We study those $(n,n)$-permutations, and more generally those balanced $(n,m)$-functions, whose component functions all admit a derivative equal to constant function 1 (this property itself implies balancedness). We call these functions quadratic-like permutations (resp. quadratic-like balanced functions) since all quadratic balanced functions have this property. We show that all Feistel permutations, all crooked permutations and (more generally) all balanced strongly plateaued functions...

We study the behavior of the multiplicative inverse function (which plays an important role in cryptography and in the study of finite fields), with respect to a recently introduced generalization of almost perfect nonlinearity (APNness), called $k$th-order sum-freedom, that extends a classic characterization of APN functions, and has also some relationship with integral attacks. This generalization corresponds to the fact that a vectorial function $F:\mathbb F_2^n\mapsto \mathbb F_2^m$...

Multivariate Cryptography is one of the main candidates for Post-quantum Cryptography. Multivariate schemes are usually constructed by applying two secret affine invertible transformations $\mathcal S,\mathcal T$ to a set of multivariate polynomials $\mathcal{F}$ (often quadratic). The secret polynomials $\mathcal{F}$ posses a trapdoor that allows the legitimate user to find a solution of the corresponding system, while the public polynomials $\mathcal G=\mathcal S\circ\mathcal...

Almost perfect nonlinear (in brief, APN) functions are vectorial functions $F:\mathbb F_2^n\rightarrow \mathbb F_2^n$ playing roles in several domains of information protection, at the intersection of computer science and mathematics. Their definition comes from cryptography and is also related to coding theory. When they are used as substitution boxes (S-boxes, which are the only nonlinear components in block ciphers), APN functions contribute optimally to the resistance against...

In symmetric cryptography, vectorial Boolean functions over finite fields F2n derive strong S-boxes. To this end, the S-box should satisfy a list of tests to resist existing attacks, such as the differential, linear, boomerang, and variants. Several tables are employed to measure an S- box’s resistance, such as the difference distribution table (DDT) and the boomerang connectivity table (BCT). Following the boomerang attacks recently revisited in terms of the boomerang switch effect, with a...

The substitution box (S-box) is often used as the only nonlinear component in symmetric-key ciphers, leading to a significant impact on the implementation performance of ciphers in both classical and quantum application scenarios by S-box circuits. Taking the Pauli-X gate, the CNOT gate, and the Toffoli gate (i.e., the NCT gate set) as the underlying logic gates, this work investigates the quantum circuit implementation of S-boxes based on the SAT solver. Firstly, we propose encoding methods...

Symmetric cryptography primitives are constructed by iterative applications of linear and nonlinear layers. Constructing efficient circuits for these layers, even for the linear one, is challenging. In 1997, Paar proposed a heuristic to minimize the number of XORs (modulo 2 addition) necessary to implement linear layers. In this study, we slightly modify Paar’s heuristics to find implementations for nonlinear Boolean functions, in particular to homogeneous Boolean functions. Additionally, we...

For any prime number $p$, we provide two classes of linear codes with few weights over a $p$-ary alphabet. These codes are based on a well-known generic construction (the defining-set method), stemming on a class of monomials and a class of trinomials over finite fields. The considered monomials are Dembowski-Ostrom monomials $x^{p^{\alpha}+1}$, for a suitable choice of the exponent $\alpha$, so that, when $p>2$ and $n\not\equiv 0 \pmod{4}$, these monomials are planar. We study the...

A Feistel Network (FN) based block cipher relies on a Substitution Box (S-Box) for achieving the non-linearity. S-Box is carefully designed to achieve optimal cryptographic security bounds. The research of the last three decades shows that considerable efforts are being made on the mathematical design of an S-Box. To import the exact cryptographic profile of an S-Box, the designer focuses on the Affine Equivalent (AE) or Extended Affine (EA) equivalent S-Box. In this research, we argue that...

In this paper, we re-investigate the Lai-Massey scheme, originally proposed in the cipher IDEA. Due to the similarity with the Feistel networks, and due to the existence of invariant subspace attacks as originally pointed out by Vaudenay at FSE 1999, the Lai-Massey scheme has received only little attention by the community. As first contribution, we propose two new generalizations of such scheme that are not (extended) affine equivalent to any generalized Feistel network proposed in the...

