Polynomial Multiplication over Finite Fields Using Field Extensions and Interpolation

A signal processing system employing digital arithmetic can be made equivalent to a set of interconnected subsystems each using finite field operations to alleviate effects of round-off noise, particularly in recursive structures. The realization of the subsystems involves the multiplication of polynomials over a finite field. Cyclic convolutions may be represented as polynomial multiplications, in a residue class ring modulo the polynomial (x <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sup> -1). Fast algorithms arise when the ring is decomposed into a direct product using its minimal ideals, the orthogonality relationships between the idempotents of the minimal ideals resembling the decomposition properties of the Chinese remainder theorem for real polynomials. The resulting component polynomial multiplications involve elements with smaller degrees. A transform domain, based upon an n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> root of unity in an extension field, offers an alternative domain for performing the component operations. Polynomial multiplications are equivalent to the vector inner product of the corresponding transform coefficients. The minimal ideal components, when mapped into the transform domain, have nonzero coefficients only in predetermined subsets, the cyclotomic subsets, which permits one coefficient to generate all others by successively forming its powers.

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.

The theory of polynomials over finite fields is important for investigating the algebraic structure of finite fields as well as for many applications. Above all, irreducible polynomials—the prime elements of the polynomial ring over a finite field—are indispensable for constructing finite fields and computing with the elements of a finite field. Section 1 introduces the notion of the order of a polynomial. An important fact is the connection between minimal polynomials of primitive elements (so-called primitive polynomials) and polynomials of the highest possible order for a given degree. Results about irreducible polynomials going beyond those discussed in the previous chapters are presented in Section 2. The next section is devoted to constructive aspects of irreducibility and deals also with the problem of calculating the minimal polynomial of an element in an extension field. Certain special types of polynomials are discussed in the last two sections. Linearized polynomials are singled out by the property that all the exponents occurring in them are powers of the characteristic. The remarkable theory of these polynomials enables us, in particular, to give an alternative proof of the normal basis theorem. Binomials and trinomials — that is, two-term and three-term polynomials—form another class of polynomials for which special results of considerable interest can be established. We remark that another useful collection of polynomials— namely, that of cyclotomic polynomials—was already considered in Chapter 2, Section 4, and that some additional information on cyclotomic polynomials is contained in Section 2 of the present chapter.

Binary extension (or Galois) fields GF(2^m) have been widely studied due to their use in several important applications, such as cryptography, error control codes and digital signal processing. These applications require efficient hardware implementations of GF(2^m) arithmetic operations, particularly multiplication, which is considered the most important and complex one. The complexity of GF(2^m) multiplication depends on the representation basis and on the defining irreducible polynomial f(y) selected for the finite field. For efficient hardware implementations, polynomial basis and irreducible trinomials or pentanomials are normally used. Any element A ∊ GF(2^m) can be represented in the polynomial basis {1,x,...,x^m-1} as A = Σm-1/i-0, a_i x^i with a_i Σ GF(2), where x is a root of the irreducible polynomial f(y) = Σ^m/_i=0^f_i y^i. Polynomial basis multiplication C = A ⋅ B requires a polynomial multiplication followed by a reduction modulo an irreducible polynomial. Mastrovito [1] proposed an efficient bit-parallel polynomial basis multiplier in which a product matrix was introduced to combine the above two steps together. A new polynomial basis multiplication method applied to irreducible trinomials was proposed in [2], where the functions S_i and T_i given by the addition of terms x_k = (a_k b_k) and z_i^j = (a_i b_j + a_j b_i), with a_i, b_i ∊ GF(2), were obtained from the decomposition of a product matrix. The addition of these functions is used for the computation of the product of two GF(2^m) elements. In [3], the above method was applied to type II irreducible pentanomials and the functions S_i and T_i were split in the form S_i = s^i_k S_i^k + ... + s^i_0 S_i^0 and T_i = t^i_k T_i^k + ... + t^i_0 T_i^0, with s^i_j, t^i_j ∊ GF(2) and k = log_2m. The terms S_i^j and T_i^j represent the sum of 2^j products a_kb_l and therefore can be implemented as a j-level complete binary tree of XOR gates. The addition in pairs of binary trees with the same depth leads to a reduction of the multiplication delay. However, splitting method imposes hard restrictions (given by the use of parenthesis in the expressions of the coordinates) for the addition of S_i^j and T_i^j terms in order to reduce the number of XOR levels. These restrictions could not be efficient for a synthesis tool in order to map that expressions into FPGA's logic blocks. If parenthesized restrictions are removed, more freedom could be given for the synthesizer to find an optimized implementation of the multiplier. In this work, efficient Xilinx FPGA implementations of GF(2^m) bit-parallel polynomial basis multipliers for irreducible trinomials are presented. Based on [2], a new general algorithm for multiplication over irreducible trinomials f(y) = y^m + y^n +1, with 1 ≤ n ≤ (m+1)/2, is proposed and the splitting method given in [3] is applied to these irreducible polynomials. Furthermore, in order to optimize the synthesis of the multipliers, a new approach for the computation of the product is used where the splitting of S_i and T_i terms is performed, but the restriction given by the addition in pairs of binary trees with the same depth has been removed. In this way, Xilinx tools are free to optimize the synthesis of the multiplier. Several GF(2^m) multipliers for different binary fields have been described in VHDL and their post-place and route implementation results in Xilinx Artix-7 have been reported. Experimental results show that the multiplier here proposed exhibits the best delay and Area×Time complexities when it is compared with similar multipliers found in the literature. Moreover, the new approach also achieves the lowest number of slices in most of the implemented multipliers.

