Field Of Prime Order Research Articles

Threshold secret sharing with geometric algebras

In this work, we propose a geometric algebra‐based variation of a well‐known threshold secret‐sharing scheme introduced by Adi Shamir in 1979. Secret sharing is a cryptographic primitive which allows a secret input to be divided into multiple shares which are then sent to a collection of parties. The shares are generated so that only “authorized” sets of shares can reconstruct the secret. In Shamir's scheme, any sufficiently large set of shares can reconstruct the secret. The minimum number of shares which can obtain the secret is called the threshold, and any number of shares smaller than the threshold reveals nothing about the secret. The shares are generated such that each party can perform computations, generating a new set of shares that, when reconstructed, are equivalent to performing those exact computations directly on the secret input data. Our variant changes the domain from which secrets are taken: A finite field with prime order is replaced by a geometric algebra over a finite field of prime order. This change preserves the important security properties of Shamir's scheme, namely, idealness (secrets and shares are chosen from the same space) and perfectness (“unauthorized” sets of shares learn nothing about the secret). Our scheme allows secret sharing to be seamlessly added to the arsenal of GA‐based applications. Our extension of Shamir's secret scheme was first worked out for geometric algebras. It appears, however, that in fact it works for other algebras, a situation worthy to be explored in future work. For definiteness, in this paper, we restrict the analysis to the case of geometric algebras.

Zigzag polynomials, Artin's conjecture and trinomials

Let Fp denote the finite field of prime order p. We introduce a new natural family of polynomials in Fp[X]. These polynomials first arose from considering faithful representations of the elementary abelian group of order p2 in characteristic p. The polynomials exhibit a number of interesting special properties. For example they satisfy a three term recursion, are closely related to zigzag zero-one sequences and form strong divisibility sequences. These polynomials are shown to be closely connected to the order of appearance of prime numbers in the Fibonacci sequence, Artin's Primitive Root Conjecture, and the factorization of trinomials over finite fields.

On the generation of some Lie-type geometries

Let Xn(K) be a building of Coxeter type Xn=An or Xn=Dn defined over a given division ring K (a field when Xn=Dn). For a non-connected set J of nodes of the diagram Xn, let Γ(K)=GrJ(Xn(K)) be the J-grassmannian of Xn(K). We prove that Γ(K) cannot be generated over any proper sub-division ring K0 of K. As a consequence, the generating rank of Γ(K) is infinite when K is not finitely generated. In particular, if K is the algebraic closure of a finite field of prime order then the generating rank of Gr1,n(An(K)) is infinite, although its embedding rank is either (n+1)2−1 or (n+1)2.

Character sums estimates and an application to a problem of Balog

We prove new bounds for sums of multiplicative characters over sums of set with small doubling and applying this result we break the square--root barrier in a problem of Balog concerning products of differences in a field of prime order.

2-элементы в группе автотопизмов полуполевой проективной плоскости

We investigate the well-known hypothesis of D.R. Hughes that the full collineation group of non-Desarguesian semifield projective plane of a finite order is solvable (the question 11.76 in Kourovka notebook was written down by N.D. Podufalov). The spread set method is used to construct the semifield projective planes with cyclic 2- subgroup of autotopisms in the case of linear space of any dimension over the field of prime order. This study completes the analogous considerations of elementary abelian 2-subgroups. We obtain the natural restriction to the order of 2-element for the semifield planes for odd and even order. It is proved that some projective linear groups can not be the autotopism subgroups for the infinite series of semifield planes. The matrix representation of Baer involution allows us to define the geometric property of autotopism of order 4. We can choose the base of a linear space such that the matrix representation of these autotopisms is convenient and uniform, it does not depend on the space dimension. The minimal counter-example is constructed to explain the restriction to the plane order. As a corollary, we proved the solvability of the full collineation group when the nonDesarguesian semifield plane has a certain even order and all its Baer subplanes are also non-Desarguesain. The main results can be used as technical for the further studies of the subgroups of even order in an autotopism group for a finite non-Desarguesian semifield plane. The results obtained are useful to investigate the semifield planes with the autotopism subgroups from J.G. Thompson’s list of minimal simple groups.