In 2002, Chow et al. initiated the formal study of white-box cryptography and introduced the CEJO framework. Since then, various white-box designs based on their framework have been proposed, all of them broken. Ranea and Preneel proposed a different method in 2020, called self-equivalence encodings and analyzed its security for AES. In this paper, we apply this method to generate the first academic white-box Speck implementations using self-equivalence encodings. Although we focus on Speck...

Since the first white-box implementation of AES published twenty years ago, no significant progress has been made in the design of secure implementations against an attacker with full control of the device. Designing white-box implementations of existing block ciphers is a challenging problem, as all proposals have been broken. Only two white-box design strategies have been published this far: the CEJO framework, which can only be applied to ciphers with small S-boxes, and self-equivalence...

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≤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, x1+(x2+1)x3. We provide some general constructions, and also study the affine equivalence between rotation-symmetric S-boxes and describe the corresponding relationship...

The SM4 block cipher was first released in 2006 as SMS4 used in the Chinese national standard WAPI, and became a Chinese national standard in 2016 and an ISO international standard in 2021. White-box cryptography aims primarily to protect the secret key used in a cryptographic software implementation in the white-box scenario that assumes an attacker to have full access to the execution environment and execution details of an implementation. Since white-box cryptography has many real-life...

Affine message authentication code (AMAC) (CRYPTO'14) is a group-based MAC with a specific algebraic structure. Downgradable AMAC (DAMAC) (CT-RSA'19) is an AMAC with a functionality that we can downgrade a message with an authentication tag while retaining validity of the tag. In this paper, we revisit DAMAC for two independent applications, namely downgradable identity-based signatures (DIBS) and trapdoor sanitizable signatures (TSS) (ACNS'08). DIBS are the digital signature analogue of...

Extended Affine (EA) equivalence is the equivalence relation between two vectorial Boolean functions $F$ and $G$ such that there exist two affine permutations $A$, $B$, and an affine function $C$ satisfying $G = A \circ F \circ B + C$. While a priori simple, it is very difficult in practice to test whether two functions are EA-equivalent. This problem has two variants: EA-testing deals with figuring out whether the two functions can be EA-equivalent, and EA-recovery is about recovering the...

The notion of almost perfect nonlinear (APN) function is important, mathematically and cryptographically. Much still needs to be understood on the structure and the properties of APN functions. For instance, finding an APN permutation in an even number of variables larger than 6 would be an important theoretical and practical advance. A way to progress on a notion is to introduce and study generalizations making sense from both theoretical and practical points of view. The introduction and ...

All academic methods to secure software implementations of block ciphers against adversaries with full control of the device have been broken. Despite the huge progress in the cryptanalysis of these white-box implementations, no recent progress has been made on the design side. Most of the white-box designs follow the CEJO framework, where each round is encoded by composing it with small random permutations. While several generic attacks have been proposed on the CEJO framework, no generic...

We present a constructive SAT-based algorithm to determine the multiplicative complexity of a Boolean function, i.e., the smallest number of AND gates in any logic network that consists of 2-input AND gates, 2-input XOR gates, and inverters. In order to speed-up solving time, we make use of several symmetry breaking constraints; these exploit properties of XAGs that may be useful beyond the proposed SAT-based algorithm. We further propose a heuristic post-optimization algorithm to reduce the...

We consider algebraic affine and projective curves of Edwards [3, 9] over the finite field ${{\text{F}}_{{{p}^{n}}}}$. It is known that many modern cryptosystems [11] can be naturally transformed into elliptic curves [5]. We research Edwards algebraic curves over a finite field, which are one of the most promising supports of sets of points which are used for fast group operations \cite{Bir}. We construct a new method for counting the order of an Edwards curve over a finite field. It should...

We characterize the ANF and the univariate representation of any vectorial function as parts of the ANF and bivariate representation of the Boolean function equal to its graph indicator. We show how this provides, when $F$ is bijective, the expression of $F^{-1}$ and/or allows deriving properties of $F^{-1}$. We illustrate this with examples and with a tight upper bound on the algebraic degree of $F^{-1}$ by means of that of $F$. We characterize by the Fourier-Hadamard transform, by the ANF,...

Differential branch number and linear branch number are critical for the security of symmetric ciphers. The recent trend in the designs like PRESENT block cipher, ASCON authenticated encryption shows that applying S-boxes that have nontrivial differential and linear branch number can significantly reduce the number of rounds. As we see in the literature that the class of 4 x 4 S-boxes have been well-analysed, however, a little is known about the n x n S-boxes for n >= 5. For instance, the...

