Large Finite Fields Research Articles

Shifted convolution and the Titchmarsh divisor problem over 𝔽q[t

In this paper, we solve a function field analogue of classical problems in analytic number theory, concerning the autocorrelations of divisor functions, in the limit of a large finite field.

The autocorrelation of the Möbius function and Chowla's conjecture for the rational function field in characteristic 2.

We prove a function field version of Chowla's conjecture on the autocorrelation of the Möbius function in the limit of a large finite field of characteristic 2, extending previous work in odd characteristic.

Prime polynomial values of linear functions in short intervals

In this paper we establish a function field analogue of a conjecture in number theory which is a combination of several famous conjectures, including the Hardy–Littlewood prime tuple conjecture, conjectures on the number of primes in arithmetic progressions and in short intervals, and the Goldbach conjecture. We prove an asymptotic formula for the number of simultaneous prime polynomial values of n linear functions, in the limit of a large finite field.

Performance analysis of network coding‐based content distribution in vehicularad‐hocnetworks

The authors investigate the content distribution among the vehicles of a cluster in a vehicular ad-hoc network, exploiting network coding. The vehicles collaborate to disseminate the coded data packets, received from a roadside info-station based on IEEE 802.11 medium access control (MAC) protocol. Two types of network coding are considered: random linear network coding (RLNC) over a large finite field and random XORed network coding (RXNC). An analytical model is proposed to address the effect of random access MAC as well as the correlation among received coded packets on the performance of content distribution. First, a p-persistent carrier sense multiple access approximation for IEEE 802.11 MAC is adopted, to derive the expected amount of time necessary to deliver the whole information (a data file) to the vehicles, that is, content distribution delay, based on RLNC. Second, the content distribution delay for RXNC is investigated. In this respect, the authors determine the probability of a newly received packet to be innovative for RXNC and characterise the probability that all nodes succeed to retrieve the data file, that is, content delivery ratio. Finally, the success of content distribution process for erasure channels is assessed.

Natural Generalizations of Threshold Secret Sharing

We present new families of access structures that, similarly to the multilevel and compartmented access structures introduced in previous works, are natural generalizations of threshold secret sharing. Namely, they admit ideal linear secret sharing schemes over every large enough finite field, they can be described by a small number of parameters, and they have useful properties for the applications of secret sharing. The use of integer polymatroids makes it possible to find many new such families and it simplifies in great measure the proofs for the existence of ideal secret sharing schemes for them.

On the Chow groups of certain geometrically rational 5-folds

We give an explicit regular model for the quadric fibration studied in [8]. As an application we show that this construction furnishes a counterexample for the integral Tate conjecture in any odd characteristic for some sufficiently large finite field. We study the étale cohomology of this regular model, and as a consequence we derive that these counterexamples are not torsion.

D sets and a Sárközy theorem for countable fields

We establish that if ℱ is a countable field of characteristic p, U is a unitary action of ℱl on a Hilbert space ℋ, and P is an essential idempotent ultrafilter on ℱ, then for every polynomial r: ℱ → ℱl with r(0) = 0 one has $$P - {\lim _g}{U_{r(g)}}f = {P_r}f{\rm{ weakly,}}$$ , where Pr is the projection onto the closed subspace $${H_r} = \{ f \in H:{\{ {U_{r(g)}}f\} _{g \in F}}{\text{ is precompact in the strong topology\} }}{\text{.}}$$ . We then derive combinatorial consequences of this result, including results for sufficiently large finite fields.

A new class of superregular matrices and MDP convolutional codes

This paper deals with the problem of constructing superregular matrices that lead to MDP convolutional codes. These matrices are a type of lower block triangular Toeplitz matrices with the property that all the square submatrices that can possibly be nonsingular due to the lower block triangular structure are nonsingular. We present a new class of matrices that are superregular over a sufficiently large finite field F. Such construction works for any given choice of characteristic of the field F and code parameters (n,k,δ) such that (n−k)|δ. We also discuss the size of F needed so that the proposed matrices are superregular.

