What a lovely hat

This paper presents dCTIDH, a CSIDH implementation that combines two recent developments into a novel state-of-the-art deterministic implementation. We combine the approach of deterministic variants of CSIDH with the batching strategy of CTIDH, which shows that the full potential of this key space has not yet been explored. This high-level adjustment in itself leads to a significant speed-up. To achieve an effective deterministic evaluation in constant time, we introduce Wombats, a new...

Index Calculus (IC) algorithm is the most effective probabilistic algorithm for solving discrete logarithms over finite fields of prime numbers, and it has been widely applied to cryptosystems based on elliptic curves. Since the IC algorithm was proposed in 1920, the research on it has never stopped, especially discretization of prime numbers on the finite fields, both the algorithm itself and its application have been greatly developed. Of course, there has been some research on elliptic...

Folding schemes (Kothapalli et al., CRYPTO 2022) are a conceptually simple, yet powerful cryptographic primitive that can be used as a building block to realise incrementally verifiable computation (IVC) with low recursive overhead without general-purpose non-interactive succinct arguments of knowledge (SNARK). Most folding schemes known rely on the hardness of the discrete logarithm problem, and thus are both not quantum-resistant and operate over large prime fields. Existing post-quantum...

In 2016,Petit et al. first studied the implementation of the index calculus method on elliptic curves in prime finite fields, and in 2018, Momonari and Kudo et al. improved algorithm of Petit et al. This paper analyzes the research results of Petit, Momonari and Kudo, and points out the existing problems of the algorithm. Therefore, with the help of sum polynomial function and index calculus, a pseudo-index calculus algorithm for elliptic curves discrete logarithm problem over prime finite...

Any isogeny between two supersingular elliptic curves can be defined over $\mathbb{F}_{p^2}$, however, this does not imply that computing such isogenies can be done with field operations in $\mathbb{F}_{p^2}$. In fact, the kernel generators of such isogenies are defined over extension fields of $\mathbb{F}_{p^2}$, generically with extension degree linear to the isogeny degree. Most algorithms related to isogeny computations are only efficient when the extension degree is small. This leads to...

Reed-Solomon (RS) codes [RS60], representing evaluations of univariate polynomials over distinct domains, are foundational in error correction and cryptographic protocols. Traditional RS codes leverage the Fourier domain for efficient encoding and decoding via Fast Fourier Transforms (FFT). However, in fields such as the Reals and some finite prime fields, limited root-of-unity orders restrict these methods. Recent research, particularly in the context of modern STARKs [BSBHR18b], has...

Succinct arguments of knowledge allow an untrusted prover to establish that they know a witness for an NP relation. Many recent efficient constructions of such schemes work over arithmetic computations expressed in finite fields. Several common settings, however, have an extremely simple representation when expressed over the integers (e.g., RSA signatures/accumulators, range checks for committed values, computations over rational numbers). Efficient arguments of knowledge working natively...

In this article, we derive the weight distribution of linear codes stemming from a subclass of (vectorial) $p$-ary plateaued functions (for a prime $p$), which includes all the explicitly known examples of weakly and non-weakly regular plateaued functions. This construction of linear codes is referred in the literature as the first generic construction. First, we partition the class of $p$-ary plateaued functions into three classes $\mathscr{C}_1, \mathscr{C}_2,$ and $\mathscr{C}_3$,...

In recent years a new class of symmetric-key primitives over $\mathbb{F}_p$ that are essential to Multi-Party Computation and Zero-Knowledge Proofs based protocols has emerged. Towards improving the efficiency of such primitives, a number of new block ciphers and hash functions over $\mathbb{F}_p$ were proposed. These new primitives also showed that following alternative design strategies to the classical Substitution-Permutation Network (SPN) and Feistel Networks leads to more efficient...

A common misconception is that the computational abilities of circuits composed of additions and multiplications are restricted to simple formulas only. Such arithmetic circuits over finite fields are actually capable of computing any function, including equality checks, comparisons, and other highly non-linear operations. While all those functions are computable, the challenge lies in computing them efficiently. We refer to this search problem as arithmetization. Arithmetization is a key...

We show that the data storage scheme [IEEE/ACM Trans. Netw., 2023, 31(4), 1550-1565] is flawed due to the false secret sharing protocol, which requires that some random $4\times 4$ matrixes over the finite field $F_p$ (a prime $p$) are invertible. But we find its mathematical proof for invertibility is incorrect. To fix this flaw, one needs to check the invertibility of all 35 matrixes so as to generate the proper 7 secret shares.

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 $F:\mathbb F_2^n\mapsto \mathbb F_2^m$...

Shaped prime moduli are often considered for use in elliptic curve and isogeny-based cryptography to allow for faster modular reduction. Here we focus on the most common choices for shaped primes that have been suggested, that is pseudo-Mersenne, generalized Mersenne and Montgomery-friendly primes. We consider how best to to exploit these shapes for maximum efficiency, and provide an open source tool to automatically generate, test and time working high-level language finite-field code....