Multiplicative complexity (MC) is defined as the minimum number of AND gates required to implement a function with a circuit over the basis (AND, XOR, NOT). Boolean functions with MC 1 and 2 have been characterized in Fischer and Peralta ( 2002) and Find et al. (2017), respectively. In this work, we identify the affine equivalence classes for functions with MC 3 and 4. In order to achieve this, we utilize the notion of the dimension $dim(f)$ of a Boolean function in relation to its linearity...

We revisit the definition of Transparency Order (TO) and that of Modified Transparency Order (MTO) as well, which were proposed to measure the resistance of an S-box against Differential Power Analysis (DPA). We spot a definitional flaw in original TO, which is proved to have significantly affected the soundness of TO and hinder it to be a good quantitative security criterion. Regretfully, the flaw itself remains virtually undiscovered in MTO, either. Surprisingly, MTO overlooks this flaw...

For all vectorial boolean functions up to dimension 4, we present canonical representatives for all extended affine (EA) and CCZ equivalence classes. We include the size of each class, as well as its algebraic degree and extended Walsh spectrum. We also answer the following questions: How large are these classes? Which of these classes contain bijective functions? And how are these classes grouped into CCZ equivalence classes?

At Eurocrypt'18, Cid, Huang, Peyrin, Sasaki, and Song introduced a new tool called Boomerang Connectivity Table (BCT) for measuring the resistance of a block cipher against the boomerang attack which is an important cryptanalysis technique introduced by Wagner in 1999 against block ciphers. Next, Boura and Canteaut introduced an important parameter related to the BCT for cryptographic Sboxes called boomerang uniformity. The purpose of this paper is to present a brief state-of-the-art on the...

Two vectorial Boolean functions are ``CCZ-equivalent'' if there exists an affine permutation mapping the graph of one to the other. It preserves many of the cryptographic properties of a function such as its differential and Walsh spectra, which is why it could be used by Dillon et al. to find the first APN permutation on an even number of variables. However, the meaning of this form of equivalence remains unclear. In fact, to the best of our knowledge, it is not known how to partition a...

In this paper we study the affine equivalence problem, where given two functions $\vec{F},\vec{G}: \{0,1\}^n \rightarrow \{0,1\}^n$, the goal is to determine whether there exist invertible affine transformations $A_1,A_2$ over $GF(2)^n$ such that $\vec{G} = A_2 \circ \vec{F} \circ A_1$. Algorithms for this problem have several well-known applications in the design and analysis of Sboxes, cryptanalysis of white-box ciphers and breaking a generalized Even-Mansour scheme. We describe a new...

S-boxes, typically the only nonlinear part of a block cipher, are the heart of symmetric cryptographic primitives. They significantly impact the cryptographic strength and the implementation characteristics of an algorithm. Due to their simplicity, quadratic vectorial Boolean functions are preferred when efficient implementations for a variety of applications are of concern. Many characteristics of a function stay invariant under affine equivalence. So far, all 6-bit Boolean functions, 3-...

The multiplicative complexity of a Boolean function is the minimum number of 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. showed that $n$-variable Boolean functions can be implemented with at most $n-1$ AND gates for $n\leq 5$. A counting argument can be used to show that, for $n \geq...

Cellular Automata (CA) represent an interesting approach to design Substitution Boxes (S-boxes) having good cryptographic properties and low implementation costs. From the cryptographic perspective, up to now there have been only ad-hoc studies about specific kinds of CA, the best known example being the $\chi$ nonlinear transformation used in Keccak. In this paper, we undertake a systematic investigation of the cryptographic properties of S-boxes defined by CA, proving some upper bounds on...

C.~Carlet, P.~Charpin, V.~Zinoviev in 1998 defined the associated Boolean function $\gamma_F(a,b)$ in $2n$ variables for a given vectorial Boolean function $F$ from $\mathbb{F}_2^n$ to itself. It takes value~$1$ if $a\neq {\bf 0}$ and equation $F(x)+F(x+a)=b$ has solutions. This article defines the differentially equivalent functions as vectorial functions having equal associated Boolean functions. It is an open problem of great interest to describe the differential equivalence class for a...

In this paper, we determine the Walsh spectra of three classes of quadratic APN functions and we prove that the class of quadratic trinomial APN functions constructed by Gölo\u glu is affine equivalent to Gold functions.

