Class Of Finite Fields Research Articles

Digit-Level Serial-In Parallel-Out Multiplier Using Redundant Representation for a Class of Finite Fields

Two digit-level finite field multipliers in $\mathbb {F}_{2^{m}}$ using redundant representation are presented. Embedding $\mathbb {F}_{2^{m}}$ in cyclotomic field $\mathbb {F}_{2}^{(n)}$ causes a certain amount of redundancy and consequently performing field multiplication using redundant representation would require more hardware resources. Based on a specific feature of redundant representation in a class of finite fields, two new multiplication algorithms along with their pertaining architectures are proposed to alleviate this problem. Considering area-delay product as a measure of evaluation, it has been shown that both the proposed architectures considerably outperform existing digit-level multipliers using the same basis. It is also shown that for a subset of the fields, the proposed multipliers are of higher performance in terms of area-delay complexities among several recently proposed optimal normal basis multipliers. The main characteristics of the postplace&route application specific integrated circuit implementation of the proposed multipliers for three practical digit sizes are also reported.

Hexagonal metric for linear codes over a finite field

This paper consider Hexagonal-metric codes over certain class of finite fields. The Hexagonal metric as defined by Huber is a non-trivial metric over certain classes of finite fields. Hexagonal-metric codes are applied in coded modulation scheme based on hexagonal-like signal constellations. Since the development of tight bounds for error correcting codes using new distance is a research problem, the purpose of this note is to generalize the Plotkin bound for linear codes over finite fields equipped with the Hexagonal metric. By means of a two-step method, the author presents a geometric method to construct finite signal constellations from quotient lattices associated to the rings of Eisenstein-Jacobi (EJ) integers and their prime ideals and thus naturally label the constellation points by elements of a finite field. The Plotkin bound is derived from simple computing on the geometric figure of a finite field.

An alternative class of irreducible polynomials for optimal extension fields

Optimal extension fields (OEF) are a class of finite fields used to achieve efficient field arithmetic, especially required by elliptic curve cryptosystems (ECC). In software environment, OEFs are preferable to other methods in performance and memory requirement. However, the irreducible binomials required by OEFs are quite rare. Sometimes irreducible trinomials are alternative choices when irreducible binomials do not exist. Unfortunately, trinomials require more operations for field multiplication and thereby affect the efficiency of OEF. To solve this problem, we propose a new type of irreducible polynomials that are more abundant and still efficient for field multiplication. The proposed polynomial takes the advantage of polynomial residue arithmetic to achieve high performance for field multiplication which costs O(m 3/2) operations in $${\mathbb{F}_p}$$ . Extensive simulation results demonstrate that the proposed polynomials roughly outperform irreducible binomials by 20% in some finite fields of medium prime characteristic. So this work presents an interesting alternative for OEFs.

Research of fast modular multiplier for a class of finite fields

A new structure of bit-parallel Polynomial Basis (PB) multiplier is proposed, which is based on a fast modular reduction method. The method was recommended by the National Institute of Standards and Technology (NIST). It takes advantage of the characteristics of irreducible polynomial, i.e., the degree of the second item of irreducible polynomial is far less than the degree of the polynomial in the finite fields GF(2m). Deductions are made for a class of finite field in which trinomials are chosen as irreducible polynomials. Let the trinomial be xm + xh +1, where 1 ≤ k ≤ [m/1]. The proposed structure has shorter critical path than the best known one up to date, while the space requirement keeps the same. The structure is practical, especially in real time cryptographic applications.

Asymptotic classes of finite structures

In this paper we consider classes of finite structures where we have good control over the sizes of the definable sets. The motivating example is the class of finite fields: it was shown in [1] that for any formula in the language of rings, there are finitely many pairs (d, μ) ∈ ω × Q>0 so that in any finite field F and for any ā ∈ Fm the size |ø(Fn,ā)| is “approximately” μ|F|d. Essentially this is a generalisation of the classical Lang-Weil estimates from the category of varieties to that of the first-order-definable sets.

