Computing multiplicative inverses in finite fields by long division

Elliptic curve cryptography is a rather new, efficient technology for security. However, its implementation is complex and software versions can be prohibitively slow. The main original contribution of this paper is the proposition of a highly parameterizable soft intellectual property core that implements all the operations needed to perform elliptic curve cryptography in hardware. This core supports several standardized elliptic curves. Here, finite field operations use only affine coordinates, which keeps interoperability with software implementations. Also, core operators such as the finite field multiplier and inversion are highly configurable, to enable fulfilling constraints of area or performance, according to what is relevant to each application. The core can thus serve several different purposes, through the setting of parameters that pick the adequate elliptic curve and others that control the finite field multiplier and inversion operator characteristics. Results obtained by synthesizing the core with different parameters for Xilinx FPGAs and 65nm CMOS ASICs show that the exploitable design space is ample: implementations can take from 17 to 5,800μs to execute a point multiplication, taking from 5.7 to 162.2 thousand LUTs in FPGAs or from 16,6 to 360,7 thousand gates in ASICs.

A celebrated theorem of Wedderburn says that a finite associative division ring is a finite (commutative) field. An elegant proof of this, due to E. Artin [3, p. 72], reduces the problem to showing the reduced (or generic) norm has a nontrivial zero, and then applies the theorem (conjectured by Artin and proved by Chevalley) that a polynomial with coefficients in a finite field and zero constant term has a nontrivial zero in that field if its degree is less than the number of variables. The Wedderburn theorem was generalized by A. A. Albert to finite strictly power-associative division rings of characteristic # 2 in [1, p. 301 ] and [2, p. i1 ] (see also [5 ]). The proof reduced the powerassociative case to the associative case by using the properties of Jordan division algebras. It is the purpose of this note to show that Artin's method carries over directly to the power-associative case using the generic norm introduced by N. Jacobson [4]. By avoiding Jordan algebras one is able to extend Albert's result to the characteristic 2 case. The usual definition of a division ring requires that left and right multiplications by a nonzero element be bijective. If the ring is finite it is necessarily an algebra, over a finite field 4, which is algebraic (even finite-dimensional) and without zero divisors. Also, as we shall see later, the algebra necessarily has a unit element. We wish to consider something slightly more general. For our purposes we say an algebraic power-associative algebra is a division algebra if it is unital and no element xO 0 is a zero divisor in the subalgebra (D[x] it generates (since 4<[x] is finite-dimensional, this is equivalent to saying x is invertible in b [x]). Our conditions are less restrictive, since the Jordan algebra of a quadratic form can be a division algebra in our sense with zero divisors. With this definition we have

Unified systolic array architecture for finite field multiplication and inversion

Algebraic feedback shift registers (AFSRs) [] are pseudo-random sequence generators that generalize linear feedback shift registers (LFSRs) and feedback with carry shift registers (FCSRs). With a general setting, AFSRs can result in sequences over an arbitrary finite field. It is well known that the sequences generated by LFSRs can be synthesized by either the Berlekamp-Massey algorithm or the extended Euclidean algorithm. There are three approaches to solving the synthesis problem for FCSRs, one based on the Euclidean algorithm [], one based on the theory of approximation lattices [] and Xu’s algorithm which is also used for some AFSRs []. Xu’s algorithm, an analog of the Berlekamp-Massey algorithm, was proposed by Xu and Klapper to solve the AFSR synthesis problem. In this paper we describe an approximation algorithm that solves the AFSR synthesis problem based on low-dimensional lattice basis reduction []. It works for AFSRs over quadratic integer rings \(\mathbb {Z}[\sqrt{D}]\) with quadratic time complexity. Given the first \(2\varphi _\pi (\mathbf {a})+c\) elements of a sequence \(\mathbf {a}\), it finds the smallest AFSR that generates \(\mathbf {a}\), where \(\varphi _\pi (\mathbf {a})\) is the \(\pi \)-adic complexity of \(\mathbf {a}\) and \(c\) is a constant.

Inversion in finite fields using logarithmic depth

The fundamental theorem of algebra states that every non-constant polynomial with complex coefficients can be factorized on linear factors with coefficients from the same field. However, this is not true for numerous other equally important and interesting fields such as, for example, finite fields. All existing finite fields are of the q=pk order, where p is a prime number and k is a positive integer. Any two finite fields of the same order are mutually isomorfic, therefore the field of q order is unique; it is also known as »the Galois field«, named after the French mathematician Evariste Galois (1811-1832). This thesis deals with factorization of polynomials over finite fields. The general rule states that some polynomials are not reducible, yet each polynomial with coefficients of the field can be uniquely factorized into irreducible factors. There are various known algorithms that can determine whether or not a polynomial is reducible and – in the case of reducible polynomials – also find this factorization. There are several such algorithms, however, this thesis shall focus on the Berlekamp's algorithm. The introductory chapter introduces the historical development of the various approaches to polynomial factorization and presents some modern applications that are currently in use. The second chapter focuses on basic concepts and characteristics of algebraic structures and some classic results that are crucial for the understanding of the algorithm (e.g. groups, rings, fields, division rings and polynomial rings.). The algorithm also uses the Chinese remainder theorem for polynomials and Euclidean algorithm for calculating the greatest common divisor of two polynomials. The first part of Chapter 3 (i.e. the main chapter of the thesis) demonstrates 4 naive methods for the finding of polynomial factorization. For each of the methods we first introduced factorization of a polynomial over real numbers and then continued with an example of a polynomial with coefficients in the finite field. The sieve method, for instance, builds a list of irreducible polynomials of low degrees. Irreducible polynomials up to (and including) degree 6 over the finite fields Z2, Z3, Z5 and Z7 can be calculated using our own code made in the VisualC# programming language. These naive methods are followed by a detailed description of the Berlekamp's algorithm with theorems, proofs and various examples. At the end of section we present our implementation of the algorithm in the computational algebra system MAGMA.

