Finite Field Arithmetic Research Articles

Finite Field Arithmetic for Cryptography

Cryptography is one of the most prominent application areas of the finite field arithmetic. Almost all public-key cryptographic algorithms including the recent algorithms such as elliptic curve and pairing-based cryptography rely heavily on finite field arithmetic, which needs to be performed efficiently to meet the execution speed and design space constraints. These objectives constitute massive challenges that necessitate interdisciplinary research efforts that will render the best algorithms, architectures, implementations, and design practices. This paper aims to provide a concise perspective on designing architectures for efficient finite field arithmetic for usage in cryptography. We present different architectures, methods and techniques for fast execution of cryptographic operations as well as high utilization of resources in the realization of cryptographic algorithms. While it is difficult to have a complete coverage of all related work, this paper aims to reflect the current trends and important implementation issues of finite field arithmetic in the context of cryptography.

Concurrent Error Detection in Finite-Field Arithmetic Operations Using Pipelined and Systolic Architectures

In this work, we consider detection of errors in polynomial, dual, and normal bases arithmetic operations. Error detection is performed by recomputing with the shifted operand method, while the operation unit is in use. This scheme is efficient for pipelined architectures, particularly systolic arrays. Additionally, one semisystolic multiplier for each of the polynomial, dual, type I, and type II optimal normal bases is presented. The results show that for having better or similar space and time overheads compared to a number of related previous work, the multipliers have generally a higher error-detection capability, e.g., the error-detection capability of the RESO-based scheme for single and multiple stuck-at faults in a polynomial basis multiplier is 100 percent. Finally, we also comment on how RESO can be used for concurrent error correction to deal with transient faults.

Hardware/Software Co-Design Implementations of Elliptic Curve Cryptosystems

This paper presents a survey of hardware/software co-design implementations of elliptic curve cryptosystems. A critical study of the underlying finite field, the representation basis, and the partitioning schemes of these implementations is conducted. The study shows that all implementations are implemented over binary fields GF(2) and the implementations that use polynomial basis are more than implementations that use normal basis for finite field arithmetic. The study also shows that the best partitioning scheme, among the surveyed implementations,

On Efficient Implementation of Accumulation in Finite Field Over $GF(2^{m})$ and its Applications

Finite field accumulation is the simplest of all the finite field operations, but at the same time, it is one of the most frequently encountered operations in finite field arithmetic. In this paper, we present a simple but highly useful modification of the conventional hardware implementation of accumulation in finite field over GF(2m) . The critical path, as well as, the hardware-complexity are reduced in the proposed design by performing the accumulation operation using m number of T flip-flops instead of using a combination of m number of XOR gates with equal number of D flip-flops in dependent loop structures. The conventional design is found to involve nearly 39% more area, 53% more delay, and 40% more maximum ac power consumption compared with the proposed accumulator. The proposed finite field accumulator is used further for the implementation of serial/parallel polynomial-basis finite field multiplication and bit-serial inter-conversion between polynomial basis representation and normal basis representation over GF(2m). The area-time complexity of the proposed bit-serial/parallel multiplier is less than half of the best of the corresponding existing structures. The structure proposed for digit-serial/parallel multiplication for trinomials is found to involve nearly 56% less area-time complexity compared with the best of the corresponding existing multipliers; and the existing design of bit-serial basis conversion is found to involve nearly twice area-time complexity compared with the proposed design using the proposed finite field accumulator.

