Prime Field Research Articles

The slice spectral sequence for a motivic analogue of the connective K(1)$K(1)$‐local sphere

AbstractWe compute the 2‐adic effective slice spectral sequence (ESSS) for the motivic stable homotopy groups of , a motivic analogue of the connective ‐local sphere over prime fields of characteristic not two. Together with the analogous computation over algebraically closed fields, this yields information about the motivic ‐local sphere over arbitrary base fields of characteristic not two. To compute the spectral sequence, we prove several results that may be of independent interest. We describe the ‐differentials in the slice spectral sequence in terms of the motivic Steenrod operations over general base fields, building on analogous results of Ananyevskiy, Röndigs, and Østvær for the very effective cover of Hermitian K‐theory. We also explicitly describe the coefficients of certain motivic Eilenberg–MacLane spectra and compute the ESSS for the very effective cover of Hermitian K‐theory over prime fields.

Solving Elliptic Curve Discrete Logarithm Problem on Twisted Edwards Curves Using Quantum Annealing and Index Calculus Method

Abstract This paper presents an approach to solving the elliptic curve discrete logarithm problem on alternative curve models over prime fields using a quantum annealing and index calculus method. Part of the algorithm, relation searching, is transformed into the Quadratic Unconstrained Boolean Optimization (QUBO) problem and then is efficiently solved using the D-Wave computer by quantum annealing. As Faugère et al. showed, twisted Edwards curves, because of their symmetric shape, allow us to obtain solutions of relations searching step using Groebner basis faster than in the case of Weierstrass curves. Because of symmetries, a system of equations of relations searching step for twisted Edwards curves has many symmetric solutions. Using the Groebner basis and having many system solutions makes it easier to find any of them. The same is true using quantum annealing - it is easier to find any solution to the QUBO problem if many are correct. In this paper, we used this observation to find out that a properly constructed QUBO problem for the relations searching step for twisted Edwards curves allows us to find a solution faster for the same size of the base field than in the case of Weierstrass curves. Using the presented approach, we solved the discrete logarithm problem using quantum annealing and index calculus method for elliptic curve discrete logarithm problem defined on twisted Edwards curve over a field 𝔽1021 with order equal to 4 · 241. It is now the biggest field and size of the group, where the elliptic curve discrete logarithm problem was solved using quantum methods.

On a polynomial bound for the orbital diameter of primitive affine groups

Let VG be a finite primitive affine permutation group, where V is a vector space of dimension d over the prime field F p and G is an irreducible linear group on V. We prove that if p divides |G|, then the diameters of all nondiagonal orbital graphs of VG are at most 9 d 3 . This improves an earlier exponential bound by A. Maróti and the author.

Discrete Logarithm Factory

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 discrete logarithms in a constant proportion of the finite fields of the given approximate size and extension degree. We combine this idea with two other variants of NFS, namely the tower and special variant. This combination improves the asymptotic complexity. We also notice that combining our approach with the MNFS variant would be an unnecessary complication as all the potential gain of MNFS is subsumed by our Factory variant anyway. Furthermore, we demonstrate how Chebotarev's density theorem allows to compute the density of finite fields that can be solved with a given precomputation. Finally, we provide experimental data in order to assess the practical reach of our approach.

Area-Time-Efficient Secure Comb Scalar Multiplication Architecture Based on Recoding

With the development of mobile communication, digital signatures with low latency, low area, and high security are in increasing demand. Elliptic curve cryptography (ECC) is widely used because of its security and lightweight. Elliptic curve scalar multiplication (ECSM) is the basic arithmetic in ECC. Based on this background information, we propose our own research objectives. In this paper, a low-latency and low-area ECSM architecture based on the comb algorithm is proposed. The detailed methodology is as follows. The recoding-k algorithm and randomization-Z algorithm are used to improve security, which can resist sample power analysis (SPA) and differential power analysis (DPA). A low-area multi-functional architecture for comb is proposed, which takes into account different stages of the comb algorithm. Based on this, the data dependency is considered and the comb architecture is optimized to achieve a uniform and efficient execution pattern. The interleaved modular multiplication algorithm and modified binary inverse algorithm are used to achieve short clock cycle delay and high frequency while taking into account the need for a low area. The proposed architecture has been implemented on Xilinx Virtex-7 series FPGA to perform ECSM on 256-bits prime field GF(p). In the hardware architecture with only 7351 slices of resource usage, a single ECSM only takes 0.74 ms, resulting in an area-time product (ATP) of 5.41. The implementation results show that our design can compete with the existing state-of-the-art engineering in terms of performance and has higher security. Our design is suitable for computing scenarios where security and computing speed are required. The implementation of the overall architecture is of great significance and inspiration to the research community.

