n}$ such that $F(x+a)-G(x)$ is a permutation for every $a\in\mathbb{F}_{p^n}$. We show that necessarily $G(x) = \wp(F(x))$ for some complete mapping $-\wp$ of $\mathbb{F}_{p^n}$, and call the permutation $F$ a perfect $\wp$ nonlinear (P$\wp$N) function. If $\wp(x) = cx$, then $F$ is a PcN function, which have been considered in the literature, lately. With a binary operation on $\mathbb{F}_{p^n}\times\mathbb{F}_{p^n}$ involving $\wp$, we obtain a quasigroup, and show that the graph of a P$\wp$N function $F$ is a difference set in the respective quasigroup. We further point to variants of symmetric designs obtained from such quasigroup difference sets. Finally, we analyze an equivalence (naturally defined via the automorphism group of the respective quasigroup) for P$\wp$N functions, respectively, the difference sets in the corresponding quasigroup.

Starting with the multiplication of elements in $\mathbb{F}_{q}^2$ which is consistent with that over $\mathbb{F}_{q^2}$, where $q$ is a prime power, via some identification of the two environments, we investigate the $c$-differential uniformity for bivariate functions $F(x,y)=(G(x,y),H(x,y))$. By carefully choosing the functions $G(x,y)$ and $H(x,y)$, we present several constructions of bivariate functions with low $c$-differential uniformity. Many P$c$N and AP$c$N functions can be produced from our constructions., Comment: Low $c$-differential uniformity, perfect and almost perfect $c$-nonlinearity, the bivariate function

The notion of $c$-differential uniformity has recently received a lot of attention since its proposal~\cite{Ellingsen}, and recently a characterization of perfect $c$-nonlinear functions in terms of difference sets in some quasigroups was obtained in~\cite{AMS22}. Independent of their applications as a measure for certain statistical biases, the construction of functions, especially permutations, with low $c$-differential uniformity is an interesting mathematical problem in this area, and recent work has focused heavily in this direction. We provide a few classes of permutation polynomials with low $c$-differential uniformity. The used technique involves handling various Weil sums, as well as analyzing some equations in finite fields, and we believe these can be of independent interest.

We propose a quantum algorithm (in the form of a quantum oracle) that estimates the closeness of a given Boolean function to one that satisfies the ``strict avalanche criterion'' (SAC). This algorithm requires $n$ queries of the Boolean function oracle, where $n$ is the number of input variables, this is fewer than the queries required by the classical algorithm to perform the same task. We compare our approach with other quantum algorithms that may be used for estimating the closeness to SAC and it is shown our algorithm verifies SAC with the fewest possible calls to quantum oracle and requires the fewest samples for a given confidence bound.

In the paper, we concentrate on the map $x\mapsto x^{\frac{p^n+1}{2}}$ on $\mathbb{F}_{p^n}$ and using combinatorial and number theory techniques, we compute its detailed $c$-differential spectrum for all values of $c\neq 1$ (the spectrum for $c=1$ is known).

In this paper we generalize Dillon's switching method to characterize the exact $c$-differential uniformity of functions constructed via this method. More precisely, we modify some PcN/APcN and other functions with known $c$-differential uniformity in a controllable number of coordinates to render more such functions. We present several applications of the method in constructing PcN and APcN functions with respect to all $c\neq 1$. As a byproduct, we generalize some result of [Y. Wu, N. Li, X. Zeng, {\em New PcN and APcN functions over finite fields}, Designs Codes Crypt. 89 (2021), 2637--2651]. Computational results rendering functions with low differential uniformity, as well as, other good cryptographic properties are sprinkled throughout the paper., Comment: 21 pages

In this paper, we investigate permutation rotation-symmetric (shift-invariant) vectorial Boolean functions on $n$ bits that are liftings from Boolean functions on $k$ bits, for $k\leq n$. These functions generalize the well-known map used in the current Keccak hash function, which is generated via the Boolean function on $3$ variables, $x_1+(x_2+1)x_3$. We provide some general constructions, and also study the affine equivalence between rotation-symmetric S-boxes and describe the corresponding relationship between the Boolean function they are associated with., Comment: 27 pages; revised version where several inaccuracies have been fixed and a few new observations added