We introduce an efficient way of performing polynomial multiplication in a class of finite fields GF(pm) in the frequency domain. The Fast Fourier Transform (FFT) based frequency domain multiplication technique, originally proposed for integer multiplication, provides an extremely efficient method for multiplication with the best known asymptotic complexity, i.e. O(n log n log log n). Unfortunately, the original FFT method bears significant overhead due to the conversions between the time and the frequency domains, which makes it impractical to perform multiplication of relatively short (160 - 1024 bits) integer operands as used in many applications. In this work, we introduce an efficient way of performing polynomial multiplication in finite fields using the FFT. We show that, with careful selection of parameters, all the multiplications required for the FFT computations can be avoided and polynomial multiplication in finite fields can be achieved with only O(m) multiplications in addition to O(m log m) simple shift, addition and subtraction operations. We show that, especially in constrained devices where multiplication is expensive, polynomial multiplication in the suggested finite fields using the FFT outperforms both the schoolbook and Karatsuba methods for practically small finite fields, e.g., relevant to elliptic curve cryptography.

This paper investigates the feasibility of a unified processor architecture to enable error coding flexibility and secure communication in low power Internet of Things (IoT) wireless networks. Error coding flexibility for wireless communication allows IoT applications to exploit the large tradeoff space in data rate, link distance and energy-efficiency. As a solution, we present a light-weight Galois Field (GF) processor to enable energy-efficient block coding and symmetric/asymmetric cryptography kernel processing for a wide range of GF sizes (2m, m = 2, 3, ..., 233) and arbitrary irreducible polynomials. Program directed connections among primitive GF arithmetic units enable dynamically configured parallelism to efficiently perform either four-way SIMD 5- to 8-bit GF operations, including multiplicative inverse, or a wide bit-width (e.g., 32-bit) GF product in a single cycle. To illustrate our ideas, we synthesized our GF processor in a 28nm technology. Compared to a baseline software implementation optimized for a general purpose ARM M0+ processor, our processor exhibits a 5-20 x speedup for a range of error correction codes and symmetric/asymmetric cryptography applications. Additionally, our proposed GF processor consumes 431μW at 0.9V and 100MHz, and achieves 35.5pJ/b energy efficiency while executing AES operations at 12.2Mbps. We achieve this within an area of 0.01mm2.

In this paper, we present three algebraic methods for constructing structured nonbinary LDPC codes over finite fields, among which the first method is based on the multiplicative inverses of nonzero elements in finite fields and gives a class of quasi-cyclic codes with girth 6, the second method gives a class of (3, rho) quasi-cyclic codes with girth 8, the third method gives a class of structured codes with cycles limited. The codes given in examples perform well over AWGN channel and have better performances or far lower computational complexities than the corresponding random Mackay codes or codes algebraically constructed by Shu Lin.

Let $G = Z_2^n \, ( n > 1 )$ denote the additive group of the Galois field $GF ( 2^n )$ and $G^ * $ denote the nonzero elements in G. We consider partitions of $G^ * $ into disjoint subsets $S_1 ,S_2 , \cdots ,S_l $ of cardinalities $k_1 ,k_2 , \cdots ,k_l $ respectively such that the sum of the elements in each of the sets $S_i $ is 0. We prove by constructive methods that such partitions exist if and only if: (1) $k + k_2 + \cdots + k_l = 2^n - 1$ and (2) $k_i \geqq 3$ for all i. Applications of these constructions to the construction of single error correcting perfect mixed codes are discussed.

To solve a problem with a dynamic programming algorithm, one must reformulate the problem such that its solution can be formed from solutions to overlapping subproblems. Because overlapping subproblems may not be apparent in the specification, it is desirable to obtain the algorithm directly from the specification. We describe a semi-automatic synthesizer of linear-time dynamic programming algorithms. The programmer supplies a declarative specification of the problem and the operators that might appear in the solution. The synthesizer obtains the algorithm by searching a space of candidate algorithms; internally, the search is implemented with constraint solving. The space of candidate algorithms is defined with a program template reusable across all linear-time dynamic programming algorithms, which we characterize as first-order recurrences. This paper focuses on how to write the template so that the constraint solving process scales to real-world linear-time dynamic programming algorithms. We show how to reduce the space with (i)~symmetry reduction and (ii)~domain knowledge of dynamic programming algorithms. We have synthesized algorithms for variants of maximal substring matching, an assembly-line optimization, and the extended Euclid algorithm. We have also synthesized a problem outside the class of first-order recurrences, by composing three instances of the algorithm template.

This paper presents the hardware architectures for inversion in GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> ), in order to be used in elliptic curve cryptosystems. In this case, the Extended Euclidean Algorithm (EEA) and the Itoh-Tsujii Algorithm (ITA) are implemented. The simulation were carried out using Quartus II v. 9,0 and the design were synthesized on the Stratix III EP3SE50F780C2. The simulation results show a very good performance using small area.

A matrix interpretation of the extended Euclidean algorithm

It is known that, given an RSA modulus, n = p q , the public key e and the corresponding private key d satisfy the modulo congruence e d ≡ 1 mod ϕ n , where ϕ n = p − 1 q − 1 . Usually, the private key d can be computed efficiently using the extended Euclidean algorithm, and it is common knowledge that the private key is unique in the sense of modular ϕ n . This paper shows that there exist multiple private keys d ; they all satisfy that d &lt; ϕ n . This paper also presents the exact relationship between an RSA public key and a corresponding private key.

Generic hardness of inversion on ring and its relation to self-bilinear map