The transform domain characterization of linear cyclic codes over finite fields using Discrete Fourier Transform (DFT) over an appropriate extension field is well known. In this paper, we extend this transform domain characterization for linear quasi-cyclic codes over finite fields. We show how one can derive a lower bound on the minimum Hamming distance of a quasi-cyclic code and decode the code upto that minimum Hamming distance using this characterization.

1: Rings. 2: Euclidean rings. 3: Highest common factor. 4: The four-squares theorem. 5: Fields and polynomials. 6: Unique factorization domains. 7: Field of quotients of an integral domain. 8: Factorization of polynomials. 9: Fields and field extensions. 10: Finite cyclic groups and finite fields. 11: Algebraic numbers. 12: Ruler and Compass constructions. 13: Homomorphisms, ideals and factor rings. 14: Principal ideal domains and a method for constructing fields. 15: Finite fields. Solutions to selected exercises. References

In this chapter we shall analyze arbitrary extension fields of a field Φ. A study of finite dimensional extension fields and a partial study of algebraic extensions has been made in Chapter I. In this chapter our primary concern will be with infinite dimensional extensions and we shall begin again with the algebraic ones. We define algebraically closed fields and prove the existence of an algebraic closure of any field. We shall extend the classical Galois theory to apply to infinite dimensional normal and separable extensions. After this we shall consider arbitrary extension fields and we shall show that these can be built up in two stages: first a purely transcendental one and then on top of this an algebraic extension. The invariant of this mode of generating a field is the transcendency degree which is the cardinal number of a transcendency basis. We shall obtain conditions for the existence of a transcendency basis such that the extension is separable algebraic over the purely transcendental extension determined by the basis. We shall also give a definition of separability of an extension field that generalizes the notion of algebraic separability. The notion of a derivation plays an important role in these considerations.KeywordsPrime IdealStructure TheoryGalois GroupHigh DerivationAlgebraic ClosureThese 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.

Generators for primary closures of Galois fields

We study the minimal number of ramified primes in Galois extensions of rational function fields over finite fields with prescribed finite Galois group. In particular, we obtain a general conjecture in analogy with the well studied case of number fields, which we establish for abelian, symmetric, and alternating groups in many cases.