In this paper, we construct some piecewise defined functions, and study their $c$-differential uniformity. As a by-product, we improve upon several prior results. Further, we look at concatenations of functions with low differential uniformity and show several results. For example, we prove that given $\beta_i$ (a basis of $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$), some functions $f_i$ of $c$-differential uniformities $\delta_i$, and $L_i$ (specific linearized polynomials defined in terms of $\beta_i$), $1\leq i\leq n$, then $F(x)=\sum_{i=1}^n\beta_i f_i(L_i(x))$ has $c$-differential uniformity equal to $\prod_{i=1}^n \delta_i$.

EFRST20, the notion of $c$-differentials was introduced as a potential expansion of differential cryptanalysis against block ciphers utilizing substitution boxes. Drawing inspiration from the technique of higher order differential cryptanalysis, in this paper we propose the notion of higher order $c$-derivatives and differentials and investigate their properties. Additionally, we consider how several classes of functions, namely the multiplicative inverse function and the Gold function, perform under higher order $c$-differential uniformity., Comment: 14 pages, to appear in International Conference on Security and Privacy, November 2021

Boolean functions are important primitives in different domains of cryptology, complexity and coding theory. In this paper, we connect the tools from cryptology and complexity theory in the domain of Boolean functions with low polynomial degree and high sensitivity. It is well known that the polynomial degree of of a Boolean function and its resiliency are directly connected. Using this connection we analyze the polynomial degree-sensitivity values through the lens of resiliency, demonstrating existence and non-existence results of functions with low polynomial degree and high sensitivity on small number of variables (upto 10). In this process, borrowing an idea from complexity theory, we show that one can implement resilient Boolean functions on a large number of variables with linear size and logarithmic depth. Finally, we extend the notion of sensitivity to higher order and note that the existing construction idea of Nisan and Szegedy (1994) can provide only constant higher order sensitivity when aiming for polynomial degree of $n-\omega(1)$. In this direction, we present a construction with low ($n-\omega(1)$) polynomial degree and super-constant $\omega(1)$ order sensitivity exploiting Maiorana-McFarland constructions, that we borrow from construction of resilient functions. The questions we raise identify novel combinatorial problems in the domain of Boolean functions.

We consider the boomerang uniformity of an infinite class of (locally-APN) power maps and show that its boomerang uniformity over the finite field $\F_{2^n}$ is $2$ and $4$, when $n \equiv 0 \pmod 4$ and $n \equiv 2 \pmod 4$, respectively. As a consequence, we show that for this class of power maps, the differential uniformity is strictly greater than its boomerang uniformity., Comment: 11 pages

In a prior paper \cite{EFRST20}, two of us, along with P. Ellingsen, P. Felke and A. Tkachenko, 1defined 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., Comment: 19 pages. arXiv admin note: text overlap with arXiv:2003.13019

Here, we give a complete description of the c-Boomerang Connectivity Table for the Gold function over finite fields of even characteristic, by using double Weil sums. In the process we generalize a result of Boura and Canteaut (IACR Trans. Symmetric Cryptol. 2018(3) : 290-310, 2018) for the classical boomerang uniformity., Comment: 20 pages

Modifying the binary inverse function in a variety of ways, like swapping two output points has been known to produce a $4$-differential uniform permutation function. Recently, in \cite{Li19} it was shown that this swapped version of the inverse function has boomerang uniformity exactly $10$, if $n\equiv 0\pmod 6$, $8$, if $n\equiv 3\pmod 6$, and 6, if $n\not\equiv 0\pmod 3$. Based upon the $c$-differential notion we defined in \cite{EFRST20} and $c$-boomerang uniformity from \cite{S20}, in this paper we characterize the $c$-differential and $c$-boomerang uniformity for the $(0,1)$-swapped inverse function in characteristic~$2$: we show that for all~$c\neq 1$, the $c$-differential uniformity is upper bounded by~$4$ and the $c$-boomerang uniformity by~$5$ with both bounds being attained for~$n\geq 4$., Comment: 25 pages

