Highly Regular Architectures for Finite Field Computation Using Redundant Basis

Recently the study and implementation of elliptic curve cryptosystems (ECC) have developed rapidly and its achievements have become a center of attraction. ECC has the advantage of high-speed processing in software even on restricted environments such as smart cards. In this paper, we concentrate on implementation of ECC over a field of prime characteristic on a 16-bit microcomputer M16C (10MHz). We report a practical software implementation of a cryptographic library which supports 160-bit elliptic curve DSA (ECDSA) signature generation, verification and SHA-1 on the processor. This library also includes general integer arithmetic routines for applicability to other cryptographic algorithms. We successfully implemented the library in 4Kbyte code/data size including SHA-1, and confirmed a speed of 150msec for generating an ECDSA signature and 630msec for verifying an ECDSA signature on M16C.

We present an efficient exponentiation algorithm in a finite field GF(q n ) using a Gauss period of type (n,1). Though the Gauss period α of type (n,1) in GF(q n ) is never primitive, a computational evidence says that there always exists a sparse polynomial (especially, a trinomial) of α which is a primitive element in GF(q n ). Our idea is easily generalized to the field determined by a root of unity over GF(q) with redundant basis technique. Consequently, we find primitive elements which yield a fast exponentiation algorithm for many finite fields GF(q n ), where a Gauss period of type (n,k) exists only for larger values of k or the existing Gauss period is not primitive and has large index in the multiplicative group GF(q n )×.

The simplest field extensions of a field F are those in which we adjoin an nth root of an element c ∈F. To investigate these, it will be valuable for us to consider roots of unity. After all, if α n = c, then the other nth roots of c are of the form ωα, where ω is some nth root of unity, and so it will be relevant whether or not F contains such roots of unity. Note that R, for example, contains the square roots of unity, namely, 1 and — 1, but it contains no higher roots of unity (other than 1) because they are all complex.

This contribution describes three algorithms for efficient implementations of elliptic curve cryptosystems. The first algorithm is an entirely new approach which accelerates the multiplications of points which is the core operation in elliptic curve public-key systems. The algorithm works in conjunction with the k-ary or sliding window method. The algorithm explores computational advantages by computing repeated point doublings directly through closed formulae rather than from individual point doublings. This approach reduces the number of inversions in the underlying finite field at the cost of extra multiplications. For many practical implementations, where field inversion is at least four times as costly as field multiplication, the new approach proofs to be faster than traditional point multiplication methods. The second algorithm deals with efficient inversion in composite Galois fields of the form GF((2n)n). Based on an idea of Itoh and Tsujii, we optimize the algorithm for software implementation of elliptic curves. The algorithm reduced inversion in the composite field to inversion in the subfield GF(2n). The third algorithm describes the application of the Karatsuba-Ofman Algorithm to multiplication in GF((2n)n). We provide a detailed complexity analysis of the algorithm for the case that subfield arithmetic is performed through table look-up. We apply all three algorithms to an implementation of an elliptic curve system over GF((216)11). We provide absolute performance measures for the field operations and for an entire point multiplication.

Let K[G] denote the group ring of G over the field K. In this note we characterize those group rings in which all left ideals are right ideals. Let R be a ring. We say that R is l.i.r.i. if every left ideal is a right ideal. A ring is l.a.r.i. if every left annihilator is a right ideal. Our notation follows that of [2]. The main results are THEOREM I. Let K be a field and let G be a nonabelian periodic group. Then if K[G] is l.a.r.i. one of the following occurs (i) Char K = 0 and G is a Hamiltonian group such that for each odd exponent, n, of G the quaternion algebra over the field K(4), where 4 is a primitive nth root of unity, is a division ring. (ii) Char K = 2 and K does not contain any primitive cube root of unity. Moreover G = Q x A, where Q is the quaternion group of order 8 and A is abelian in which each element has odd order and if n is an exponent for A, the least integer m > 1 satisfying 2m -1 (mod n) is odd. Conversely if K[G] satisfies either (i) or (ii), then K[G] is l.i.r.i. and, in particular, it is l.a.r.i. Observe that if char K > 2 and G is periodic, then K[GI is l.a.r.i. if and only if G is abelian. THEOREM II. Let K[G] denote the group ring over a nonabelian group. Then the following are equivalent (i) K[G] is l.i.r.i. (ii) G is locally finite and if a,f8 E K[GI with a,8 = 0, then /3a = 0. (iii) G is locally finite and K[G] is l.a.r.i. If we combine the above theorems we get necessary and sufficient conditions for K[GI to be l.i.r.i. By using the antiautomorphism of K[G] given by ,xeG k.x H* IeG kxxwe see that K[G] is l.i.r.i. (l.a.r.i.) if and only if K[G] is r.i.l.i. (r.a.l.i.). Received by the editors February 2, 1978. AMS (MOS) subject classifications (1970). Primary 16A26.

During the last few years, a considerable effort has been devoted to the development of reconfigurable computers, machines that are based on the close interoperation of traditional microprocessors and Field Programmable Gate Arrays (FPGAs). Several prototype machines of this type have been designed, and demonstrated significant speedups compared to conventional workstations for computationally intensive problems, such as codebreaking. Nevertheless, the efficient use and programming of such machines is still an unresolved problem. In this paper, we demonstrate an efficient implementation of an Elliptic Curve scalar multiplication over GF(2/sup m/), using one of the leading reconfigurable computers available on the market, SRC-6E. We show how the hardware architecture and programming model of this reconfigurable computer has influenced the choice of the algorithm partitioning strategy for this application. A detailed analysis of the control, data transfer, and reconfiguration overheads is given in the paper, together with the performance comparison of our implementation against an optimized microprocessor implementation.