A Study of Suitability and Effectiveness of Various Implementation Options Of Finite Field Arithmetic on Elliptic Curve Crypto System

A Study of Suitability and Effectiveness of Various Implementation Options Of Finite Field Arithmetic on Elliptic Curve Crypto System

Dense Linear Algebra over Word-Size Prime Fields

In the past two decades, some major efforts have been made to reduce exact (e.g. integer, rational, polynomial) linear algebra problems to matrix multiplication in order to provide algorithms with optimal asymptotic complexity. To provide efficient implementations of such algorithms one need to be careful with the underlying arithmetic. It is well known that modular techniques such as the Chinese remainder algorithm or the p -adic lifting allow very good practical performance, especially when word size arithmetic is used. Therefore, finite field arithmetic becomes an important core for efficient exact linear algebra libraries. In this article, we study high performance implementations of basic linear algebra routines over word size prime fields: especially matrix multiplication; our goal being to provide an exact alternate to the numerical BLAS library. We show that this is made possible by a careful combination of numerical computations and asymptotically faster algorithms. Our kernel has several symbolic linear algebra applications enabled by diverse matrix multiplication reductions: symbolic triangularization, system solving, determinant, and matrix inverse implementations are thus studied.

Horner's rule-based multiplication over GF(p) and GF(pn ): a survey

This paper aims at surveying multipliers based on Horner's rule for finite field arithmetic. We present a generic architecture based on five processing elements and introduce a classification of several algorithms based on our model. We provide the readers with a detailed description of each scheme which should allow them to write a VHDL description or a VHDL code generator.

On a class of equations with special degrees over finite fields

One of the main objectives of arithmetic of finite fields is to study the value of N(f = 0). In general, it is difficult to give an explicit formula for N(f = 0). Hence there are various estimates for N(f = 0). The degree of f , denoted deg f , plays an important role in these estimates. An elementary upper bound for N(f = 0) (see [10, p. 147]) is N(f = 0) ≤ q deg f. Let ordp be the p-adic additive valuation normalized so that ordp p = 1. The classical Chevalley–Warning theorem asserts that ordp(N(f = 0)) ≥ 1 if n > deg f . Further, Ax [2] showed that

Elliptic Curve Cryptography Engineering

In recent years, elliptic curve cryptography (ECC) has gained widespread exposure and acceptance, and has already been included in many security standards. Engineering of ECC is a complex, interdisciplinary research field encompassing such fields as mathematics, computer science, and electrical engineering. In this paper, we survey ECC implementation issues as a prominent case study for the relatively new discipline of cryptographic engineering. In particular,we show that the requirements of efficiency and security considered at the implementation stage affect not only mere low-level, technological aspects but also, significantly, higher level choices, ranging from finite field arithmetic up to curve mathematics and protocols.

Reverse multiple recursive random number generators

Every multiple recursive generator (MRG) has an associated reverse MRG that generates the same sequence but in reverse order. An explicit formula for the recurrence of this reverse MRG is derived based on finite field arithmetic. These two MRGs are equal in both periodicity and spectral value. This property could be exploited to reduce the number of candidates when searching for good MRG parameters.

Implementation of Cryptosystems Based on Tate Pairing

Tate pairings over elliptic curves are important in cryptography since they can be used to construct efficient identity-based cryptosystems, and their implementation dominantly determines the efficiencies of the cryptosystems. In this paper, the implementation of a cryptosystem is provided based on the Tate pairing over a supersingular elliptic curve of MOV degree 3. The implementation is primarily designed to re-use low-level codes developed in implementation of usual elliptic curve cryptosystems. The paper studies how to construct the underlying ground field and its extension to accelerate the finite field arithmetic, and presents a technique to speedup the time-consuming powering in the Tate pairing algorithm.

Elements of Prescribed Order, Prescribed Traces and Systems of Rational Functions Over Finite Fields

Let k ≥ 1 and f1 .... , fr ∈ Fqk (x) be a system of rational functions forming a strongly linearly independent set over a finite field Fq. Let γ1 ..... γr ∈ Fq be arbitrarily prescribed elements. We prove that for all sufficiently large extensions Fqkm, there is an element ξ ∈ Fqkm of prescribed order such that TrFqkm/Fq (fi (ξ))= γi for i = 1, ..., r, where TrFqkm/Fq is the relative trace map from Fqkm onto Fq. We give some applications to BCH codes, finite field arithmetic and ordered orthogonal arrays. We also solve a question of Helleseth et al. (Hypercubic 4 and 5-designs from Double-Error-Correcting codes, Des. Codes. Cryptgr. 28(2003). pp. 265-282) completely.