Building upon the observation that the newly defined~\cite{EFRST20} concept of $c$-differential uniformity is not invariant under EA or CCZ-equivalence~\cite{SPRS20}, we showed in~\cite{SG20} that adding some appropriate linearized monomials increases the $c$-differential uniformity of the inverse function, significantly, for some~$c$. We continue that investigation here. First, by analyzing the involved equations, we find bounds for the uniformity of the Gold function perturbed by a single monomial, exhibiting the discrepancy we previously observed on the inverse function. Secondly, to treat the general case of perturbations via any linearized polynomial, we use characters in the finite field to express all entries in the $c$-Differential Distribution Table (DDT) of an $(n,n)$-function on the finite field $\F_{p^n}$, and further, we use that method to find explicit expressions for all entries of the $c$-DDT of the perturbed Gold function (via an arbitrary linearized polynomial)., Comment: 22 pages

In this paper we characterize the $c$-Boomerang Connectivity Table (BCT), $c\neq 0$ (thus, including the classical $c=1$ case), for all monomial function $x^d$ in terms of characters and Weil sums on the finite field~$\F_{p^n}$. We further simplify these expressions for the Gold functions $x^{p^k+1}$ for all $1\leq k

We propose a quantum algorithm to estimate the Gowers $U_2$ norm of a Boolean function, and extend it into a second algorithm to distinguish between linear Boolean functions and Boolean functions that are $\epsilon$-far from the set of linear Boolean functions, which seems to perform better than the classical BLR algorithm. Finally, we outline an algorithm to estimate Gowers $U_3$ norms of Boolean functions.

Drawing inspiration from Nyberg's paper~\cite{Nyb91} on perfect nonlinearity and the $c$-differential notion we defined in~\cite{EFRST20}, in this paper we introduce the concept of $c$-differential bent functions in two different ways (thus extending Kumar et al.~\cite{Ku85} classical definition). We further extend the notion of perfect $c$-nonlinear introduced in~\cite{EFRST20}, also in two different ways, and show that, in both cases, the concepts of $c$-differential bent and perfect $c$-nonlinear are equivalent (under some natural restriction of the parameters). Some constructions of functions with these properties are also provided; one such construction provides a large class of PcN functions with respect to all $c$ in some subfield of the field under consideration. We also show that both our classes of $0$-differential bents are supersets of permutation polynomials, and that Maiorana-McFarland bent functions are not differential bent (of the first kind)., Comment: 24 pages

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 convey 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., Comment: 12 pages, IEEE Transactions on Information Theory, Accepted

While the classical differential uniformity ($c=1$) is invariant under the CCZ-equivalence, the newly defined \cite{EFRST20} concept of $c$-differential uniformity, in general is not invariant under EA or CCZ-equivalence, as was observed in \cite{SPRS20}. In this paper, we find an intriguing behavior of the inverse function, namely, that adding some appropriate linearized monomials increases the $c$-differential uniformity significantly, for some~$c$. For example, adding the linearized monomial $x^{p^d}$, where $d$ is the largest nontrivial divisor of $n$, increases the mentioned $c$-differential uniformity from~$2$ or $3$ (for $c\neq 0$) to $\geq p^{d}+2$, which in the case of AES' inverse function on $\F_{2^8}$ is a significant value of~$18$., Comment: 11 pages. arXiv admin note: text overlap with arXiv:2004.11859

In this paper, for a positive integer $n\ge 1$, we look at the size and prime factors of the iterates of the Ramanujan $\tau$ function applied to $n$., Comment: 14 pages

We defined in~\cite{EFRST20} a new multiplicative $c$-differential, and the corresponding $c$-differential uniformity and we characterized the known perfect nonlinear functions with respect to this new concept, as well as the inverse in any characteristic. The work was continued in~\cite{RS20}, investigating the $c$-differential uniformity for some further APN functions. Here, we extend the concept to the boomerang uniformity, introduced at Eurocrypt '18 by Cid et al.~\cite{Cid18}, to evaluate S-boxes of block ciphers, and investigate it in the context of perfect nonlinearity and related functions., Comment: 31 pages, 1 figure

