Finite Fields Exist and are Unique

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.

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.

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.

On the diophantine equation x(x + 1)(x + 2)…(x + (m − 1)) =g(y)

Is every irreducible polynomial reducible after a polynomial substitution?

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.

Factorization of a class of polynomials over finite fields

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\((a^{-1})=0\).

Association schemes of conjugate matrices

The dynamics of permutations on irreducible polynomials

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 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.

Recently S. D. Cohen resolved a conjecture of Chowla and Zassenhaus (1968) in the affirmative by showing that, if f ( x ) f(x) and g ( x ) g(x) are integral polynomials of degree n ≥ 2 n \geq 2 and p is a prime exceeding ( n 2 − 3 n + 4 ) 2 {({n^2} - 3n + 4)^2} for which f and g are both permutation polynomials of the finite field F p {F_p} , then their difference h = f − g h = f - g cannot be such that h ( x ) = c x h(x) = cx for some integer c not divisible by p. In this note we provide a significant generalization by proving that, if h is not a constant in F p {F_p} and t is the degree of h, then t ≥ 3 n / 5 t \geq 3n/5 and, provided t ≤ n − 3 t \leq n - 3 , t and n are not relatively prime. In a sense this measures the isolation of permutation polynomials of the same degree over large finite prime fields.