Bose–Chaudhuri–Hocquenghem (BCH) codes are broadly used to correct errors in flash memory systems and digital communications. These codes are cyclic block codes and have their arithmetic fixed over the splitting field of their generator polynomial. There are many solutions proposed using CPUs, hardware, and Graphical Processing Units (GPUs) for the BCH decoders. The performance of these BCH decoders is of ultimate importance for systems involving flash memory. However, it is essential to have a flexible solution to correct multiple bit errors over the different finite fields (GF(2)). In this paper, we propose a pragmatic approach to decode BCH codes over the different finite fields using hardware circuits and GPUs in tandem. We propose to employ hardware design for a modified syndrome generator and GPUs for a key-equation solver and an error corrector. Using the above partition, we have shown the ability to support multiple bit errors across different BCH block codes without compromising on the performance. Furthermore, the proposed method to generate modified syndrome has zero latency for scenarios where there are no errors. When there is an error detected, the GPUs are deployed to correct the errors using the iBM and Chien search algorithm. The results have shown that using the modified syndrome approach, we can support different multiple finite fields with high throughput.

A novel method for blind reconstruction of binary Bose–Chaudhuri–Hocquenghem codes is proposed. Compared to previously reported works, a new approach to find the goal generator polynomial is employed. First, using the feature that each codeword polynomial of a t‐error‐correcting Bose–Chaudhuri–Hocquenghem code has the same 2t consecutive roots over Galois field, a new set of candidate generator polynomials is introduced. Then, this set in a random situation to find the correct generator polynomial is investigated. Monte Carlo simulations demonstrate the superiority of the proposed reconstruction algorithm compared to the previous methods.

Counting reducible and singular bivariate polynomials

Linear codes over F/sub p/ often admit a natural representation as trace codes of codes that are defined over an extension field F/sub p/m. In the paper, the authors obtain estimates for the weights of subcodes of such trace codes. Their main result is a far-reaching generalization of the Carlitz-Uchiyama bound for the duals of binary BCH codes. In particular, they prove sharp bounds for the generalized Hamming weights of a large class of codes, including duals of BCH codes, classical Goppa codes, Melas codes, and arbitrary cyclic codes of length n=p/sup m/-1. The main tool is the theory of algebraic functions over finite fields, in particular the Hasse-Weil bound for the number of places of degree one. >

Modern large scale distributed storage systems play a central role in data center and cloud storage, while node failure in data center is common. The lost data in failure node must be recovered efficiently. Locally repairable codes (LRCs) are designed to solve this problem. The locality of an LRC is the number of nodes that participate in recovering the lost data from node failure, which characterizes the repair efficiency. An LRC is called optimal if its minimum distance attains Singleton-type upper bound [1]. In this paper, using basic techniques of linear algebra over finite field, infinite optimal LRCs over extension fields are derived from a given optimal LRC over base field (or small field). Next, this paper investigates the relation between near-MDS codes with some constraints and LRCs, further, proposes an algorithm to determine locality of dual of a given linear code. Finally, based on near-MDS codes and the proposed algorithm, those obtained optimal LRCs are shown.

A general structure of the Welch-Gong (WG) stream cipher family is based on filtering an m-sequence of degree l over a finite field $\ensuremath{{\mathbb{F}}}_{2^m}$ where the filtering function is a WG transformation from $\ensuremath{{\mathbb{F}}}_{2^m}$ to $\ensuremath{{\mathbb{F}}}_{2}$ . For a fixed m and l, the linear span of the filtering sequence can be enhanced by increasing the algebraic degree of the WG transformations. This can be accomplished by the composition of a WG transformation with a monomial permutation, which is called the decimation of a WG transformation. In this paper, we first present the new exponent set of WG transformations, and show the existence of exponents derived from the new exponent set for which a decimated WG transformation achieves the maximum algebraic degree. As a result, the linear span of keystreams produced by a decimated WG cipher can be maximized and calculated theoretically. We then give a description of a decimated WG stream cipher which is built upon an LFSR and a decimated WG transformation over an extension field. The randomness properties of keystreams produced by a decimated WG cipher are derived based on the new exponent set. We also discuss the selection criteria for choosing the optimal parameters for the WG cipher family in order to achieve the maximum level of security. Finally, we present the optimal parameters for the WG transformations over $\ensuremath{{\mathbb{F}}}_{2^m}, 7\leq m \leq 16$ based on the proposed criteria.