Low complexity bit parallel architectures for polynomial basis multiplication over GF(2m)

Representing the field elements with respect to the polynomial (or standard) basis, we consider bit parallel architectures for multiplication over the finite field GF(2m). In this effect, first we derive a new formulation for polynomial basis multiplication in terms of the reduction matrix Q. The main advantage of this new formulation is that it can be used with any field defining irreducible polynomial. Using this formulation, we then develop a generalized architecture for the multiplier and analyze the time and gate complexities of the proposed multiplier as a function of degree m and the reduction matrix Q. To the best of our knowledge, this is the first time that these complexities are given in terms of Q. Unlike most other articles on bit parallel finite field multipliers, here we also consider the number of signals to be routed in hardware implementation and we show that, compared to the well-known Mastrovito's multiplier, the proposed architecture has fewer routed signals. The proposed generalized architecture is further optimized for three special types of polynomials, namely, equally spaced polynomials, trinomials, and pentanomials. We have obtained explicit formulas and complexities of the multipliers for these three special irreducible polynomials. This makes it very easy for a designer to implement the proposed multipliers using hardware description languages like VHDL and Verilog with minimum knowledge of finite field arithmetic.

Practical fast algorithm for finite field arithmetics using group rings

This paper studies a fast algorithm for finite field arithmetics, by repre- senting a finite field as a residue of a group ring of a finite cyclic group, where the frobenius (q-th power) operation is e‰ciently computable. When the characteristic of the field is greater than 2, our algorithm is often much faster than a standard method (NTL) in computing inverse and power. For example, ours is roughly 23.6 times faster in computing power in F8191136 than NTL. The implementation contains a new scheme for computing powers, which is applicable for any group if the q-th power operation is negligibly fast.

Efficient Arithmetic in Finite Field Extensions with Application in Elliptic Curve Cryptography

This contribution focuses on a class of Galois field used to achieve fast finite field arithmetic which we call an Optimal Extension Field (OEF), first introduced in [3]. We extend this work by presenting an adaptation of Itoh and Tsujii's algorithm for finite field inversion applied to OEFs. In particular, we use the facts that the action of the Frobenius map in GF (pm) can be computed with only m-1 subfield multiplications and that inverses in GF (p) may be computed cheaply using known techniques. As a result, we show that one extension field inversion can be computed with a logarithmic number of extension field multiplications. In addition, we provide new extension field multiplication formulas which give a performance increase. Further, we provide an OEF construction algorithm together with tables of Type I and Type II OEFs along with statistics on the number of pseudo-Mersenne primes and OEFs. We apply this new work to provide implementation results using these methods to construct elliptic curve cryptosystems on both DEC Alpha workstations and Pentium-class PCs. These results show that OEFs when used with our new inversion and multiplication algorithms provide a substantial performance increase over other reported methods.