In the implementation of isogeny-based schemes, V\'{e}lu's formulas are essential for constructing and evaluating odd degree isogenies. Bernstein et al. proposed an approach known as $\surd$elu, which computes an $\ell$-isogeny at a cost of $\tilde{\mathcal{O}}(\sqrt{\ell})$ finite field operations. This paper presents two key improvements to enhance the efficiency of the implementation of $\surd$\'{e}lu from two aspects: optimizing the partition involved in $\surd$\'{e}lu and speeding...

Traditional STARKs require a cyclic group of a smooth order in the field. This allows efficient interpolation of points using the FFT algorithm, and writing constraints that involve neighboring rows. The Elliptic Curve FFT (ECFFT, Part I and II) introduced a way to make efficient STARKs for any finite field, by using a cyclic group of an elliptic curve. We show a simpler construction in the lines of ECFFT over the circle curve $x^2 + y^2 = 1$. When $p + 1$ is divisible by a large power of...

Since 2015, there has been a significant decrease in the asymptotic complexity of computing discrete logarithms in finite fields. As a result, the key sizes of many mainstream pairing-friendly curves have to be updated to maintain the desired security level. In PKC'20, Guillevic conducted a comprehensive assessment of the security of a series of pairing-friendly curves with embedding degrees ranging from $9$ to $17$. In this paper, we focus on pairing-friendly curves with embedding degrees...

Fully homomorphic encryption (FHE) is an advanced cryptography technique to allow computations (i.e., addition and multiplication) over encrypted data. After years of effort, the performance of FHE has been significantly improved and it has moved from theory to practice. The transciphering framework is another important technique in FHE to address the issue of ciphertext expansion and reduce the client-side computational overhead. To apply the transciphering framework to the CKKS FHE scheme,...

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...

GLS254 is an elliptic curve defined over a finite field of characteristic 2; it contains a 253-bit prime order subgroup, and supports an endomorphism that can be efficiently computed and helps speed up some typical operations such as multiplication of a curve element by a scalar. That curve offers on x86 and ARMv8 platforms the best known performance for elliptic curves at the 128-bit security level. In this paper we present a number of new results related to GLS254: - We describe...

Recently, many cryptographic primitives such as homomorphic encryption (HE), multi-party computation (MPC) and zero-knowledge (ZK) protocols have been proposed in the literature which operate on prime field $\mathbb{F}_p$ for some large prime $p$. Primitives that are designed using such operations are called arithmetization-oriented primitives. As the concept of arithmetization-oriented primitives is new, a rigorous cryptanalysis of such primitives is yet to be done. In this paper, we...

Zero-knowledge proofs are widely used in real-world applications for authentication, access control, blockchains, and cryptocurren- cies, to name a few. A core element in zero-knowledge proof systems is the underlying hash function, which plays a vital role in the effi- ciency of the proof system. While the traditional hash functions, such as SHA3 or BLAKE3 are efficient on CPU architectures, they perform poorly within zero-knowledge proof systems. This is pri- marily due to the...

Consider the problem of efficiently evaluating isogenies $\phi: \mathcal{E} \to \mathcal{E}/H$ of elliptic curves over a finite field $\mathbb{F}_q$, where the kernel \(H = \langle{G}\rangle\) is a cyclic group of odd (prime) order: given \(\mathcal{E}\), \(G\), and a point (or several points) $P$ on $\mathcal{E}$, we want to compute $\phi(P)$. This problem is at the heart of efficient implementations of group-action- and isogeny-based post-quantum cryptosystems such as CSIDH. Algorithms...

Pairings are useful tools in isogeny-based cryptography and have been used in SIDH/SIKE and other protocols. As a general technique, pairings can be used to move problems about points on curves to elements in finite fields. However, until now, their applicability was limited to curves over fields with primes of a specific shape and pairings seemed too costly for the type of primes that are nowadays often used in isogeny-based cryptography. We remove this roadblock by optimizing pairings for...

The Number Field Sieve and its variants are the best algorithms to solve the discrete logarithm problem in finite fields (except for the weak small characteristic case). The Factory variant accelerates the computation when several prime fields are targeted. This article adapts the Factory variant to non-prime finite fields of medium and large characteristic. A precomputation, solely dependent on an approximate finite field size and an extension degree, allows to efficiently compute...

The enumeration of all outputs of a given multivariate polynomial is a fundamental mathematical problem and is incorporated in some algebraic attacks on multivariate public key cryptosystems. For a degree-$d$ polynomial in $n$ variables over the finite field with $q$ elements, solving the enumeration problem classically requires $O\left(\tbinom{n+d}{d} \cdot q^n\right)$ operations. At CHES 2010, Bouillaguet et al. proposed a fast enumeration algorithm over the binary field $\mathbb{F}_2$....

In this paper we study non-trivial self-pairings with cyclic domains that are compatible with isogenies between elliptic curves oriented by an imaginary quadratic order $\mathcal{O}$. We prove that the order $m$ of such a self-pairing necessarily satisfies $m \mid \Delta_\mathcal{O}$ (and even $2m \mid \Delta_\mathcal{O} $ if $4 \mid \Delta_\mathcal{O}$ and $4m \mid \Delta_\mathcal{O}$ if $8 \mid \Delta_\mathcal{O}$) and is not a multiple of the field characteristic. Conversely, for each $m$...