In [13] for a given vectorial Boolean function $F$ from $\mathbb{F}_2^n$ to itself it was defined an associated Boolean function $\gamma_F(a,b)$ in $2n$ variables that takes value~$1$ iff $a\neq{\bf 0}$ and equation $F(x)+F(x+a)=b$ has solutions. In this paper we introduce the notion of differentially equivalent functions as vectorial functions that have equal associated Boolean functions. It is an interesting open problem to describe differential equivalence class of a given APN...

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 ciphers with a small number of AND gates are preferred in protocols for fully homomorphic encryption, multi-party computation and zero-knowledge proofs. In 2002, Fischer and Peralta showed that the number of $n$-variable Boolean functions with multiplicative complexity one equals $2\binom{2^n}{3}$....

Motivated by the development of Side-Channel Analysis (SCA) countermeasures which can provide security up to a certain order, defeating higher-order attacks has become amongst the most challenging issues. For instance, Threshold Implementation (TI) which nicely solves the problem of glitches in masked hardware designs is able to avoid first-order leakages. Hence, its extension to higher orders aims at counteracting SCA attacks at higher orders, that might be limited to univariate scenarios....

This paper investigates the effect of affine transformations of the Sbox on the maximal expected differential probability MEDP and linear potential MELP over two rounds of a substitution-permutation network, when the diffusion layer is linear over the finite field defined by the Sbox alphabet. It is mainly motivated by the fact that the 2-round MEDP and MELP of the AES both increase when the AES Sbox is replaced by the inversion in $GF(2^8)$. Most notably, we give new upper bounds on these...

In this paper we consider permutations on n symbols as bijections on Z/nZ. Treating permutations this way facilitates us with additional structures such as group, ring defined in the set Z/nZ. We explore some of the properties of permutations arising out of this treatment. We propose two properties viz. affine equivalence and non-linearity for permutations on the lines similar to there description given in the case of functions. We also establish some results which are quite similar to those...

White-box cryptography is an obfuscation technique for protecting secret keys in software implementations even if an adversary has full access to the implementation of the encryption algorithm and full control over its execution platforms. This concept was presented by Chow et al. with white-box implementations of DES and AES in 2002. The strategy used in the implementations has become a design principle for subsequent white-box implementations. However, despite its practical importance,...

We consider the Fourier Entropy-Influence (FEI) conjecture in the context of cryptographic Boolean functions. We show that the FEI conjecture is true for the functions satisfying the strict avalanche criterion, which forms a subset of asymptotic log--density~$1$ in the set of all Boolean functions. Further, we prove that the FEI conjecture is satisfied for plateaued Boolean functions, monomial algebraic normal form (with the best involved constant), direct sums, as well as concatenations of...

In FSE 2010, Nandi proved a sufficient condition of pseudo random function (PRF) for affine domain extensions (ADE), wide class of block cipher based domain extensions. This sufficient condition is satisfied by all known blockcipher based ADE constructions, however, it is not a characterization of PRF. In this paper we completely characterize the ADE and show that {\em message authentication code (MAC) and weakly collision resistant (WCR) are indeed equivalent to PRF}. Note that a PRF is...

In [eprint.iacr.org/2012/663] method of virtual isomorphisms of ciphers was applied for differential/linear cryptanalysis of AES. It was shown that AES seems to be weak against those attacks. That result can be generalized to AES-like ciphers, which diffusion map is a block matrix, and its block size is the same as the S-box size. S-box is possibly weak if it is affine equivalent to a substitution that has the same cycling type as an affine substitution. Class of possibly weak S-boxes is...

In the last decade, algebraic and fast algebraic attacks are regarded as the most successful attacks on LFSR-based stream ciphers. Since the notion of algebraic immunity was introduced, the properties and constructions of Boolean functions with maximum algebraic immunity have been researched in a large number of papers. However, there are few results with respect to Boolean functions with provable good immunity against fast algebraic attacks. In previous literature, only Carlet-Feng...

Finding the key of symmetric cipher takes computing common zero of polynomials, which define ideal and corresponding variety, usually considered over algebraically closed field. The solution is the point of the variety over prime field; it is defined by a sum of the polynomial ideal and the field ideal that defines prime field. Some authors use partitioning of this sum and reducing syzygies of polynomial ideal modulo field ideal. We generalize this method and consider polynomial ideal as a...

