Non-prime Finite Field Research Articles

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.

Matroid Lifts and Representability

A 1965 result of Crapo shows that every elementary lift of a matroid $M$ can be constructed from a linear class of circuits of $M$. In a recent paper, Walsh generalized this construction by defining a rank-$k$ lift of a matroid $M$ given a rank-$k$ matroid $N$ on the set of circuits of $M$, and conjectured that all matroid lifts can be obtained in this way. In this sequel paper we simplify Walsh's construction and show that this conjecture is true for representable matroids but is false in general. This gives a new way to certify that a particular matroid is non-representable, which we use to construct new classes of non-representable matroids.
 Walsh also applied the new matroid lift construction to gain graphs over the additive group of a non-prime finite field, generalizing a construction of Zaslavsky for these special groups. He conjectured that this construction is possible on three or more vertices only for the additive group of a non-prime finite field. We show that this conjecture holds for four or more vertices, but fails for exactly three.

Constructing CM Fields for NFS to Accelerate DL Computation in Non-Prime Finite Fields

The hardness of discrete logarithm problem (DLP) over finite fields is the security foundation of many cryptographic protocols. When the characteristic is not small, the state-of-the-art algorithms for solving DLP are the number field sieve (NFS) and its variants. In the relation collection step, to translate the relations between prime ideals to those of elements one needs to use the Schirokauer map. Besides, if the number field has non-trivial automorphisms, one can use them to accelerate the factor-base logarithms computation. However, the Schirokauer map is not compatible with automorphisms. To exploit automorphism efficiently, we focus on the method to construct fields in NFS such that the fields on both sides have non-trivial automorphisms with the logarithms of units being zero. 1. We construct two families of CM polynomials of arbitrary even degree with small coefficients, corresponding to the automorphisms being <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> ↦ – <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> or <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> ↦ 1/ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> . 2. We show how to combine these polynomials with the JLSV <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and Conjugation polynomial selection methods on both sides. 3. We also generalize our method to the multiple number field sieve and the extended tower number field sieve.

Bijective Cremona transformations of the plane

We study the birational self-maps of the projective plane over finite fields that induce permutations on the set of rational points. As a main result, we prove that no odd permutation arises over a non-prime finite field of characteristic two, which completes the investigation initiated by Cantat about which permutations can be realized this way. Main ingredients in our proof include the invariance of parity under groupoid conjugations by birational maps, and a list of generators for the group of such maps.

A New Matroid Lift Construction and an Application to Group-Labeled Graphs

A well-known result of Brylawski constructs an elementary lift of a matroid $M$ from a linear class of circuits of $M$. We generalize this result by constructing a rank-$k$ lift of $M$ from a rank-$k$ matroid on the set of circuits of $M$. We conjecture that every lift of $M$ arises via this construction.&#x0D; We then apply this result to group-labeled graphs, generalizing a construction of Zaslavsky. Given a graph $G$ with edges labeled by a group, Zaslavsky's lift matroid $K$ is an elementary lift of the graphic matroid $M(G)$ that respects the group-labeling; specifically, the cycles of $G$ that are circuits of $K$ coincide with the cycles that are balanced with respect to the group-labeling. For $k \geqslant 2$, when does there exist a rank-$k$ lift of $M(G)$ that respects the group-labeling in this same sense? For abelian groups, we show that such a matroid exists if and only if the group is isomorphic to the additive group of a non-prime finite field.

Constructive Asymptotic Bounds of Locally Repairable Codes via Function Fields

Locally repairable codes have been investigated extensively in recent years due to practical applications in distributed and cloud storage systems. However, there are few asymptotic constructions of locally repairable codes in the literature. In this paper, we provide a new explicit asymptotic construction of locally repairable codes over arbitrary finite fields from local expansions of functions at a rational place. This construction gives a Tsfasman-Vladut-Zink type bound for locally repairable codes. Its main advantage is that there are no constraints on both locality and alphabet size. Furthermore, we show that the Gilbert-Varshamov type bound of locally repairable codes over non-prime finite fields can be improved for sufficiently large alphabet sizes.