We study satisfiability modulo the theory of finite fields and give a decision procedure for this theory. We implement our procedure for prime fields inside the cvc5 SMT solver. Using this theory, we con- struct SMT queries that encode translation validation for various zero knowledge proof compilers applied to Boolean computations. We evalu- ate our procedure on these benchmarks. Our experiments show that our implementation is superior to previous approaches (which encode...

We present several new heuristic algorithms to compute class polynomials and modular polynomials modulo a prime $p$ by revisiting the idea of working with supersingular elliptic curves. The best known algorithms to this date are based on ordinary curves, due to the supposed inefficiency of the supersingular case. While this was true a decade ago, the recent advances in the study of supersingular curves through the Deuring correspondence motivated by isogeny-based cryptography has...

This paper presents the first functional encryption (FE) scheme for the attribute-weighted sum (AWS) functionality that supports the uniform model of computation. In such an FE scheme, encryption takes as input a pair of attributes (x,z) where the attribute x is public while the attribute z is private. A secret key corresponds to some weight function f, and decryption recovers the weighted sum f(x)z. This is an important functionality with a wide range of potential real life applications,...

We define the membership function of a set as the function that determines whether an input is an element of the set. Canetti, Rothblum, and Varia showed how to obfuscate evasive membership functions of hyperplanes over a finite field of order an exponentially large prime, assuming the hardness of a modified decisional Diffie-Hellman problem. Barak, Bitansky, Canetti, Kalai, Paneth, and Sahai extended their work from hyperplanes to hypersurfaces of bounded degree, assuming multilinear maps....

Binary elliptic curves are elliptic curves defined over finite fields of characteristic 2. On software platforms that offer carryless multiplication opcodes (e.g. pclmul on x86), they have very good performance. However, they suffer from some drawbacks, in particular that non-supersingular binary curves have an even order, and that most known formulas for point operations have exceptional cases that are detrimental to safe implementation. In this paper, we show how to make a prime order...

We address three main open problems concerning the use of radical isogenies, as presented by Castryck, Decru and Vercauteren at Asiacrypt 2020, in the computation of long chains of isogenies of fixed, small degree between elliptic curves over finite fields. Firstly, we present an interpolation method for finding radical isogeny formulae in a given degree $N$, which by-passes the need for factoring division polynomials over large function fields. Using this method, we are able to push the...

There are many recent results on reverse-engineering (potentially hidden) structure in cryptographic S-boxes. The problem of recovering structure in the other main building block of symmetric cryptographic primitives, namely, the linear layer, has not been paid that much attention so far. To fill this gap, in this work, we develop a systematic approach to decomposing structure in the linear layer of a substitution-permutation network (SPN), covering the case in which the specification of the...

We describe how any function over a finite field $\mathbb{F}_{p^n}$ can be represented in terms of the values of its derivatives. In particular, we observe that a function of algebraic degree $d$ can be represented uniquely through the values of its derivatives of order $(d-1)$ up to the addition of terms of algebraic degree strictly less than $d$. We identify a set of elements of the finite field, which we call the degree $d$ extension of the basis, which has the property that for any...

Zero-knowledge proof systems are usually designed to support computations for circuits over $\mathbb{F}_2$ or $\mathbb{F}_p$ for large $p$, but not for computations over $\mathbb{Z}_{2^k}$, which all modern CPUs operate on. Although $\mathbb{Z}_{2^k}$-arithmetic can be emulated using prime moduli, this comes with an unavoidable overhead. Recently, Baum et al. (CCS 2021) suggested a candidate construction for a designated-verifier zero-knowledge proof system that natively runs over...

Fully homomorphic encryption~(FHE) has flourished since it was first constructed by Gentry~(STOC 2009). Single instruction multiple data~(SIMD) gave rise to efficient homomorphic operations on vectors in \((\mathbb{F}_{t^d})^\ell\), for prime \(t\). RLWE instantiated with cyclotomic polynomials of the form \(X^{2^N}+1\) dominate implementations of FHE due to highly efficient fast Fourier transformations. However, this choice yields very short SIMD plaintext vectors and high degree extension...

In this paper, we investigate the cardinality, denoted by $(j_1,j_2,j_3,j_4)_2$, of the intersection of $(\mathcal{C}^{(2)}_{j_1}-1)\cap(\mathcal{C}^{(2)}_{j_2}-2)\cap(\mathcal{C}^{(2)}_{j_3}-3) \cap(\mathcal{C}^{(2)}_{j_4}-4)$ for $j_1,j_2,j_3,j_4\in\{0,1\}$, where $\mathcal{C}^{(2)}_0, \mathcal{C}^{(2)}_1$ are the cyclotomic classes of order two over the finite field $\mathbb{F}_{p^n}$, $p$ is an odd prime and $n$ is a positive integer. By making most use of the results on cyclotomic...

Permutation polynomials of finite fields have many applications in Coding Theory, Cryptography and Combinatorics. In the first part of this paper we present a new family of local permutation polynomials based on a class of symmetric subgroups without fixed points, the so called e-Klenian groups. In the second part we use the fact that bivariate local permutation polynomials define Latin Squares, to discuss several constructions of Mutually Orthogonal Latin Squares (MOLS) and, in...