In this chapter we explore the feasibility of efficient implementation of an arithmetic processor for performing elliptic curve computations over finite fields. For a secure system, it is evident from the results of Chapter 5 that the curve and underlying field should be judiciously chosen. However we should point out that for a given underlying field there are a large number of suitable elliptic curve to choose from. If the logarithm problem in supersingular curves is indeed as hard as the logarithm problem in finite fields, and/or if the logarithm problem in non-supersingular curves is indeed intractable, then the systems discussed here are both efficient and secure and very attractive for practical usage.

A family of elliptic curves for cryptographic use is proposed for which the determination of the order of the corresponding algebraic group is much easier than in the general case. This makes it easier to meet the cryptographic requirement that this order have a large prime factor. Another advantage of this familiy is that the group operation simplifies slightly. Explicit numerical examples are given that are suitable for practical use.KeywordsElliptic CurveCyclic GroupAlgebraic GroupFinite FieldElliptic CurfThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper we show that a semi-commutative Galois extension of associative unital algebra by means of an element \(\tau \), which satisfies \(\tau ^N={\mathbbm {1}}\) (\({\mathbbm {1}}\) is the identity element of an algebra and \(N\ge 2\) is an integer) induces a structure of graded q-differential algebra, where q is a primitive Nth root of unity. A graded q-differential algebra with differential d, which satisfies \(d^N=0, N\ge 2\), can be viewed as a generalization of graded differential algebra. The subalgebra of elements of degree zero and the subspace of elements of degree one of a graded q-differential algebra together with a differential d can be considered as a first order noncommutative differential calculus. In this paper we assume that we are given a semi-commutative Galois extension of associative unital algebra, then we show how one can construct the graded q-differential algebra and when this algebra is constructed we study its first order noncommutative differential calculus. We also study the subspaces of graded q-differential algebra of degree greater than one which we call the higher order noncommutative differential calculus induced by a semi-commutative Galois extension of associative unital algebra. We also study the subspaces of graded q-differential algebra of degree greater than one which we call the higher order noncommutative differential calculus induced by a semi-commutative Galois extension of associative unital algebra. Finally we show that a reduced quantum plane can be viewed as a semi-commutative Galois extension of a fractional one-dimensional space and we apply the noncommutative differential calculus developed in the previous sections to a reduced quantum plane.

We construct a q-analog of exterior calculus with a differential d satisfying dN = 0, where N ≥ 2 and q is a primitive Nth root of unity, on a noncommutative space and introduce a notion of a q-differential k-form. A noncommutative space we consider is a reduced quantum plane. Our construction of a q-analog of exterior calculus is based on a generalized Clifford algebra with four generators and on a graded q-differential algebra. We study the structure of the algebra of q-differential forms on a reduced quantum plane and show that the first order calculus induced by the differential d is a coordinate calculus. The explicit formulae for partial derivatives of this first order calculus are found.

I recently came across a very appealing result about the product of the lengths of the chords of a regular polygon with n sides inscribed in a unit circle. If one vertex is selected and the product of the chord lengths from that vertex to each of the others is evaluated, the result is just n . There is a simple proof. Consider such a polygon and denote its vertices by the points A 0 , A 1 , … , A n . Suppose that O is the centre of the circle and the origin of the complex plane. The roots of the equation are just where ω is a primitive nth root of unity. Each of these roots represents one of the vertices. Suppose for convenience that the real number 1 is represented by A 0 .

We propose a generalization of the concept of connection form by means of a graded q-differential algebra Wq, where q is a primitive Nth root of unity, and develop the concept of curvature N-form for this generalization of the connection form. The Bianchi identity for a curvature N-form is proved. We study an Wq-connection on module and prove that every projective module admits an Wq-connection. If the module is equipped with a Hermitian structure, we introduce a notion of an Wq-connection consistent with the Hermitian structure.

An elliptic semiplane is a λ-fold of a symmetric 2-(v,k,λ) design, where parallelism is transitive. We prove existence and uniqueness of a 3-fold cover of a 2-(15,7,3) design, and give several constructions. Then we prove that the automorphism group is 3.Alt(7). The corresponding bipartite graph is a minimal graph with valency 7 and girth 6, which has automorphism group 3.Sym(7). A polynomial with real coefficients is called formally positive if all of the coefficients are positive. We conjecture that the determinant of a matrix appearing in the proof of the van der Waerden conjecture due to Egorychev is formally positive in all cases, and we prove a restricted version of this conjecture. This is closely related to a problem concerning a certain generalization of Latin rectangles. Let ω be a primitive nth root of unity over GF(2), and let m_i(x) be the minimal polynomial of ω^i. The code of length n = 2^r-1 generated by m_1(x)m_t(x) is denoted C^t_r. We give a recursive formula for the number of codewords of weight 4 in C^11_r for each r.

In this paper we consider a Galois-Fourier type of transform suitable for the study of a class of codes which can be seen as a generalization of the Goppa codes and which encompasses among others the Chien-Choy, Srivastava and related codes. This class is formally equivalent to Helgert’s family of so-called alternant codes and, as such, this novel approach might provide some new insight into their properties. Such a transform satisfies all the classical properties of a discrete Galois-Fourier transform without being necessarily generated through a primitive Nth root of unity. As a direct application of this technique, we demonstrate in a compact manner how the basic properties of alternant codes can be derived within the framework of this approach and new lower boouds on the guaranteed minimum distance are derived. Moreover it is shown how Blahut’s transform scheme can be applied to their decoding. Due to the availability of efficient GF transformers, it is believed that such a scheme will prove ...

Even-order subgroups of finite-dimensional division rings