Arithmetization-oriented APN permutations

AbstractRecently, 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 the prime field $${\mathbb {F}}_p$$ 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 investigate arithmetization-oriented APN functions. More precisely, we investigate APN permutations in the CCZ-classes of known families of APN power functions over the prime field $${\mathbb {F}}_p$$ F p . Moreover, we present a class of binomial permutation having differential uniformity at most 5 defined via the quadratic character over finite fields of odd characteristic. Computationally it is confirmed that the latter family contains new APN permutations for some small parameters. We conjecture it to contain an infinite subfamily of APN permutations.

P -adic reconstruction of rational functions in multiloop amplitudes

Numerical reconstruction techniques are widely employed in the calculation of multiloop scattering amplitudes. In recent years, it has been observed that the rational functions in multiloop calculations greatly simplify under partial fractioning. In this article, we present a technique to reconstruct rational functions directly in partial-fractioned form, by evaluating the functions at special integer points chosen for their properties under a p-adic metric. As an application, we apply this technique to reconstruct the largest rational function in the integration-by-parts reduction of one of the rank-5 integrals appearing in two-loop five-point full-color massless amplitude calculations in quantum chromodynamics. The number of required numerical probes (per prime field) is found to be around 25 times smaller than in conventional techniques, and the obtained result is 130 times smaller. The reconstructed result displays signs of additional structure that could be used to further reduce its size and the number of required probes. Published by the American Physical Society 2024

Monolith: Circuit-Friendly Hash Functions with New Nonlinear Layers for Fast and Constant-Time Implementations

Hash functions are a crucial component in incrementally verifiable computation (IVC) protocols and applications. Among those, recursive SNARKs and folding schemes require hash functions to be both fast in native CPU computations and compact in algebraic descriptions (constraints). However, neither SHA-2/3 nor newer algebraic constructions, such as Poseidon, achieve both requirements. In this work we overcome this problem in several steps. First, for certain prime field domains we propose a new design strategy called Kintsugi, which explains how to construct nonlinear layers of high algebraic degree which allow fast native implementations and at the same time also an efficient circuit description for zeroknowledge applications. Then we suggest another layer, based on the Feistel Type-3 scheme, and prove wide trail bounds for its combination with an MDS matrix. We propose a new permutation design named Monolith to be used as a sponge or compression function. It is the first arithmetization-oriented function with a native performance comparable to SHA3-256. At the same time, it outperforms Poseidon in a circuit using the Merkle tree prover in the Plonky2 framework. Contrary to previously proposed designs, Monolith also allows for efficient constant-time native implementations which mitigates the risk of side-channel attacks.

The growth recurrence and Gelfand-Kirillov base of the ordinary cusp

We introduce the Gelfand-Kirillov base for a numerical semigroup ring over the prime field of characteristic p p , where p p is prime, and show its existence for the semigroup ring of the ordinary cusp by establishing a growth recurrence with respect to Frobenius.

Matrix Factorization and Some Fast Discrete Transforms

In this paper, three discrete transforms related to vector spaces over finite fields are studied. For our purposes, and according to the properties of the finite fields, the most suitable transforms are as follows: for binary fields, this is the Walsh–Hadamard transform; for odd prime fields, the Vilenkin–Chrestenson transform; and for composite fields, the trace transform. A factorization of the transform matrices using Kronecker power is given so that the considered discrete transforms are reduced to the fast discrete transforms. Examples and applications are also presented of the considered transforms in coding theory for calculating the weight distribution of a linear code.