Let $q$ be an odd prime power and ${\mathbb F}_{q^3}$ be the finite field with $q^3$ elements. In this paper, we propose two classes of permutation trinomials of ${\mathbb F}_{q^3}$ for an arbitrary odd characteristic based on the multivariate method and some subtle manipulation of solving equations with low degrees over finite fields. Moreover, we demonstrate that these two classes of permutation trinomials are QM inequivalent to all known permutation polynomials over ${\mathbb F}_{q^3}$....

An important cryptographic operation on elliptic curves is hashing to a point on the curve. When the curve is not of prime order, the point is multiplied by the cofactor so that the result has a prime order. This is important to avoid small subgroup attacks for example. A second important operation, in the composite-order case, is testing whether a point belongs to the subgroup of prime order. A pairing is a bilinear map e : G1 × G2 → GT where G1 and G2 are distinct subgroups of prime order...

Compressed $\Sigma$-Protocol Theory (CRYPTO 2020) presents an ``alternative'' to Bulletproofs that achieves the same communication complexity while adhering more elegantly to existing $\Sigma$-protocol theory, which enables their techniques to be directly applicable to other widely used settings in the context of ``plug \& play'' algorithmics. Unfortunately, their techniques are restricted to arithmetic circuits over \emph{prime} fields, which rules out the possibility of using more...

This note explains how to guarantee the membership of a point in the prime-order subgroup of an elliptic curve (over a finite field) satisfying some moderate conditions. For this purpose, we apply the Tate pairing on the curve, however it is not required to be pairing-friendly. Whenever the cofactor is small, the new subgroup test is much more efficient than other known ones, because it needs to compute at most two $n$-th power residue symbols (with small $n$) in the basic field. More...

Fully Homomorphic Encryption (FHE) gives the ability to evaluate any function over encrypted data. However, despite numerous improvements during the last decade, the computational overhead caused by homomorphic computations is still very important. As a consequence, optimizing the way of performing the computations homomorphically remains fundamental. Several popular FHE schemes such as BGV and BFV encode their data, and thus perform their computations, in finite fields. In this work, we...

Pairing computations on elliptic curves with odd prime degrees are rarely studied as low efficiency. Recently, Clarisse, Duquesne and Sanders proposed two new curves with odd prime embedding degrees: \textit{BW13-P310} and \textit{BW19-P286}, which are suitable for some special cryptographic schemes. In this paper, we propose efficient methods to compute the optimal ate pairing on this types of curves, instantiated by the \textit{BW13-P310} curve. We first extend the technique of lazy...

This article proposes four optimizations of indifferentiable hashing onto (prime-order subgroups of) ordinary elliptic curves over finite fields $\mathbb{F}_{\!q}$. One of them is dedicated to elliptic curves $E$ without non-trivial automorphisms provided that $q \equiv 2 \ (\mathrm{mod} \ 3)$. The second deals with $q \equiv 2, 4 \ (\mathrm{mod} \ 7)$ and an elliptic curve $E_7$ of $j$-invariant $-3^3 5^3$. The corresponding section plays a rather theoretical role, because (the quadratic...

We propose a new hash function Reinforced Concrete, which is the first generic purpose hash that is fast both for a zero-knowledge prover and in native x86 computations. It is suitable for a various range of zero-knowledge proofs and protocols, from set membership to generic purpose verifiable computation. Being up to 15x faster than its predecessor Poseidon hash, Reinforced Concrete inherits security from traditional time-tested schemes such as AES, whereas taking the zero-knowledge...

Zero-knowledge proofs are highly flexible cryptographic protocols that are an important building block for many secure systems. Typically, these are defined with respect to statements that are formulated as arithmetic operations over a fixed finite field. This inflexibility is a disadvantage when it comes to complex programs, as some fields are more amenable to express certain operations than others. At the same time, there do not seem to be many proofs with a programming model similar to...

Let \(q~=~2^n\), and let \(\mathcal{E} / \mathbb{F}_{q^{\ell}}\) be a generalized Galbraith--Lin--Scott (GLS) binary curve, with $\ell \ge 2$ and \((\ell, n) = 1\). We show that the GLS endomorphism on \(\mathcal{E} / \mathbb{F}_{q^{\ell}}\) induces an efficient endomorphism on the Jacobian \(\mathrm{Jac}_\mathcal{H}(\mathbb{F}_q)\) of the genus-\(g\) hyperelliptic curve \(\mathcal{H}\) corresponding to the image of the GHS Weil-descent attack applied to \(\mathcal{E} /...

With the rise of lattice cryptography, (negacyclic) convolution has received increased attention. E.g., the NTRU scheme internally employs cyclic polynomial multiplication, which is equivalent to the standard convolution, on the other hand, many Ring-LWE-based cryptosystems perform negacyclic polynomial multiplication. A method by Crandall implements an efficient negacyclic convolution over a finite field of prime order using an extended Discrete Galois Transform (DGT) – a finite field...