Quantitative structure of stable sets in finite abelian groups

We prove an arithmetic regularity lemma for stable subsets of finite abelian groups, generalising our previous result for high-dimensional vector spaces over finite fields of prime order. A qualitative version of this generalisation was recently obtained by the first author in joint work with Conant and Pillay, using model-theoretic techniques. In contrast, the approach in the present paper is highly quantitative and relies on several key ingredients from arithmetic combinatorics.

Size of free groups in varieties generated by finite groups

The number of distinct [Formula: see text]-variable word maps on a finite group [Formula: see text] is the order of the rank [Formula: see text] free group in the variety generated by [Formula: see text]. For a group [Formula: see text], the number of word maps on just two variables can be quite large. We improve upon previous bounds for the number of word maps over a finite group [Formula: see text]. Moreover, we show that our bound is sharp for the number of 2-variable word maps over the affine group over fields of prime order and over the alternating group on five symbols.

A finite analogue of the ring of algebraic numbers

We construct an analogue of the ring of algebraic numbers, living in a quotient of the product of all finite fields of prime order. We use this ring to deduce some results about linear recurrent sequences.

Kummer for Genus One Over Prime-Order Fields

This work considers the problem of fast and secure scalar multiplication using curves of genus one defined over a field of prime order. Previous work by Gaudry and Lubicz (Finite Fields Appl 15(2):246–260, 2009) had suggested the use of the associated Kummer line to speed up scalar multiplication. In the present work, we explore this idea in detail. The first task is to obtain an elliptic curve in Legendre form which satisfies necessary security conditions such that the associated Kummer line has small parameters and a base point with small coordinates. It turns out that the ladder step on the Kummer line supports parallelism and can be implemented very efficiently in constant time using the single-instruction multiple-data (SIMD) operations available in modern processors. For the 128-bit security level, this work presents three Kummer lines denoted as \(K_1:=\mathsf{KL2519(81,20)}\), \(K_2:=\mathsf{KL25519(82,77)}\) and \(K_3:=\mathsf{KL2663(260,139)}\) over the three primes \(2^{251}-9\), \(2^{255}-19\) and \(2^{266}-3\), respectively. Implementations of scalar multiplications for all three Kummer lines using Intel intrinsics have been done, and the code is publicly available. Timing results on the Skylake and the Haswell processors of Intel indicate that both fixed base and variable base scalar multiplications for \(K_1\) and \(K_2\) are faster than those achieved by Sandy2x, which is a highly optimised SIMD implementation in assembly of the well-known Curve25519. On Skylake, both fixed base and variable base scalar multiplications for \(K_3\) are faster than Sandy2x, whereas on Haswell, fixed base scalar multiplication for \(K_3\) is faster than Sandy2x while variable base scalar multiplication for both \(K_3\) and Sandy2x takes roughly the same time. In practical terms, the particular Kummer lines that are introduced in this work are serious candidates for deployment and standardisation. We further illustrate the usefulness of the proposed Kummer lines by instantiating the quotient Digital Signature Algorithm on all the three Kummer lines.

On Isometric Error-Correcting Codes over Finite Fields of Prime Order

Error-correcting codes obtained from each other via a Hamming distance preserving transformation, called isometry, are equivalent. In this paper, we consider three different isometries which yield permutationally equivalent linear codes, monomially equivalent linear codes, and equivalent nonlinear codes, respectively. For each case, we derive some new necessary and sufficient conditions for equivalence using the concept of weight vector of a code or distance matrix of a code. Results hold for error-correcting codes over a finite field of prime order.

On consecutive primitive nth roots of unity modulo q

Given n∈N, we study the conditions under which a finite field of prime order q will have adjacent elements of multiplicative order n. In particular, we analyze the resultant of the cyclotomic polynomial Φn(x) with Φn(x+1), and exhibit Lucas and Mersenne divisors of this quantity. For each n≠1,2,3,6, we prove the existence of a prime qn for which there is an element α∈Zqn where α and α+1 both have multiplicative order n. Additionally, we use algebraic norms to set analytic upper bounds on the size and quantity of these primes.