Algebraic lattices coming from $${\mathbb {Z}}$$-modules generalizing ramified prime ideals in odd prime degree cyclic number fields

Algebraic lattices coming from $${\mathbb {Z}}$$-modules generalizing ramified prime ideals in odd prime degree cyclic number fields

On the 𝓐-generators of the polynomial algebra as a module over the Steenrod algebra, I

Abstract Let 𝓟 n := H *((ℝP ∞) n ) ≅ ℤ2[x 1, x 2, …, x n ] be the graded polynomial algebra over ℤ2, where ℤ2 denotes the prime field of two elements. We investigate the Peterson hit problem for the polynomial algebra 𝓟 n , viewed as a graded left module over the mod-2 Steenrod algebra, 𝓐. For n > 4, this problem is still unsolved, even in the case of n = 5 with the help of computers. In this article, we study the hit problem for the case n = 6 in the generic degree dr = 6(2 r − 1) + 4.2 r with r an arbitrary non-negative integer. By considering ℤ2 as a trivial 𝓐-module, then the hit problem is equivalent to the problem of finding a basis of ℤ2-vector space ℤ2 ⊗𝓐𝓟 n . The main goal of the current article is to explicitly determine an admissible monomial basis of the ℤ2 vector space ℤ2 ⊗𝓐𝓟6 in some degrees. As an application, the behavior of the sixth Singer algebraic transfer in the degree 6(2 r − 1) + 4.2 r is also discussed at the end of this paper.

Modular invariants of a vector and a covector for some elementary abelian p-groups

Let F p be the prime field of order p > 0 and G be an elementary abelian p-group. For some n-dimensional cohyperplane G-representations V over F p , we show that F p [ V ⊕ V * ] G , the invariant ring of a vector and a covector is a complete intersection by exhibiting an explicit generating set (in fact, a SAGBI basis) and exposing all relations among the generators.

Base of exponent representation matters -- more efficient reduction of discrete logarithm problem and elliptic curve discrete logarithm problem to the QUBO problem

This paper presents further improvements in the transformation of the Discrete Logarithm Problem (DLP) and Elliptic Curve Discrete Logarithm Problem (ECDLP) over prime fields to the Quadratic Unconstrained Binary Optimization (QUBO) problem. This is significant from a cryptanalysis standpoint, as QUBO problems may be solved using quantum annealers, and the fewer variables the resulting QUBO problem has, the less time is expected to obtain a solution.The main idea presented in the paper is allowing the representation of the exponent in different bases than the typically used base 2 (binary representation). It is shown that in such cases, the reduction of the discrete logarithm problem over the prime field \(\mathbb{F}_p\) to the QUBO problem may be obtained using approximately \(1.89n^2\) logical variables for \(n\) being the bitlength of prime \(p\), instead of the \(2n^2\) which was previously the best-known reduction method. The paper provides a practical example using the given method to solve the discrete logarithm problem over the prime field \(\mathbb{F}_{47}\). Similarly, for the elliptic curve discrete logarithm problem over the prime field \(\mathbb{F}_p\), allowing the representation of the exponent in different bases than typically used base two results in a lower number of required logical variables for \(n\) being the bitlength of prime \(p\), from \(3n^3\) to \(\frac{6n^3}{\log_{2}\left(\frac{3}{4}n\right)}\) logical variables, in the case of Edwards curves.

The centralizer of~$\frak{sl}(0,n)$ in the generalized Witt Lie superalgebra over fields of prime characteristic

This paper considers centralizers of the Lie superalgebra~$\frak{sl}(0,n)$ over prime characteristic fields. Using homological methods, the centralizers of the even and odd parts of ~$\frak{sl}(0,n)$ in the generalized Witt Lie superalgebra are calculated and a summary of their structural properties is provided.

Systematic KMTNet Planetary Anomaly Search. XI. Complete Sample of 2016 Subprime Field Planets