Elliptic Curve Cryptography (ECC) was independently introduced by Koblitz and Miller in the eighties. ECC requires shorter sizes of underlying finite fields in com- parison to other public key cryptosystems such as RSA, introduced by Rivest, Shamir and Adleman. Hyperelliptic curves, a generalization of elliptic curves, require decreas- ing field size as genus increases. Hyperelliptic curves of genus g achieve equivalent security of ECC with field size 1/g times the size of field of ECC for g...

We present cryptanalytic results of an exhaustive search of all $16!$ bijective 4-bit S-Boxes. Previously affine equivalence classes have been exhaustively analyzed in 2007 work by Leander and Poschmann. We extend on this work by giving further properties of the optimal S-Box linear equivalence classes. In our main analysis we consider two S-Boxes to be cryptanalytically equivalent if they are isomorphic up to the permutation of input and output bits and a XOR of a constant in the input and...

It is well-known that affine equivalence relations keep nonlineaerity invariant for all Boolean functions. The set of all Boolean functions, $\mathcal{F}_n$, over $\bbbf_2^n$, is naturally regarded as the $2^n$ dimensional vector space, $\bbbf_2^{2^n}$. Thus, while analyzing the transformations acting on $\mathcal{F}_n$, $S_{2^{2^n}}$, the group of all bijective mappings, defined from $\bbbf_2^{2^n}$ onto itself should be considered. As it is shown in \cite{ser,ser:dog,ser:dog:2}, there...

It is proved that there does not exist a linearized polynomial $L(x)\in\mathbb{F}_{2^n}[x]$ such that $x^{-1}+L(x)$ is a permutation on $\mathbb{F}_{2^n}$ when $n\geq5$, which is proposed as a conjecture in \cite{li}. As a consequence, a permutation is EA-equivalent to the inverse function over $\mathbb{F}_{2^n}$ if and only if it is affine equivalent to it when $n\geq 5$.

One of the classes of bent Boolean functions introduced by John Dillon in his thesis is family H. While this class corresponds to a nice original construction of bent functions in bivariate form, Dillon could exhibit in it only functions which already belonged to the well- known Maiorana-McFarland class. We first notice that H can be extended to a slightly larger class that we denote by H. We observe that the bent functions constructed via Niho power functions, which four examples are known,...

This paper presents a deterministic algorithm for converting points on an ordinary elliptic curve (defined over a field of characteristic 2) to points on a complete binary Edwards curve. This avoids the problem of choosing curve parameters at random. When implemented on a large (512 bit) hardware multiplier, computation of point multiplication using this algorithm performs significantly better, in terms of code complexity, code coverage and timing, than the standard implementation. In...

\begin{document} Ciphers $y=C(x, k)$ and $Y=C_{1}(X, K)$ are isomorphic if there exists invertible computable in both directions map $y \leftrightarrow Y$, $x \leftrightarrow X$, $k \leftrightarrow K$. Cipher is vulnerable if and only if isomorphic cipher is vulnerable. Instead of computing the key of a cipher it is sufficient to find suitable isomorphic cipher and compute its key. If $\varphi $ is arbitrary substitution and $T$ is round substitution, its conjugate $T_{1}=\varphi T\varphi ^{...

We study further CCZ-equivalence of $(n,m)$-functions. We prove that for Boolean functions (that is, for $m=1$), CCZ-equivalence coincides with EA-equivalence. On the contrary, we show that for $(n,m)$- functions, CCZ-equivalence is strictly more general than EA-equivalence when $n\ge5$ and $m$ is greater or equal to the smallest positive divisor of $n$ different from 1. Our result on Boolean functions allows us to study the natural generalization of CCZ-equivalence corresponding to the...

We prove that, for bent vectorial functions, CCZ-equivalence coincides with EA-equivalence. However, we show that CCZ-equivalence can be used for constructing bent functions which are new up to CCZ-equivalence. Using this approach we construct classes of nonquadratic bent Boolean and bent vectorial functions.

It is proved that no non-quadratic Kasami bent is affine equivalent to Maiorana-MacFarland type bent functions.