Succinct non-interactive arguments of knowledge (SNARKs) enable non-interactive efficient verification of NP computations and admit short proofs. However, all current SNARK constructions assume that the statements to be proven can be efficiently represented as either Boolean or arithmetic circuits over finite fields. For most constructions, the choice of the prime field $\mathbb{F}_p$ is limited by the existence of groups of matching order for which secure bilinear maps exist. In this work...

Motivated by new applications such as secure Multi-Party Computation (MPC), Fully Homomorphic Encryption (FHE), and Zero-Knowledge proofs (ZK), the need for symmetric encryption schemes that minimize the number of field multiplications in their natural algorithmic description is apparent. This development has brought forward many dedicated symmetric encryption schemes that minimize the number of multiplications in GF(2^n) or GF(p), with p being prime. These novel schemes have lead to new...

At PQCrypto-2020, Castryck and Decru proposed CSURF (CSIDH on the surface) as an improvement to the CSIDH protocol. Soon after that, at Asiacrypt-2020, together with Vercauteren they introduced radical isogenies as a further improvement. The main improvement in these works is that both CSURF and radical isogenies require only one torsion point to initiate a chain of isogenies, in comparison to Vélu isogenies which require a torsion point per isogeny. Both works were implemented using...

Efficient Reed-Solomon code reconstruction algorithms, for example, by Guruswami and Wootters (STOC--2016), translate into local leakage attacks on Shamir secret-sharing schemes over characteristic-2 fields. However, Benhamouda, Degwekar, Ishai, and Rabin (CRYPTO--2018) showed that the Shamir secret sharing scheme over prime-fields is leakage resilient to one-bit local leakage if the reconstruction threshold is roughly 0.87 times the total number of parties. In several application scenarios,...

Sequences of consecutive Legendre and Jacobi symbols as pseudorandom bit generators were proposed for cryptographic use in 1988. Major interest has been shown towards pseudorandom functions (PRF) recently, based on the Legendre and power residue symbols, due to their efficiency in the multi-party setting. The security of these PRFs is not known to be reducible to standard cryptographic assumptions. In this work, we show that key-recovery attacks against the Legendre PRF are equivalent to...

The problem of solving explicitly the equation $P_a(X):=X^{q+1}+X+a=0$ over the finite field $\GF{Q}$, where $Q=p^n$, $q=p^k$ and $p$ is a prime, arises in many different contexts including finite geometry, the inverse Galois problem \cite{ACZ2000}, the construction of difference sets with Singer parameters \cite{DD2004}, determining cross-correlation between $m$-sequences \cite{DOBBERTIN2006} and to construct error correcting codes \cite{Bracken2009}, cryptographic APN...

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 ...

Prime integers form the basis for finite field and elliptic curve cryptography, as well as many other applications. Provable prime generation guarantees primality and is more efficient than probabilistic generation, and provides components for an efficient primality proof. This paper details a protocol which takes in the proof components from the generation process, proves primality, and as an added benefit, supplies the user with a subgroup generator.

Prime integers are the backbone of most public key cryptosystems. Attacks often go after the primes themselves, as in the case of all factoring and index calculus algorithms. Primes are time sensitive cryptographic material and should be periodically changed. Unfortunately many systems use fixed primes for a variety of reasons, including the difficulty of generating trusted, random, cryptographically secure primes. This is particularly concerning in the case of discrete log based...

Optimization of finite field arithmetic is important for the deployment of public key cryptography, particularly in the context of elliptic curve cryptography. Until now the primary concern has been operations over the prime field $\F_p$, where $p$ is a prime. With the advent of pairing-based cryptography there arises a need to also look at optimal arithmetic over extension fields $\F_{p^n}$ for small values of $n$. Here we focus on the determination of quadratic residuosity and the...

Substitution boxes (S-boxes) based on the inversion mapping in even-characteristic finite fields are widely used components in the design of cryptographic primitives such as block ciphers (notably the AES cipher). This report focuses on the inversion mapping in finite fields GF(p^n) where p is a (small) odd prime and n is a (small) integer. We compare the differential and linear profiles of S-boxes over odd- and even-characteristic fields, which also motivates the design and analysis of AES...

Often in cryptography one needs to make a consistent choice of square root in a finite field. We show that such a choice is equivalent to providing a reasonable sign function. Then we show that for $\mathbb{F}_{p^k}$ (with odd prime $p \neq 1$ and $k\neq 0$) such a sign function exists if and only if $k$ is odd.

This paper reports on the computation of a discrete logarithm in the finite field $\mathbb{F}_{2^{30750}}$, breaking by a large margin the previous record, which was set in January 2014 by a computation in $\mathbb{F}_{2^{9234}}$. The present computation made essential use of the elimination step of the quasi-polynomial algorithm due to Granger, Kleinjung and Zumbrägel, and is the first large-scale experiment to truly test and successfully demonstrate its potential when applied recursively,...

This paper deals with Montgomery-friendly primes designed for the modular reduction algorithm of Montgomery. These numbers are scattered in the literature and their properties are partially exploited. We exhibit a large family of Montgomery-friendly primes which give rise to efficient modular reduction algorithms. We develop two main uses. The first one is dedicated directly to cryptography, in particular for isogeny based approaches and more generally to Elliptic Curves Cryptography. We...