Following Shin et al. (2023b), which is a part of the “Systematic KMTNet Planetary Anomaly Search” series (i.e., a search for planets in the 2016 KMTNet prime fields), we conduct a systematic search of the 2016 KMTNet subprime fields using a semi-machine-based algorithm to identify hidden anomalous events missed by the conventional by-eye search. We find four new planets and seven planet candidates that were buried in the KMTNet archive. The new planets are OGLE-2016-BLG-1598Lb, OGLE-2016-BLG-1800Lb, MOA-2016-BLG-526Lb, and KMT-2016-BLG-2321Lb, which show typical properties of microlensing planets, i.e., giant planets orbit M-dwarf host stars beyond their snow lines. For the planet candidates, we find planet/binary or 2L1S/1L2S degeneracies, which are an obstacle to firmly claiming planet detections. By combining the results of Shin et al. (2023b) and this work, we find a total of nine hidden planets, which is about half the number of planets discovered by eye in 2016. With this work, we have met the goal of the systematic search series for 2016, which is to build a complete microlensing planet sample. We also show that our systematic searches significantly contribute to completing the planet sample, especially for planet/host mass ratios smaller than 10−3, which were incomplete in previous by-eye searches of the KMTNet archive.

The geometrically m$m$‐step solvable Grothendieck conjecture for affine hyperbolic curves over finitely generated fields

The geometrically m$m$‐step solvable Grothendieck conjecture for affine hyperbolic curves over finitely generated fields

Secured V2X communication using optimized prime field ECC architecture

The objective of this research is to ensure a high level of security in various forms of vehicular communication (V2V, V2I, and V2N) and to support vehicles in safely receiving all keys and messages from Roadside Unit (RSU), other vehicles, or the network with the support of simple cryptographic techniques. Elliptic Curve Cryptography (ECC) is one of the many cryptographic techniques that offer a desirable response to this requirement. The Koblitz curve secp256k1 is supported by the designed ECC processor for 256-bit point multiplication and point addition. A powerful illustration of the "Divide and Conquer" strategy's ability to accelerate multiplication asymptotically is given by the Karatsuba algorithm. The ECC processor's multiplication operation is accelerated further by the incorporation of pipelining. With a maximum clock frequency of 238.40MHz, the proposed Karatsuba-based ECC processor executes 256-bit single point multiplication in 0.937ms, providing 273.21kbps throughput, and taking up 8.42k slices in a Virtex-7 FPGA. Scalar multiplication is extended by incorporating a pipeline by increasing the highest clock frequency by up to 7.97%, which decreases time consumption by 9.90% and boosts throughput by 10.99%. In terms of area, operating frequency, area-delay product, and throughput, the suggested pipelined Karatsuba multiplier based ECC processor performs better than the existing designs.

The admissible monomial bases for the polynomial algebra of five variables in some types of generic degrees

Let Pk be the graded polynomial algebra F2[x1,x2,…,xk] over the prime field of two elements, F2, with the degree of each xi being 1. We study the hit problem, set up by Frank Peterson, of finding a minimal set of generators for Pk as a module over the mod-2 Steenrod algebra, A. In this paper, we explicitly determine a minimal set of A-generators for P5 in the case of the generic degrees n=2d+1−1 and n=2d+1−2 for all d⩾6.

Algebraic Attack on FHE-Friendly Cipher HERA Using Multiple Collisions

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, a new transciphering framework called the Real-to-Finite-Field (RtF) framework and a corresponding FHE-friendly symmetric-key primitive called HERA were proposed at ASIACRYPT 2021. Although HERA has a very similar structure to AES, it is considerably different in the following aspects: 1) the power map x → x3 is used as the S-box; 2) a randomized key schedule is used; 3) it is over a prime field Fp with p > 216. In this work, we perform the first third-party cryptanalysis of HERA, by showing how to mount new algebraic attacks with multiple collisions in the round keys. Specifically, according to the special way to randomize the round keys in HERA, we find it possible to peel off the last nonlinear layer by using collisions in the last-round key and a simple property of the power map. In this way, we could construct an overdefined system of equations of a much lower degree in the key, and efficiently solve the system via the linearization technique. As a esult, for HERA with 192 and 256 bits of security, respectively, we could break some parameters under the same assumption made by designers that the algebra constant ω for Gaussian elimination is ω = 2, i.e., Gaussian elimination on an n × n matrix takes O(nω) field operations. If using more conservative choices like ω ∈ {2.8, 3}, our attacks can also successfully reduce the security margins of some variants of HERA to only 1 round. However, the security of HERA with 80 and 128 bits of security is not affected by our attacks due to the high cost to find multiple collisions. In any case, our attacks reveal a weakness of HERA caused by the randomized key schedule and its small state size.