A method for constructing differentially 4-uniform quadratic hexanomials has been recently introduced by J. Dillon. We give various generalizations of this method and we deduce the constructions of new infinite classes of almost perfect nonlinear quadratic trinomials and hexanomials from $\mathbb{F}_{2^{2m}}$ to $\mathbb{F}_{2^{2m}}$. We check for $m=3$ that some of these functions are CCZ-inequivalent to power functions.

We present a method for constructing new quadratic APN functions from known ones. Applying this method to the Gold power functions we construct an APN function $x^3+\tr(x^9)$ over $\F_{2^n}$. It is proven that in general this function is CCZ-inequivalent to the Gold functions (and therefore EA-inequivalent to power functions), to the inverse and Dobbertin mappings, and in the case $n=7$ it is CCZ-inequivalent to all power mappings.

The first APN polynomials EA-inequivalent to power functions have been constructed in [1,2] by applying CCZ-equivalence to the Gold APN functions. It is a natural question whether it is possible to construct APN polynomials EA-inequivalent to power functions by using only EA-equivalence and inverse transformation on a power APN function: this would be the simplest method to construct APN polynomials EA-inequivalent to power functions. In the present paper we prove that the answer to this...

We exhibit an infinite class of almost perfect nonlinear quadratic binomials from $\mathbb{F}_{2^n}$ to $\mathbb{F}_{2^n}$ ($n\geq 12$, $n$ divisible by 3 but not by 9). We prove that these functions are EA-inequivalent to any power function and that they are CCZ-inequivalent to any Gold function and to any Kasami function. It means that for $n$ even they are CCZ-inequivalent to any known APN function, and in particular for $n=12,24$, they are therefore CCZ-inequivalent to any power...

We exhibit an infinite class of almost perfect nonlinear quadratic binomials from $\mathbb{F}_{2^{n}}$ to $\mathbb{F}_{2^{n}}$ with $n=4k$ and $k$ odd. We prove that these functions are CCZ-inequivalent to known APN power functions when $k\ne 1$. In particular it means that for $n=12,20,28$, they are CCZ-inequivalent to any power function.

The existence of $9$-variable Boolean functions having nonlinearity strictly greater than $240$ has been shown very recently (May 2006) by Kavut, Maitra and Yücel. The functions with nonlinearity 241 have been identified by a heuristic search in the class of Rotation Symmetric Boolean Functions (RSBFs). In this paper we efficiently perform the exhaustive search to enumerate the 9-variable RSBFs having nonlinearity $> 240$ and found that there are such functions with nonlinearity 241 only and...

We exhibit an infinite class of almost perfect nonlinear quadratic polynomials from $\mathbb{F}_{2^n}$ to $\mathbb{F}_{2^n}$ ($n\geq 12$, $n$ divisible by 3 but not by 9). We prove that these functions are EA-inequivalent to any power function. In the forthcoming version of the present paper we will proof that these functions are CCZ-inequivalent to any Gold function and to any Kasami function, in particular for $n=12$, they are therefore CCZ-inequivalent to power functions.

By walsh transform, autocorrelation function, decomposition, derivation and modification of truth table, some new invariants are obtained. Based on invariant theory, we get two results: first a general algorithm which can be used to judge if two boolean functions are affinely equivalent and to obtain the affine equivalence relationship if they are equivalent. For example, all 8-variable homogenous bent functions of degree 3 are classified into 2 classes; second, the classification of the...

This paper presents an efficient approach for classification of the affine equivalence classes of cosets of the first order Reed-Muller code with respect to cryptographic properties such as correlation-immunity, resiliency and propagation characteristics. First, we apply the method to completely classify all the $48$ classes into which the general affine group $AGL(2,5)$ partitions the cosets of $RM(1,5)$. Second, we describe how to find the affine equivalence classes together with their...

In this paper we ask the question what happens if we replace all the constants in Rijndael, including the replacement of the irreducible polynomial, the coefficients of the MixColumn operation, the affine transformation in the S box, etc. We show that such replacements can create new dual ciphers, which are equivalent to the original in all aspects. We present several such dual ciphers of Rijndael, such as the square of Rijndael, and dual ciphers with the irreducible polynomial replaced by...

We show the existence of a previously unknown linear redundancy property of the only nonlinear component of the AES block cipher. It is demonstrated that the outputs of the 8*8 Rijndael s-box (based on inversion in a finite field) are all equivalent under affine transformation. The method used to discover these affine relations is novel and exploits a new fundamental result on the invariance properties of local connection structure of affine equivalence classes. As well as increasing...