Faster individual discrete logarithms in finite fields of composite extension degree

Computing discrete logarithms in finite fields is a main concern in cryptography. The best algorithms in large and medium characteristic fields (e.g., $\rm {GF}(p^2)$, $\rm {GF}(p^{12})$) are the Number Field Sieve and its variants (special, high-degree, tower). The best algorithms in small characteristic finite fields (e.g., $\rm {GF}(3^{6 \cdot 509})$) are the Function Field Sieve, Joux’s algorithm, and the quasipolynomial-time algorithm. The last step of this family of algorithms is the individual logarithm computation. It computes a smooth decomposition of a given target in two phases: an initial splitting, then a descent tree. While new improvements have been made to reduce the complexity of the dominating relation collection and linear algebra steps, resulting in a smaller factor basis (database of known logarithms of small elements), the last step remains at the same level of difficulty. Indeed, we have to find a smooth decomposition of a typically large element in the finite field. This work improves the initial splitting phase and applies to any nonprime finite field. It is very efficient when the extension degree is composite. It exploits the proper subfields, resulting in a much more smooth decomposition of the target. This leads to a new trade-off between the initial splitting step and the descent step in small characteristic. Moreover it reduces the width and the height of the subsequent descent tree.

Slim NoC

Emerging chips with hundreds and thousands of cores require networks with unprecedented energy/area efficiency and scalability. To address this, we propose Slim NoC (SN): a new on-chip network design that delivers significant improvements in efficiency and scalability compared to the state-of-the-art. The key idea is to use two concepts from graph and number theory, degree-diameter graphs combined with non-prime finite fields, to enable the smallest number of ports for a given core count. SN is inspired by state-of-the-art off-chip topologies; it identifies and distills their advantages for NoC settings while solving several key issues that lead to significant overheads on-chip. SN provides NoC-specific layouts, which further enhance area/energy efficiency. We show how to augment SN with state-of-the-art router microarchitecture schemes such as Elastic Links, to make the network even more scalable and efficient. Our extensive experimental evaluations show that SN outperforms both traditional low-radix topologies (e.g., meshes and tori) and modern high-radix networks (e.g., various Flattened Butterflies) in area, latency, throughput, and static/dynamic power consumption for both synthetic and real workloads. SN provides a promising direction in scalable and energy-efficient NoC topologies.

Collecting relations for the number field sieve in

In order to assess the security of cryptosystems based on the discrete logarithm problem in non-prime finite fields, as are the torus-based or pairing-based ones, we investigate thoroughly the case in$\mathbb{F}_{p^{6}}$with the number field sieve. We provide new insights, improvements, and comparisons between different methods to select polynomials intended for a sieve in dimension 3 using a special-$\mathfrak{q}$strategy. We also take into account the Galois action to increase the relation productivity of the sieving phase. To validate our results, we ran several experiments and real computations for various polynomial selection methods and field sizes with our publicly available implementation of the sieve in dimension 3, with special-$\mathfrak{q}$and various enumeration strategies.

Towers of Function Fields over Non-prime Finite Fields

Over all non-prime finite fields, we construct some recursive towers of function fields with many rational places. Thus we obtain a substantial improvemient on all known lower bounds for Thara's quantity A(l), for l - p(n) with p prime and n > 3 odd. We relate the explicit equations to Drinfeld modular varieties.

Codes and the Cartier operator

In this article, we present a new construction of codes from algebraic curves. Given a curve over a non-prime finite field, the obtained codes are defined over a subfield. We call themCartier codessince their construction involves the Cartier operator. This new class of codes can be regarded as a natural geometric generalisation of classical Goppa codes. In particular, we prove that a well-known property satisfied by classical Goppa codes extends naturally to Cartier codes. We prove general lower bounds for the dimension and the minimum distance of these codes and compare our construction with a classical one: the subfield subcodes of Algebraic Geometry codes. We prove that every Cartier code is contained in a subfield subcode of an Algebraic Geometry code and that the two constructions have similar asymptotic performances.We also show that some known results on subfield subcodes of Algebraic Geometry codes can be proved nicely by using properties of the Cartier operator and that some known bounds on the dimension of subfield subcodes of Algebraic Geometry codes can be improved thanks to Cartier codes and the Cartier operator.