We give some classes of power maps with low $c$-differential uniformity over finite fields of odd characteristic, {for $c=-1$}. Moreover, we give a necessary and sufficient condition for a linearized polynomial to be a perfect $c$-nonlinear function and investigate conditions when perturbations of perfect $c$-nonlinear (or not) function via an arbitrary Boolean or $p$-ary function is perfect $c$-nonlinear. In the process, we obtain a class of polynomials that are perfect $c$-nonlinear for all $c\neq 1$, in every characteristic. The affine, extended affine and CCZ-equivalence is also looked at, as it relates to $c$-differential uniformity., Comment: Revised version; 22 pages; to appear in Des. Codes Cryptogr

In a prior paper [14], along with P. Ellingsen, P. Felke and A. Tkachenko, we 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 show that their c-differential uniformity increases significantly, in some cases., Comment: arXiv admin note: text overlap with arXiv:1909.03628

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) 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., Comment: 23 pages

In this paper we define a new transform on (generalized) Boolean functions, which generalizes the Walsh-Hadamard, nega-Hadamard, $2^k$-Hadamard, consta-Hadamard and all $HN$-transforms. We describe the behavior of what we call the root- Hadamard transform for a generalized Boolean function $f$ in terms of the binary components of $f$. Further, we define a notion of complementarity (in the spirit of the Golay sequences) with respect to this transform and furthermore, we describe the complementarity of a generalized Boolean set with respect to the binary components of the elements of that set., Comment: 19 pages

In this paper we define a notion of partial APNness and find various characterizations and constructions of classes of functions satisfying this condition. We connect this notion to the known conjecture that APN functions modified at a point cannot remain APN. In the second part of the paper, we find conditions for some transformations not to be partially APN, and in the process, we find classes of functions that are never APN for infinitely many extensions of the prime field $\F_2$, extending some earlier results of Leander and Rodier., Comment: 24 pages; to appear in Cryptography and Communications

In this paper we find a complete characterization of plateaued Boolean functions in terms of the associated Cayley graphs. Precisely, we show that a Boolean function $f$ is $s$-plateaued (of weight $=2^{(n+s-2)/2}$) if and only if the associated Cayley graph is a complete bipartite graph between the support of $f$ and its complement (hence the graph is strongly regular of parameters $e=0,d=2^{(n+s-2)/2}$). Moreover, a Boolean function $f$ is $s$-plateaued (of weight $\neq 2^{(n+s-2)/2}$) if and only if the associated Cayley graph is strongly $3$-walk-regular (and also strongly $\ell$-walk-regular, for all odd $\ell\geq 3$) with some explicitly given parameters., Comment: 7 pages, 1 figure, Proceedings of Africacrypt 2018

In this paper we define the (edge-weighted) Cayley graph associated to a generalized Boolean function, introduce a notion of strong regularity and give several of its properties. We show some connections between this concept and generalized bent functions (gbent), that is, functions with flat Walsh-Hadamard spectrum. In particular, we find a complete characterization of quartic gbent functions in terms of the strong regularity of their associated Cayley graph., Comment: 13 pages, 2 figures

In this paper we define a class of Boolean and generalized Boolean functions defined on $\mathbb{F}_2^n$ with values in $\mathbb{Z}_q$ (mostly, we consider $q=2^k$), which we call landscape functions (whose class containing generalized bent, semibent, and plateaued) and find their complete characterization in terms of their components. In particular, we show that the previously published characterizations of generalized bent and plateaued Boolean functions are in fact particular cases of this more general setting. Furthermore, we provide an inductive construction of landscape functions, having any number of nonzero Walsh-Hadamard coefficients. We also completely characterize generalized plateaued functions in terms of the second derivatives and fourth moments., Comment: 19 pages