Thanks to relatively small public and secret keys, the Supersingular Isogeny Key Encapsulation (SIKE) protocol made it into the third evaluation round of the post-quantum standardization project of the National Institute of Standards and Technology (NIST). Even though a large body of research has been devoted to the efficient implementation of SIKE, its latency is still undesirably long for many real-world applications. Most existing implementations of the SIKE protocol use the Montgomery...

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 present the first practical software implementation of Supersingular Isogeny Key Encapsulation (SIKE) round 2, targeting NIST's 1, 2, 3, and 5 security levels on 32-bit ARM Cortex-M4 microcontrollers. The proposed library introduces a new speed record of all SIKE Round 2 protocols with reasonable memory consumption on the low-end target platforms. We achieved this record by adopting several state-of-the-art engineering techniques as well as highly-optimized hand-crafted assembly...

In modern era of computer science there are many applications of the polynomials over finite fields especially of the polynomials over extended Galois fields GF(p^q) where p is the prime modulus and q is the extension of the said Galois field, in the generation of the modern algorithms in the computer science, the soft computation, the cryptology and the cryptanalysis and especially in generation of the S-boxes of the cryptographic block and stream ciphers. The procedure and the algorithms...

The ElGamal cryptosystem is one of the most widely used public-key cryptosystems that depends on the difficulty of computing the discrete logarithms over finite fields. Over the years, the original system has been modified and altered in order to achieve a higher security and efficiency. In this paper, a generalization for the original ElGamal system is proposed which also relies on the discrete logarithm problem. The encryption process of the scheme is improved such that it depends on the...

We study the isogenies of certain abelian varieties over finite fields with non-commutative endomorphism algebras with a view to potential use in isogeny-based cryptography. In particular, we show that any two such abelian varieties with endomorphism rings maximal orders in the endomorphism algebra are linked by a cyclic isogeny of prime degree.

Let $l$ and $k$ be two integers such that $l|k$. Define $T_l^k(X):=X+X^{p^l}+\cdots+X^{p^{l(k/l-2)}}+X^{p^{l(k/l-1)}}$ and $S_l^k(X):=X-X^{p^l}+\cdots+(-1)^{(k/l-1)}X^{p^{l(k/l-1)}}$, where $p$ is any prime. This paper gives explicit representations of all solutions in $\GF{p^n}$ to the affine equations $T_l^{k}(X)=a$ and $S_l^{k}(X)=a$, $a\in \GF{p^n}$. For the case $p=2$ that was solved very recently in \cite{MKCL2019}, the result of this paper reveals another solution.

The present work reports progress in discrete logarithm computation for the general medium prime case using the function field sieve algorithm. A new record discrete logarithm computation over a 1051-bit field having a 22-bit characteristic was performed. This computation builds on and implements previously known techniques. Analysis indicates that the relation collection and descent steps are within reach for fields with 32-bit characteristic and moderate extension degrees. It is the linear...

The survey deals with elliptic curves which are implemented in the OpenSSL 1.1.1d software library. The objective of this work is to highlight the elliptic curves which comply with the Russian national digital signature standard, namely the GOST R 34.10--2012. For this reason the paper focuses on the OpenSSL elliptic curves over a finite field of a prime order and provides the results of testing those curves for compliance with the GOST R 34.10--2012 requirements. Two cases are observed. The...

In order to obtain an efficient elliptic curve with 128-bit security and a prime order, we explore the use of finite fields $GF(p^n)$, with $p$ a small modulus (less than $2^{16}$) and $n$ a prime. Such finite fields allow for an efficient inversion algorithm due to Itoh and Tsujii, which we can leverage to make computations on an ordinary curve (short Weierstraß equation) in affine coordinates. We describe a very efficient variant of Montgomery reduction for computations modulo $p$, and...

Solving the equation $P_a(X):=X^{q+1}+X+a=0$ over finite field $\GF{Q}$, where $Q=p^n, q=p^k$ and $p$ is a prime, arises in many different contexts including finite geometry, the inverse Galois problem \cite{ACZ2000}, the construction of difference sets with Singer parameters \cite{DD2004}, determining cross-correlation between $m$-sequences \cite{DOBBERTIN2006,HELLESETH2008} and to construct error-correcting codes \cite{Bracken2009}, as well as to speed up the index calculus method for...

Isogenies on elliptic curves are of great interest in post-quantum cryptography and appeal to more and more researchers. Many protocols have been proposed such as OIDH, SIDH and CSIDH with their own advantages. We now focus on the CSIDH which based on the Montgomery curves in finite fields Fp with p=3 mod 8 whose endomorphism ring is O. We try to change the form of elliptic curves into y^2=x^3+Ax^2-x and the characteristic of the prime field into p=7 mod 8 , which induce the endomorphism...

There have been notable improvements in discrete logarithm computations in finite fields since 2015 and the introduction of the Tower Number Field Sieve algorithm (TNFS) for extension fields. The Special TNFS is very efficient in finite fields that are target groups of pairings on elliptic curves, where the characteristic is special (e.g.~sparse). The key sizes for pairings should be increased, and alternative pairing-friendly curves can be considered. We revisit the Special variant of TNFS...