A Subexponential Algorithm for Discrete Logarithms Over all Finite Fields

There are numerous subexponential algorithms for computing discrete logarithms over certain classes of finite fields. However, there appears to be no published subexponential algorithm for computing discrete logarithms over all finite fields. We present such an algorithm and a heuristic argument that there exists a $c \in {\Re _{ > 0}}$ such that for all sufficiently large prime powers ${p^n}$, the algorithm computes discrete logarithms over ${\text {GF}}({p^n})$ within expected time: ${e^{c{{(\log ({p^n})\log \log ({p^n}))}^{1/2}}}}$.

A subexponential algorithm for discrete logarithms over all finite fields

There are numerous subexponential algorithms for computing discrete logarithms over certain classes of finite fields. However, there appears to be no published subexponential algorithm for computing discrete logarithms over all finite fields. We present such an algorithm and a heuristic argument that there exists a c ∈ ℜ > 0 c \in {\Re _{ > 0}} such that for all sufficiently large prime powers p n {p^n} , the algorithm computes discrete logarithms over GF ( p n ) {\text {GF}}({p^n}) within expected time: e c ( log ⁡ ( p n ) log ⁡ log ⁡ ( p n ) ) 1 / 2 {e^{c{{(\log ({p^n})\log \log ({p^n}))}^{1/2}}}} .

Corrigendum: “Rings which admit elimination of quantifiers”

Matatyahu Rubin pointed out that the proof of Lemma 6.1 [2] works only for rings of prime or zero characteristic. This invalidates the characterization of semiprime rings with the descending chain condition on right or left ideals which admit elimination of quantifiers given in [2] and cited in the abstract [1]. Although the correct characterization is easy to derive, it is complex to state.Let be the class of finite fields. Let be the class of 2 × 2 matrix rings over a field with a prime number of elements. Let be the class of rings of the form GF(pn) ⊕ GF(pk) such that either n = k or g.c.d.(n, k) = 1 and p is a prime. Let ′ be the class of algebraically closed fields. Let P denote the set of all prime numbers together with zero. Let be the set of all ordered pairs (f, Q) where Q is a finite subset of P and f: Q → ⋃ ⋃ ⋃ such that the characteristic of the ring f(q) is q. Finally, let be the class of rings of the form ⊕q∈Qf(q) for some (f,Q) in .A corrected version of Theorem 6.2 [2] isTheorem 1. Let R be a ring with the descending chain condition on left or right ideals and without nonzero trivial ideals. Then R admits elimination of quantifiers if only if R belong to.

Rings which admit elimination of quantifiers

AbstractWe say that a ring admits elimination of quantifiers, if in the language of rings, {0, 1, +, ·}, the complete theory of R admits elimination of quantifiers.Theorem 1. Let D be a division ring. Then D admits elimination of quantifiers if and only if D is an algebraically closed or finite field.A ring is prime if it satisfies the sentence: ∀x∀y∃z (x =0 ∨ y = 0∨ xzy ≠ 0).Theorem 2. If R is a prime ring with an infinite center and R admits elimination of quantifiers, then R is an algebraically closed field.Let be the class of finite fields. Let be the class of 2 × 2 matrix rings over a field with a prime number of elements. Let be the class of rings of the form GF(pn)⊕GF(pk) such that either n = k or g.c.d. (n, k) = 1. Let be the set of ordered pairs (f, Q) where Q is a finite set of primes and such that the characteristic of the ring f(q) is q. Finally, let be the class of rings of the form ⊕q ∈ Qf(q), for some (f, Q) in .Theorem 3. Let R be a finite ring without nonzero trivial ideals. Then R admits elimination of quantifiers if and only if R belongs to.Theorem 4. Let R be a ring with the descending chain condition of left ideals and without nonzero trivial ideals. Then R admits elimination of quantifiers if and only if R is an algebraically closed field or R belongs to.In contrast to Theorems 2 and 4, we haveTheorem 5. If R is an atomless p-ring, then R is finite, commutative, has no nonzero trivial ideals and admits elimination of quantifiers, but is not prime and does not have the descending chain condition.We also generalize Theorems 1, 2 and 4 to alternative rings.