Galois towers over non-prime finite fields

We construct Galois towers with good asymptotic properties over any non-prime finite field $\mathbb F_{\ell }$; that is, we construct sequences of function fields $\mathcal {N}=(N_1 \subset N_2 \subset \cdots )$ over $\mathbb F_{\ell }$ of increasing genu

Using Symmetries in the Index Calculus for Elliptic Curves Discrete Logarithm

In 2004, an algorithm is introduced to solve the DLP for elliptic curves defined over a non-prime finite field \(\mathbb{F}_{q^{n}}\). One of the main steps of this algorithm requires decomposing points of the curve \(E(\mathbb{F}_{q^{n}})\) with respect to a factor base, this problem is denoted PDP. In this paper, we will apply this algorithm to the case of Edwards curves, the well-known family of elliptic curves that allow faster arithmetic as shown by Bernstein and Lange. More precisely, we show how to take advantage of some symmetries of twisted Edwards and twisted Jacobi intersections curves to gain an exponential factor 2ω(n−1) to solve the corresponding PDP where ω is the exponent in the complexity of multiplying two dense matrices. Practical experiments supporting the theoretical result are also given. For instance, the complexity of solving the ECDLP for twisted Edwards curves defined over \(\mathbb{F}_{q^{5}}\), with q≈264, is supposed to be ∼ 2160 operations in \(E(\mathbb{F}_{q^{5}})\) using generic algorithms compared to 2130 operations (multiplications of two 32-bits words) with our method. For these parameters the PDP is intractable with the original algorithm.The main tool to achieve these results relies on the use of the symmetries and the quasi-homogeneous structure induced by these symmetries during the polynomial system solving step. Also, we use a recent work on a new algorithm for the change of ordering of a Gröbner basis which provides a better heuristic complexity of the total solving process.

Polynomial and Normal Bases for Finite Fields

We discuss two different ways to speed up exponentiation in nonprime finite fields: on the one hand, reduction of the total number of operations, and on the other hand, fast computation of a single operation. Two data structures are particularly useful: sparse irreducible polynomials and normal bases. We report on implementation results for our methods.

Computing special powers in finite fields

We study exponentiation in nonprime finite fields with very special exponents such as they occur, for example, in inversion, primitivity tests, and polynomial factorization. Our algorithmic approach improves the corresponding exponentiation problem from about quadratic to about linear time.

Explicitly constructed extensions of the rational function field with prescribed splitting

Given a finite set of places S, we provide techniques to explicitly construct extensions of the rational function field over a non-prime finite field of constants in which all places in S split completely. Our techniques do not rely on class field theory. We then explore connections to the theory of function fields with many rational places for their genus.

Holey Segre Varieties

Holey Segre varieties are introduced, which generalize classical Segre varieties and whose existence is suggested by the fact that any non-prime finite field contains proper subfields. More precisely, a holey Segre variety is the tensor product PG(n, q) ⊗PG(m,qk ). An algebraic description of such an object is given and its structure is investigated. Furthermore, special choices of the parameters are considered, in which cases a holey Segre variety admits a partition into classical Segre varieties. Also, some holey Segre varieties are isomorphic to non-canonical subgeometries embedded in suitable Segre varieties.

On the Minimum Linear Complexity of de Bruijn Sequences over Non-prime Finite Fields

It has been conjectured that over any non-prime finite field Fpmand for any positive integer n, there exists a spannde Bruijn sequence over Fpmwhich has the minimum possible linear complexitypnm−1+n. We give a proof by construction that this conjecture is true.

Characterising the Linear Complexity of Span 1 de Bruijn Sequences over Finite Fields

We give a complete resolution to a conjecture regarding the characterisation of linear complexities of span 1 de Bruijn sequences over nonprime finite fields. This contrasts with results for prime fields, where the characterisation is equivalent to an open question concerning permutation polynomials.

An algorithm for evaluation of discrete logarithms in some nonprime finite fields

In this paper we propose an algorithm for evaluation of logarithms in the finite fields F p n F_{p^n} , where the number p n − 1 p^n-1 has a small primitive factor r r . The heuristic estimate of the complexity of the algorithm is equal to exp ⁡ ( ( c + o ( 1 ) ) ( log ⁡ p r log 2 ⁡ r ) 1 / 3 ) \exp ((c+o(1))(\log p\,r\log ^2r)^{1/3}) , where n n grows to ∞ \infty , and p p is limited by a polynomial in n n . The evaluation of logarithms is founded on a new congruence of the kind of D. Coppersmith, C ( x ) k ≡ D ( x ) C(x)^k\equiv D(x) , which has a great deal of solutions—pairs of polynomials C ( x ) , D ( x ) C(x),D(x) of small degrees.