Gaudry and Lubicz introduced the idea of Kummer line in 2009, and Karati and Sarkar proposed three Kummer lines over prime fields in 2017. In this work, we explore the problem of secure and efficient scalar multiplications on binary field using Kummer line and investigate the possibilities of speedups using Kummer line compared to Koblitz curves, binary Edwards curve and Weierstrass curves. We propose a binary Kummer line $\mathsf{BKL}251$ over binary field $\mathbb{F}_{2^{251}}$ where the...

An elliptic curve known as Curve448 defined over the finite field $\mathbb{F}_p$, where $p=2^{448}-2^{224}-1$, has been proposed as part of the Transport Layer Security (TLS) protocol, version 1.3. Elements of $\mathbb{F}_p$ can be represented using 7 limbs where each limb is a 64-bit quantity. This paper describes efficient algorithms for reduction modulo $p$ that are required for performing field arithmetic in $\mathbb{F}_p$ using 7-limb representation. A key feature of our work is that we...

This article generalizes the simplified Shallue--van de Woestijne--Ulas (SWU) method of a deterministic finite field mapping $h\!: \mathbb{F}_{\!q} \to E_a(\mathbb{F}_{\!q})$ to the case of any elliptic $\mathbb{F}_{\!q}$-curve $E_a\!: y^2 = x^3 - ax$ of $j$-invariant $1728$. In comparison with the (classical) SWU method the simplified SWU method allows to avoid one quadratic residuosity test in the field $\mathbb{F}_{\!q}$, which is a quite painful operation in cryptography with regard to...

In this paper we present a practical protocol for secure ridge regression. We develop the necessary secure linear algebra tools, using only basic arithmetic over prime fields. In particular, we will show how to solve linear systems of equations and compute matrix inverses efficiently, using appropriate secure random self-reductions of these problems. The distinguishing feature of our approach is that the use of secure fixed-point arithmetic is avoided entirely, while circumventing the need...

In this paper, we compute an algebraic decomposition of blackbox rings in the generic ring model. More precisely, we explicitly decompose a black-box ring as a direct product of a nilpotent black-box ring and local Artinian black-box rings, by computing all its primitive idempotents. The algorithm presented in this paper uses quantum subroutines for the computation of the p-power parts of a black-box ring and then classical algorithms for the computation of the corresponding primitive...

Though it is well known that the roots of any affine polynomial over finite field can be computed by a system of linear equations by using a normal base of the field, such solving approach appears to be difficult to apply when the field is fairly large. Thus, it may be of great interest to find explicit representation of the solutions independently of the field base. This was previously done only for quadratic equations over binary finite field. This paper gives explicit representation of...

We present the first practical software implementation of Supersingular Isogeny Key Encapsulation (SIKE) round 2, targeting NIST’s 1, 2, and 5 security levels on 32-bit ARM Cortex-M4 microcontrollers. The proposed library introduces a new speed record of SIKE protocol on the target platform. We achieved this record by adopting several state-of-the-art engineering techniques as well as highly-optimized hand-crafted assembly implementation of finite field arithmetic. In particular,...

For $1\leq m \leq n$, we consider a natural $m$-out-of-$n$ multi-instance scenario for a public-key encryption (PKE) scheme. An adversary, given $n$ independent instances of PKE, wins if he breaks at least $m$ out of the $n$ instances. In this work, we are interested in the scaling factor of PKE schemes, $\mathrm{SF}$, which measures how well the difficulty of breaking $m$ out of the $n$ instances scales in $m$. That is, a scaling factor $\mathrm{SF}=\ell$ indicates that breaking $m$ out of...

At ASIACRYPT 2018, Castryck, Lange, Martindale, Panny and Renes proposed CSIDH, which is a key-exchange protocol based on isogenies between elliptic curves, and a candidate for post-quantum cryptography. However, the implementation by Castryck et al. is not constant-time. Specifically, a part of the secret key could be recovered by the side-channel Attacks. Recently, Meyer, Campos and Reith proposed a constant-time implementation of CSIDH by introducing dummy isogenies and taking secret...

Let $p$ be a small prime and $n=n_1n_2>1$ be a composite integer. For the function field sieve algorithm applied to $\mathbb{F}_{p^n}$, Guillevic (2019) had proposed an algorithm for initial splitting of the target in the individual logarithm phase. This algorithm generates polynomials and tests them for $B$-smoothness for some appropriate value of $B$. The amortised cost of generating each polynomial is $O(n_2^2)$ multiplications over $\mathbb{F}_{p^{n_1}}$. In this work, we propose a new...

We consider the problem of constructing Diffie-Hellman (DH) parameters which pass standard approaches to parameter validation but for which the Discrete Logarithm Problem (DLP) is relatively easy to solve. We consider both the finite field setting and the elliptic curve setting. For finite fields, we show how to construct DH parameters $(p,q,g)$ for the safe prime setting in which $p=2q+1$ is prime, $q$ is relatively smooth but fools random-base Miller-Rabin primality testing with some...