On Additive Bases of Sets with Small Product Set

We prove that finite sets of real numbers satisfying vertical bar AA vertical bar 0 cannot have small additive bases nor can they be written as a set of sums B + C with vertical bar B vertical bar, vertical bar C vertical bar >= 2. The result can be seen as a real analog of the conjecture of Sarkozy that multiplicative subgroups of finite fields of prime order are additively irreducible.

Tiling sets and spectral sets over finite fields

We study tiling and spectral sets in vector spaces over prime fields. The classical Fuglede conjecture in locally compact abelian groups says that a set is spectral if and only if it tiles by translation. This conjecture was disproved by T. Tao in Euclidean spaces of dimensions 5 and higher, using constructions over prime fields (in vector spaces over finite fields of prime order) and lifting them to the Euclidean setting. Over prime fields, when the dimension of the vector space is less than or equal to 2 it has recently been proven that the Fuglede conjecture holds (see [6]). In this paper we study this question in higher dimensions over prime fields and provide some results and counterexamples. In particular we prove the existence of spectral sets which do not tile in Zp5 for all odd primes p and Zp4 for all odd primes p such that p≡3 mod 4. Although counterexamples in low dimensional groups over cyclic rings Zn were previously known they were usually for non-prime n or a small, sporadic set of primes p rather than general constructions. This paper is a result of a Research Experience for Undergraduates program ran at the University of Rochester during the summer of 2015 by A. Iosevich, J. Pakianathan and G. Petridis.

GENERALIZED GELFAND–GRAEV REPRESENTATIONS IN SMALL CHARACTERISTICS

Let$\mathbf{G}$be a connected reductive algebraic group over an algebraic closure$\overline{\mathbb{F}_{p}}$of the finite field of prime order$p$and let$F:\mathbf{G}\rightarrow \mathbf{G}$be a Frobenius endomorphism with$G=\mathbf{G}^{F}$the corresponding$\mathbb{F}_{q}$-rational structure. One of the strongest links we have between the representation theory of$G$and the geometry of the unipotent conjugacy classes of$\mathbf{G}$is a formula, due to Lusztig (Adv. Math.94(2) (1992), 139–179), which decomposes Kawanaka’s Generalized Gelfand–Graev Representations (GGGRs) in terms of characteristic functions of intersection cohomology complexes defined on the closure of a unipotent class. Unfortunately, the formula given in Lusztig (Adv. Math.94(2) (1992), 139–179) is only valid under the assumption that$p$is large enough. In this article, we show that Lusztig’s formula for GGGRs holds under the much milder assumption that$p$is an acceptable prime for$\mathbf{G}$($p$very good is sufficient but not necessary). As an application we show that every irreducible character of$G$, respectively, character sheaf of$\mathbf{G}$, has a unique wave front set, respectively, unipotent support, whenever$p$is good for $\mathbf{G}$.

Computational Results for Regular Difference Systems of Sets Attaining or Being Close to the Levenshtein Bound

AbstractDifference systems of sets (DSSs) are combinatorial structures arising in connection with code synchronization that were introduced by Levenshtein in 1971, and are a generalization of cyclic difference sets. In this paper, we consider a collection of m‐subsets in a finite field of prime order to be a regular DSS for an integer m, and give a lower bound on the parameter ρ of the DSS using cyclotomic numbers. We show that when we choose ‐subsets from the multiplicative group of order e, the lower bound on ρ is independent of the choice of subsets. In addition, we present some computational results for DSSs with block sizes and , whose parameter ρ attains or comes close to the Levenshtein bound for .

