Polynomials over Finite Fields

Irreducible polynomials play an important role in design of Forward Error Correction (FEC) codes for data transmission with integrity and automatic correction of data, as for example, Low-Density Parity Check codes. The usage of irreducible polynomials enables construction of non-prime-order finite fields. Most of the irreducible polynomials belong to binary Galois field. The important analytical concept is optimisation of irreducible polynomials for use in FECs in nonbinary Galois (NBG) field, leading to the development of an algorithm for LDPC that can work with nonbinary Galois fields. According to studies, the Tanner graph for ‘nonbinary Low Density Parity Check’ codes might get sparser as the field’s dimensions rise, ensuring that they do much better than their binary counterparts. A detailed discussion of representation of nonbinary irreducible polynomial and the computations involved have been illustrated. The concept has been tried for NB-LDPC codes. To prove the notion, computational complexity is found from different parameters such as performance of error correction capability, complexity cost and simulation time taken. Such detail study makes the NBG fields-based FEC very suitable for high-speed data transmission with self-error correction.

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 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.

Some relations between the irreducible polynomials over a finite field and its quadratic extension

The S-Box and Inverse S-Box in the tradition advance encryption standard is fixed and it is made by the composite field arithmatic to find the multiplicative inverse in the finite field GF (2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">8</sup> ). But in traditional AES the use of irreducible polynomial m(x) = x <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">8</sup> + x <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> + x <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> + x+1 to find out multiplicative inverse, is known to the attacker. So if we can make use of different irreducible polynomial every time to the finite field of GF (2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">8</sup> ) and send this the receiver combined with the secret key, then every time a new irreducible polynomial is used and a random S-Box is generated. Hence, the security of the algorithm is enhanced. In this paper, we devise an algorithm, which make use of different irreducible polynomial to finite field of GF (2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">8</sup> ) to make random S-Box and Inverse S-Box.

On the construction of irreducible and primitive polynomials from [formula omitted] to [formula omitted

Factorization of a class of polynomials over finite fields

In this paper, we present a method to compute all the irreducible and primitive polynomials of degree m over a finite field. We also describe two concrete implementations of our method with respective time complexities O(m2 + m log m) and O(m2 + log m). These implementations, using in parallel different devices introduced to operate in these fields [1], [7], allows us to reduce the time complexity of our method below that of the best previously known methods [3]. Our algorithm is especially well-suited for applications using large finite fields.

Let us recapitulate our results so far. We know that there is, for every prime p, a finite field with p elements, viz. the integers (mod p). This field we have denoted by F p . Furthermore, we know by Theorem 4.1 that provided there is an irreducible polynomial of degree m over F p , there exists a field with p m elements. On the other hand, by Theorem 5.1, there can be no finite field with a number of elements which is not a power of some prime number. This leaves two obvious questions: A. For which values of p and m do irreducible polynomials exist? B. Are there any other kinds of finite fields, i.e., ones that are not constructed by using Theorem 4.1?

We present the formula for the number of monic irreducible polynomials of degree n over the finite field \({\mathbb {F}}_q\) where the coefficients of \(x^{n-1}\) and x vanish for \(n\ge 3\). In particular, we give a relation between rational points of algebraic curves over finite fields and the number of elements \(a\in {\mathbb {F}}_{q^n}\) for which Trace\((a)=0\) and Trace\((a^{-1})=0\).

This paper propose the method of constructing the highly efficiency adder and multiplier systems over finite fields. The addition arithmetic operation over finite field is simple comparatively because that addition arithmetic operation is analyzed by each digit modP summation independently. But in case of multiplication arithmetic operation, we generate maximum k = 2m − 2° of αk terms, therefore we decrease k into m − 1 using irreducible primitive polynomial. We propose two method of control signal generation for the purpose of performing above decrease process. One method is the combinational logic expression and the other method is universal signal generation.

The dynamics of permutations on irreducible polynomials

Let F q \mathbb {F}_q be a finite field. Given two irreducible polynomials f , g f,g over F q \mathbb {F}_q , with deg ⁡ f \deg f dividing deg ⁡ g \deg g , the finite field embedding problem asks to compute an explicit description of a field embedding of F q [ X ] / f ( X ) \mathbb {F}_q[X]/f(X) into F q [ Y ] / g ( Y ) \mathbb {F}_q[Y]/g(Y) . When deg ⁡ f = deg ⁡ g \deg f = \deg g , this is also known as the isomorphism problem. This problem, a special instance of polynomial factorization, plays a central role in computer algebra software. We review previous algorithms, due to Lenstra, Allombert, Rains, and Narayanan, and propose improvements and generalizations. Our detailed complexity analysis shows that our newly proposed variants are at least as efficient as previously known algorithms, and in many cases significantly better. We also implement most of the presented algorithms, compare them with the state of the art computer algebra software, and make the code available as an open source. Our experiments show that our new variants consistently outperform available software.

Known results on permutation polynomials with coefficients in a finite field are extended to algebras of the form Lv=K[X]/(p(X)v)), where K is a finite field, p(X) belongs to K[X] and is an irreducible polynomial, v = 1,2..., and to the algebra of power series L[[Z]]. Finally analogues of Dickson polynomials in these algebras are studied.

Multiplication and squaring are main finite field operations in cryptographic computations and designing efficient multipliers and squarers affect the performance of cryptosystems. In this paper, we consider the Montgomery multiplication in the binary extension fields and study different structures of bit-serial and bit-parallel multipliers. For each of these structures, we study the role of the Montgomery factor, and then by using appropriate factors, propose new architectures. Specifically, we propose two bit-serial multipliers for general irreducible polynomials, and then derive bit-parallel Montgomery multipliers for two important classes of irreducible polynomials. In this regard, first we consider trinomials and provide a way for finding efficient Montgomery factors which results in a low time complexity. Then, we consider type-II irreducible pentanomials and design two bit-parallel multipliers which are comparable to the best finite field multipliers reported in the literature. Moreover, we consider squaring using this family of irreducible polynomials and show that this operation can be performed very fast with the time complexity of two XOR gates.