A Simple Slide Rule for Finite Fields

We describe how an index table for a finite field can be constructed easily by hand, provided that a primitive polynomial for that field is given. Let q be a prime number and let n be a positive integer. Addition in the finite field Fqn of qn elements is very easy if one views it as an n-dimensional vector space over Fq (the integers modulo q). Likewise, multiplication in Fqn is very easy as its multiplicative group F*n is cyclic. In order to illustrate arithmetic in Fqn, involving both operations, most undergraduate algebra texts use the representation of Fqn as the factor ring of Fq [x] modulo the principal ideal generated by the minimal polynomial of Fqn over Fq. Certainly, this illuminates the concept of factor rings, but actual computations can become tedious, as reduction modulo the ideal requires long division. A very useful tool for finite field arithmetic is an index table. Before we explain this, let us briefly recall some theory of finite fields. As F*n is cyclic, every nonzero b E Fqn can be written as a power b = ai of a generator a of that group, with a unique integer exponent 0 < i < qn _ 1. This exponent is called the index of the element b, denoted by i = ind,, (b). Such a generator a is a called a primitive element of Fqnl; it is a root (in Fqn) of a primitive polynomial f (x) E Fq [x]. Conversely, every root of a primitive polynomial is a primitive element. Suppose that we are given a primitive polynomial f (x) for Fqn over Fq. Then we can tie together the multiplicative structure of Fqn with its additive structure. We fix a root a of f, and the former is just the cyclic group generated by a. The latter is a vector space, having a basis B = I 1, a, ... ., an-l1 . So every nonzero field element can be viewed as a power of a and as a linear combination of powers of a with coefficients in Fq, The index table establishes the correspondence, listing for every exponent 0 < i < qn _ 1 (as a column) the coordinates of the vector ai with respect to B. It works like a logarithm table, as the exponentiation laws for rings imply that ind,x (b . c) = ind,x (b) + ind,x (c). Hence, all arithmetic in Fqnl becomes easy: addition is vector addition modulo q, and multiplication is carried out using the table. But where do we get the index table from? For example, the standard reference for finite fields conveniently contains nearly four pages of index tables [3, pp. 546-549]. However, we can construct our own tables very easily by hand (if n and q are not too large), provided that we know a primitive polynomial f of Fqn over Fq. This construction is the aim of this Note. The reference [3] contains many primitive polynomials, too. Before we illustrate the method, let us make the notion of an index table precise:

A General Method for Establishing Geometric Inequalities in a Triangle

q can be any prime power. The material necessary to see this can be found in [3]. A different approach to finite field arithmetic can be found in [4]. The latter approach takes less space, but the table approach is faster, once the table is constructed. And it is very easy to implement. We have not addressed the question of existence and constructability of primitive polynomials. Existence can be shown rather easily using M6bius inversion in number theory [1, Theorem 16.9]. They can be constructed using Conway polynomials [2].

An architecture for computing Zech's logarithms in GF(2/sup m/)

In this paper, a new method for calculation of Zech's logarithm in GF(2/sup m/) is presented. For a given element, the logarithm is calculated by bit operations performed on its binary representation. No look-up tables are used. The proposed method makes feasible the implementation of an universal-type device for finite field arithmetic.

Efficient normal basis multipliers in composite fields

It is well-known that a class of finite fields GF(2/sup n/) using an optimal normal basis is most suitable for a hardware implementation of arithmetic in finite fields. In this paper, we introduce composite fields of some hardware-applicable properties resulting from the normal basis representation and the optimal condition. We also present a hardware architecture of the proposed composite fields including a bit-parallel multiplier.

A new division algorithm based on lookahead of partial-remainder (LAPR) for high-speed/low-power coding applications

A new polynomial division algorithm in finite field GF(2/sup m/) based on the lookahead of partial-remainder (LAPR) is proposed. Since our algorithm is based on partial division on group basis and lookahead technique exploiting the linearity in finite field arithmetic, it is possible to completely eliminate polynomial multiplications leading to highly increased throughput per unit time. The inherent regularity and feedforward nature of our algorithm make it possible to be fully pipelined. When pipelined, its throughput is one quotient and one remainder per clock cycle, regardless of the degree of dividend polynomial, which is orders of magnitude faster than the conventional architecture using linear feedback shift register. An area-efficient sequential architecture based on LAPR is also presented, Although the throughput rate of sequential architecture is lower than that of the pipelined one, it is still higher than that of any division architecture ever reported. They are shown to be efficient, regular, and easily expandable, and hence, naturally suitable for very large scale integration implementation. In systems requiring modest speed, the high-speed nature of our proposed architecture can be traded for low power consumption by reducing clock rate. We verified the general validity of the division algorithm based on LAPR by mathematical manipulation and simulation. The superiority of our proposed architecture compared with other reported ones is demonstrated with regard to its throughput, latency delays, and power.