Tight lower bounds for linear 2-query LCCs over finite fields

A Locally Correctable Code (LCC) is an error correcting code that has a probabilistic self-correcting algorithm that, with high probability, can correct any coordinate of the codeword by looking at only a few other coordinates, even if a ? fraction of the coordinates is corrupted. LCCs are a stronger form of LDCs (Locally Decodable Codes) which have received a lot of attention recently due to their many applications and surprising constructions. In this work, we show a separation between linear 2-query LDCs and LCCs over finite fields of prime order. Specifically, we prove a lower bound of the form pΩ(?d) on the length of linear 2-query LCCs over Fp, that encode messages of length d. Our bound improves over the known bound of 2Ω(?d) [8,10,6] which is tight for LDCs. Our proof makes use of tools from additive combinatorics which have played an important role in several recent results in theoretical computer science. We also obtain, as corollaries of our main theorem, new results in incidence geometry over finite fields. The first is an improvement to the Sylvester-Gallai theorem over finite fields [14] and the second is a new analog of Beck's theorem over finite fields. The paper also contains an appendix, written by Sergey Yekhanin, showing that there do exist nonlinear LCCs of size 2O(d) over Fp, thus highlighting the importance of the linearity assumption for our result.

An improved incidence bound for fields of prime order

Let P be a set of points and L a set of lines in Fp2, with |P|,|L|≤N and N<p. We show that P and L generate no more than CN32−1806+o(1) incidences for some absolute constant C. This improves on the previously best-known bound of CN32−110,678.

List-Decoding Multiplicity Codes

We study the list-decodability of multiplicity codes. These codes, which are based on evaluations of high-degree polynomials and their derivatives, have rate approaching 1 while simultaneously allowing for sublinear-time error correction. In this paper, we show that multiplicity codes also admit powerful and local algorithms that work even in the presence of a large error fraction. In other words, we give algorithms for recovering a polynomial given several evaluations of it and its derivatives, where possibly many of the given evaluations are incorrect. Our first main result shows that univariate multiplicity codes over fields of prime order can be list-decoded up to the so-called list-decoding capacity. Specifically, we show that univariate multiplicity codes of rate R over fields of prime order can be list-decoded from a (1 R e) fraction of errors in polynomial time (for constant R;e). This resembles the behavior of the Folded Reed-Solomon Codes of Guruswami and Rudra (Trans. Info. Theory 2008). The algorithm is based on constructing a differential equation of which the desired codeword is a solution; this differential equation is then solved using a power-series approach (a variation of Hensel lifting) along with other algebraic ideas. Our second main result is a algorithm for decoding multivariate multiplicity codes up to their Johnson radius. The key ingredient of this algorithm is the construction of a special family of algebraically-repelling curves passing through the points of F m ; no moderate-degree multivariate polynomial over F m can simultaneously vanish on all these A version of this paper was posted online as an Electronic Colloq. on Computational Complexity Tech. Report (20). Supported in part by a Sloan Fellowship and NSF grant CCF-1253886.

On a certain sum-product estimate in fields of prime order

Let \(\mathbb{F}_p \) be the finite field of p elements with p prime. If A is a subset of \(\mathbb{F}_p \) and g is an element of \(\mathbb{F}_p^* \) with order ν, then $$\max \{ |A + g \cdot A|,|A \cdot A|\} \gg \left( {\frac{\nu } {{\nu + |A|^2 }}} \right)^{1/12} |{\rm A}|^{13/12} $$ .

Generalised array low‐density parity‐check codes

In this study, using Group Permutation Low-Density Parity-Check (GP-LDPC) codes, the authors generalise the concept of array Low-Density Parity-Check (LDPC) codes from fields of prime order to those of prime power order. In fact, they consider the additive group of the finite field GF( q ), q a prime power, as the underlying group for the GP-LDPC code construction and since when q is a prime, the author's code construction method coincides with that of quasi-cyclic array LDPC codes, they call their codes, generalised array LDPC (GA-LDPC) codes. First, they prove that, like array LDPC codes, GA-LDPC codes are quasi-cyclic codes. Then, they analyse the girth of GA-LDPC codes in a way similar to that for array LDPC codes and introduce some shortened GA-LDPC codes with girths 8, 10 and 12. For many values of g , J and L , the lengths of ( J , L )-regular shortened GA-LDPC codes of girth g and rate at least 1 - J / L , constructed in this study, are smaller than the lengths of ( J , L )-regular LDPC codes of girth g and rate at least 1 - J / L , constructed in the literature. Also, simulation results show that GA-LDPC codes perform well with the iterative message-passing decoding.