Low-lying Zeros of Quadratic Dirichlet L-Functions, Hyper-elliptic Curves and Random Matrix Theory

The statistics of low-lying zeros of quadratic Dirichlet L-functions were conjectured by Katz and Sarnak to be given by the scaling limit of eigenvalues from the unitary symplectic ensemble. The n-level densities were found to be in agreement with this in a certain neighborhood of the origin in the Fourier domain by Rubinstein in his Ph.D. thesis in 1998. An attempt to extend the neighborhood was made in the Ph.D. thesis of Peng Gao (n-level density of the low-lying zeros of quadratic Dirichlet L-functions, 2005), who under GRH gave the density as a complicated combinatorial factor, but it remained open whether it coincides with the Random Matrix Theory factor. For n ≤ 7 this was recently confirmed by Levinson and Miller. We resolve this problem for all n, not by directly doing the combinatorics, but by passing to a function field analogue, of L-functions associated to hyper-elliptic curves of given genus g over a field of q elements. We show that the answer in this case coincides with Gao’s combinatorial factor up to a controlled error. We then take the limit of large finite field size q → ∞ and use the Katz–Sarnak equidistribution theorem, which identifies the monodromy of the Frobenius conjugacy classes for the hyperelliptic ensemble with the group USp(2g). Further taking the limit of large genus g→ ∞ allows us to identify Gao’s combinatorial factor with the RMT answer.

THE AUTOCORRELATION OF THE MOBIUS FUNCTION AND CHOWLA'S CONJECTURE FOR THE RATIONAL FUNCTION FIELD

We prove a function field version of Chowla's conjecture on the autocorrelation of the Mobius function in the limit of a large finite field.

Hardy–Littlewood Tuple Conjecture Over Large Finite Fields

We prove the following function field analog of the Hardy–Littlewood conjecture (which generalizes the twin prime conjecture) over large finite fields. Let n and r be positive integers and q an odd prime power. For a tuple of distinct polynomials of degree <n let π(q,n;a) be the number of monic polynomials of degree n such that f+a 1 ,…,f+a r are simultaneously irreducible. We prove that as and n,r fixed.

Electric field dependence of charge carrier hopping transport within the random energy landscape in an organic field effect transistor

We extended our analytical effective medium theory [Phys. Rev. B 81, 045202 (2010)] to describe the temperature-dependent hopping charge carrier mobility at arbitrary electric fields in the large carrier density regime. Special emphasis was made to analyze the influence of the lateral electric field on the Meyer--Neldel (MN) phenomenon observed when studying the charge mobilities in thin-film organic field-effect transistors (OFET). Our calculations are based on the average hopping transition time approach, generalized for large carrier concentration limit finite fields, and taking into account also spatial energy correlations. The calculated electric field dependences of the hopping mobility at large carrier concentrations are in good agreement with previous computer simulations data. The shift of the MN temperature in an OFET upon applied electric field is shown to be a consequence of the spatial energy correlation in the organic semiconductor film. Our calculations show that the phenomenological Gill equation is clearly inappropriate for describing conventional charge carrier transport at low carrier concentrations. On the other hand a Gill-type behavior has been observed in a temperature range relevant for measurements of the charge carrier mobility in OFET structures. Since the present model is not limited to zero-field mobility, it allows a more accurate evaluation of important material parameters from experimental data measured at a given electric field. In particular, we showed that both the MN and Gill temperature can be used for estimating the width of the density of states distribution.

Fully homomorphic SIMD operations

At PKC 2010 Smart and Vercauteren presented a variant of Gentry's fully homomorphic public key encryption scheme and mentioned that the scheme could support SIMD style operations. The slow key generation process of the Smart---Vercauteren system was then addressed in a paper by Gentry and Halevi, but their key generation method appears to exclude the SIMD style operation alluded to by Smart and Vercauteren. In this paper, we show how to select parameters to enable such SIMD operations. As such, we obtain a somewhat homomorphic scheme supporting both SIMD operations and operations on large finite fields of characteristic two. This somewhat homomorphic scheme can be made fully homomorphic in a naive way by recrypting all data elements separately. However, we show that the SIMD operations can be used to perform the recrypt procedure in parallel, resulting in a substantial speed-up. Finally, we demonstrate how such SIMD operations can be used to perform various tasks by studying two use cases: implementing AES homomorphically and encrypted database lookup.

Ideal Hierarchical Secret Sharing Schemes

Hierarchical secret sharing is among the most natural generalizations of threshold secret sharing, and it has attracted a lot of attention since the invention of secret sharing until nowadays. Several constructions of ideal hierarchical secret sharing schemes have been proposed, but it was not known what access structures admit such a scheme. We solve this problem by providing a natural definition for the family of the hierarchical access structures and, more importantly, by presenting a complete characterization of the ideal hierarchical access structures, that is, the ones admitting an ideal secret sharing scheme. Our characterization is based on the well-known connection between ideal secret sharing schemes and matroids and, more specifically, on the connection between ideal multipartite secret sharing schemes and integer polymatroids. In particular, we prove that every hierarchical matroid port admits an ideal linear secret sharing scheme over every large enough finite field. Finally, we use our results to present a new proof for the existing characterization of the ideal weighted threshold access structures.