Recently there has been a significant progress on the tower number field sieve (TNFS) method, reducing the complexity of the discrete logarithm problem (DLP) in finite field extensions of composite degree. These new variants of the TNFS attacks have a major impact on pairing-based cryptography and particularly on the selection of the underlying elliptic curve groups and extension fields. In this paper we revise the criteria for selecting pairing-friendly elliptic curves considering these new...

Elliptic curve cryptography requires efficient arithmetic over the underlying field. In particular, fast implementation of multiplication and squaring over the finite field is required for efficient projective coordinate based scalar multiplication as well as for inversion using Fermat’s little theorem. In the present work we consider the problem of obtaining efficient algorithms for field multiplication and squaring. From a theoretical point of view, we present a number of algorithms for...

Since 2016 and the introduction of the exTNFS (extended Tower Number Field Sieve) algorithm, the security of cryptosystems based on non- prime finite fields, mainly the paring and torus-based one, is being reassessed. The feasibility of the relation collection, a crucial step of the NFS variants, is especially investigated. It usually involves polynomials of degree one, i.e., a search space of dimension two. However, exTNFS uses bivariate polynomials of at least four coefficients. If sieving...

This paper addresses fast scalar multiplication for elliptic curves over finite fields. In the first part of the paper, we obtain several efficiently computable formulas for basic elliptic curves arithmetic in the family of twisted Edwards curves over prime fields. Our $2Q+P$ formula saves about $2.8$ field multiplications, and our $5P$ formula saves about $4.2$ field multiplications in standard projective coordinate systems, compared to the latest existing results. In the second part of the...

Most multi-party computation protocols allow secure computation of arithmetic circuits over a finite field, such as the integers modulo a prime. In the more natural setting of integer computations modulo $2^{k}$, which are useful for simplifying implementations and applications, no solutions with active security are known unless the majority of the participants are honest. We present a new scheme for information-theoretic MACs that are homomorphic modulo $2^k$, and are as efficient as the...

If $q$ is a prime and $n$ is a positive integer then any two finite fields of order $q^n$ are isomorphic. Elements of these fields can be thought of as polynomials with coefficients chosen modulo $q$, and a notion of length can be associated to these polynomials. A non-trivial isomorphism between the fields, in general, does not preserve this length, and a short element in one field will usually have an image in the other field with coefficients appearing to be randomly and uniformly...

We examine how two parallel modes of operation for Authenticated Encryption (namely CTR+PMAC and OTR mode) work when evaluated in a multi-party computation engine. These two modes are selected because they suit the PRFs examined in previous works. In particular the modes are highly parallel, and do not require evaluation of the inverse of the underlying PRF. In order to use these modes one needs to convert them from their original instantiation of being defined on binary blocks of data, to...

In this paper, a new algorithm to solve the discrete logarithm problem is presented which is similar to the usual baby-step giant-step algorithm. Our algorithm exploits the order of the discrete logarithm in the multiplicative group of a finite field. Using randomization with parallelized collision search, our algorithm indicates some weakness in NIST curves over prime fields which are considered to be the most conservative and safest curves among all NIST curves.

Fix an ordinary abelian variety defined over a finite field. The ideal class group of its endomorphism ring acts freely on the set of isogenous varieties with same endomorphism ring, by complex multiplication. Any subgroup of the class group, and generating set thereof, induces an isogeny graph on the orbit of the variety for this subgroup. We compute (under the Generalized Riemann Hypothesis) some bounds on the norms of prime ideals generating it, such that the associated graph has good...

In the past two years there have been several advances in Number Field Sieve (NFS) algorithms for computing discrete logarithms in finite fields $\mathbb{F}_{p^n}$ where $p$ is prime and $n > 1$ is a small integer. This article presents a concise overview of these algorithms and discusses some of the challenges with assessing their impact on keylengths for pairing-based cryptosystems.

Isogeny based post-quantum cryptography is one of the most recent addition to the family of quantum resistant cryptosystems. In this paper, we propose an efficient modular multiplication algorithm for primes of the form $p = 2 \cdot 2^a \cdot 3^b - 1$ with b even, typically used in such cryptosystem. Our modular multiplication algorithm exploits the special structure present in such primes. We compare the efficiency of our technique with Barrett reduction and Montgomery multiplication. Our C...

Addition chain calculations play a critical role in determining the efficiency of cryptosystems based on isogenies on elliptic curves. However, finding a minimal length addition chain is not easy; a generalized version of the problem, in which one must find a chain that simultaneously forms each of a sequence of values, is NP-complete. For the special primes used in such cryptosystems, finding fast addition chains for finite field arithmetic such as inversion and square root is also not...

We give a systematic overview of techniques to compute efficient arithmetic modulo $2^xp^y\pm 1$. This is useful for computations in the supersingular isogeny Diffie-Hellman (SIDH) key-exchange protocol which is one of the more recent contenders in the post-quantum public-key arena. One of the main computational bottlenecks in this key-exchange protocol is computing modular arithmetic in a finite field defined by a prime of this special shape. Recent implementations already use this special...