Efficient software implementations of large finite fields GF (2 n ) for secure storage applications

Finite fields are widely used in constructing error-correcting codes and cryptographic algorithms. In practice, error-correcting codes use small finite fields to achieve high-throughput encoding and decoding. Conversely, cryptographic systems employ considerably larger finite fields to achieve high levels of security. We focus on developing efficient software implementations of arithmetic operations in reasonably large finite fields as needed by secure storage applications. In this article, we study several arithmetic operation implementations for finite fields ranging from GF (2 32 ) to GF (2 128 ). We implement multiplication and division in these finite fields by making use of precomputed tables in smaller fields, and several techniques of extending smaller field arithmetic into larger field operations. We show that by exploiting known techniques, as well as new optimizations, we are able to efficiently support operations over finite fields of interest. We perform a detailed evaluation of several techniques, and show that we achieve very practical performance for both multiplication and division. Finally, we show how these techniques find applications in the implementation of HAIL, a highly available distributed cloud storage layer. Using the newly implemented arithmetic operations in GF (2 64 ), HAIL improves its performance by a factor of two, while simultaneously providing a higher level of security.

Multiuser Cooperative Diversity Through Network Coding Based on Classical Coding Theory

In this paper, we propose and analyze a generalized construction of distributed network codes for a network consisting of M users sending independent information to a common base station through independent block fading channels. The aim is to increase the diversity order of the system without reducing its throughput. The proposed scheme, called generalized dynamic-network codes (GDNC), is a generalization of the dynamic-network codes (DNC) recently proposed by Xiao and Skoglund. The design of the network codes that maximize the diversity order is recognized as equivalent to the design of linear block codes over a nonbinary finite field under the Hamming metric. We prove that adopting a systematic generator matrix of a maximum distance separable block code over a sufficiently large finite field as the network transfer matrix is a sufficient condition for full diversity order under link failure model. The proposed generalization offers a much better tradeoff between rate and diversity order compared to the DNC. An outage probability analysis showing the improved performance is carried out, and computer simulations results are shown to agree with the analytical results.

Irreducible values of polynomials

Schinzel's Hypothesis H is a general conjecture in number theory on prime values of polynomials that generalizes, e.g., the twin prime conjecture and Dirichlet's theorem on primes in arithmetic progression. We prove a quantitative arithmetic analog of this conjecture for polynomial rings over pseudo algebraically closed fields. This implies results over large finite fields via model theory. A main tool in the proof is an irreducibility theorem à la Hilbert.

Reduced-Complexity Decoders of Long Reed-Solomon Codes Based on Composite Cyclotomic Fourier Transforms

Long Reed-Solomon (RS) codes are desirable for digital communication and storage systems due to their improved error performance, but the high computational complexity of their decoders is a key obstacle to their adoption in practice. As discrete Fourier transforms (DFTs) can evaluate a polynomial at multiple points, efficient DFT algorithms are promising in reducing the computational complexities of syndrome based decoders for long RS codes. In this paper, we first propose partial composite cyclotomic Fourier transforms (CCFTs) and then devise syndrome based decoders for long RS codes over large finite fields based on partial CCFTs. The new decoders based on partial CCFTs achieve a significant saving of computational complexities for long RS codes. Since partial CCFTs have modular and regular structures, the new decoders are suitable for hardware implementations. To further verify and demonstrate the advantages of partial CCFTs, we implement in hardware the syndrome computation block for a $(2720, 2550)$ shortened RS code over GF$(2^{12})$. In comparison to previous results based on Horner's rule, our hardware implementation not only has a smaller gate count, but also achieves much higher throughputs.

Performance of algebraic graphs based stream-ciphers using large finite fields

Algebraic graphs D(n, q) and their analog graphs D(n, K), where K is a finite commutative ring were used successfully in Coding Theory (as Tanner graphs for the construction of LDPC codes and turbo-codes) and in Cryptography (stream-ciphers, public-keys and tools for the key-exchange protocols. Many properties of cryptography algorithms largely depend on the choice of finite field Fq or commutative ring K. For practical implementations the most convenient fields are F and rings modulo Z modulo 2m. In this paper the reader can find the first results about the comparison of D(n, 2m) based stream-ciphers for m = 8, 16, 32 implemented in C++. They show that performance (speed) of algorithms gets better when m is increased.

On a family of preimage-resistant functions

AbstractIn the present paper we define a new hash function, based on inhomogeneous polynomials. First we define a large family of polynomials over finite fields and we prove that the members of this family are nearly permutational polynomials. Then we define a subfamily of the above family, such that the elements in the subfamily are easy to evaluate. We prove that (working in a large enough finite field) finding a preimage by chance of such a function is computationally infeasible, and we mention that methods for solving the equation corresponding to the preimage problem for such polynomials are also out of reach.