Implementation of Polynomial - ONB I Basis Conversion

The theory of finite fields has important applications in coding theory and cryptography. Two type of basis of finite field € GF (2 n ) over € GF (2) are of particular interest, polynomial basis and optimal normal basis (type I and II) of the form € β, β 2 , ..., β p^n−1 { } for some element € α , € β in . Choosing between optimal normal basis and polinomial basis depends on the application. This paper presents an implementation of an efficient method to convert from the representation of a field element in one basis to the representation of a field element in another basis. With this method, it is possible to extend an implementation in one basis so that it supports other choices of basis.

Proceedings of the 3rd International Conference on Computation for Science and Technology, 2015

Elliptic Curve Cryptography is generally are implemented over prime fields or binary fields. Arithmetic in binary fields can be classified according to the basis representation being used. Two of the most common basis used in binary fields are polynomial basis and normal basis. The optimal normal basis is especially known to be more efficient than polynomial basis because the inversion can be achieved by performing repeated multiplication using the method of Itoh and Tsujii, and squaring can be executed by performing only one cyclic shift operation. In previous research we have built algorithms and implementations on basis conversion between polynomial basis and normal basis. In this paper we will present implementation of arithmetic operation algorithms on both polynomial basis and optimal normal basis representation and the conversion method between them. We will also give comparison of time and space between implementation by using optimal basis representation with conversion and without conversion.

2014 IEEE 55th Annual Symposium on Foundations of Computer Science, 2014

In this paper, we present a new basis of polynomial over finite fields of characteristic two and then apply it to the encoding/decoding of Reed-Solomon erasure codes. The proposed polynomial basis allows that h-point polynomial evaluation can be computed in O(h log 2 (h)) finite field operations with small leading constant. As compared with the canonical polynomial basis, the proposed basis improves the arithmetic complexity of addition, multiplication, and the determination of polynomial degree from O(h log 2 (h) log 2 log 2 (h)) to O(h log 2 (h)). Based on this basis, we then develop the encoding and erasure decoding algorithms for the (n = 2 r , k) Reed-Solomon codes. Thanks to the efficiency of transform based on the polynomial basis, the encoding can be completed in O(n log 2 (k)) finite field operations, and the erasure decoding in O(n log 2 (n)) finite field operations. To the best of our knowledge, this is the first approach supporting Reed-Solomon erasure codes over characteristic-2 finite fields while achieving a complexity of O(n log 2 (n)), in both additive and multiplicative complexities. As the complexity leading factor is small, the algorithms are advantageous in practical applications.

This article addresses an efficient hardware implementations for multiplication over finite field GF(2 233). Multiplication in GF(2 n) is very commonly used in cryptography and error correcting codes. An efficient hardware could reduce the cost and development for these applications. This work presents the hardware implementation of polynomial basis. In this case, the multipliers were designed using bit-serial multiplication , bit-parallel multiplication, PCA based serial multiplication and PCA parallel based multiplication algorithms, the synthesis and simulation were carried out using Quartus II v.5.0 of Altera, and the designs were synthesized on the Stratix II EP2S60F1020C3. The simulation results show that the multipliers designed present a very good performance using small area.

2020

Finite field is a wide topic in mathematics. Consequently, none can talk about the whole contents of finite fields. That is why this research focuses on small content of finite fields such as polynomials computational, ring of integers modulo p where p is prime or a power of prime. Most of the times, books which talk about finite fields are rarely to be found, therefore one can know how arithmetic computational on small finite fields works and be able to extend to the higher order. This means how integer and polynomial arithmetic operations are done for Z p such as addition, subtraction, division and multiplication in Z p followed by reduction of p (modulo p). Since addition is the same as subtraction and division is treated as the inverse of the multiplication, thus in this paper, only addition and multiplication arithmetic operations are applied for the considered small finite fields (Z 2 − Z 17 ). With polynomials, one can learn from this paper how arithmetic computational throug...

International Journal of Mathematics & …, 2013

The theory of composite fields has important applications in cryptography. The use of composite field characteristic of dividing one big chunk of operation into smaller ones, allows us to implement in limited resources with adequate level of security. Two type of basis of composite field GF ((2 n ) m ) over GF (2 n ) are of particular interest, polynomial basis of the form {1, α, α 2 , · · · , α m−1 } and type I and type II optimal normal basis of the form {α, α 2 , · · · , α 2 m−1 } for some element α in GF ((2 n ) m ). Choosing between optimal normal basis and polynomial basis depends on the application. This paper presents a method to convert from the representation of a composite field element in one basis to the representation of a composite field element in another basis.

Acta Applicandae Mathematicae, 2006

In this work, we present a survey of efficient techniques for software implementation of finite field arithmetic especially suitable for cryptographic applications. We discuss different algorithms for three types of finite fields and their special versions popularly used in cryptography: Binary fields, prime fields and extension fields. Implementation details of the algorithms for field addition/subtraction, field multiplication, field reduction and field inversion for each of these fields are discussed in detail. The efficiency of these different algorithms depends largely on the underlying microprocessor architecture. Therefore, a careful choice of the appropriate set of algorithms has to be made for a software implementation depending on the performance requirements and available resources. Mathematics Subject Classifications 12-02 • 12E30 • 12E10 Key words field arithmetic • cryptography • efficient implementation • binary field arithmetic • prime field arithmetic • extension field arithmetic • optimal extension fields

2006

Self-reciprocal irreducible monic (srim) polyn omials over finite fields have been studied in the past. These polynomials can be studied in the context of quad ratic transformation of irreducible polynomials over finite fields. In this talk we present the generalization of some of the results known about srim polynomials to polynomials obtained by quadratic transformation of irreducible polynomials over finite fields. Speaker:Dan Bernstein (University of Illinois at Chicago) Title: Faster factorization into coprimes Abstract: How quickly can we factor a set of univariate polyn mials into coprimes? See http://cr.yp.to/coprimes.html for examples and applications. Bach, Driscoll, and Shallit chieved time n in 1990, wheren is the number of input coefficients; I achieved time n(lg n) in 1995; much more recently I achieved time n(lg n). Speaker:Antonia Bluher (National Security Agency) Title: Hyperquadratic elements of degree 4 Abstract: I will describe joint work with Alain Lasjaunias a ...

The chapter introduces a comparative analysis of the complexity of the Tate pairing operation on a supersingular elliptic curve and the complexity of the final exponentiation in the tripartite key agreement cryptographic protocol. The analysis takes into account a possibility of using different bases of finite fields in combination. Operations of multiplication and multiple squaring in the field GFð2 n Þ and its 4-degree extension, of Tate pairing on supersingular elliptic curve and of final exponentiation are considered separately and in combination. We conclude that the best complexity bound for the pairing and the final exponentiation in the cryptographically significant field GFð2 191 Þ is provided by the combination of the polynomial basis of this field and 1-type optimal basis of the field expansion.

IEEE Transactions on Information Theory, 2016

Recently, a new polynomial basis over binary extension fields was proposed such that the fast Fourier transform (FFT) over such fields can be computed in the complexity of order O(n lg(n)), where n is the number of points evaluated in FFT. In this work, we reformulate this FFT algorithm such that it can be easier understood and be extended to develop frequencydomain decoding algorithms for (n = 2 m , k) systematic Reed-Solomon (RS) codes over F2m , m ∈ Z + , with n − k a power of two. First, the basis of syndrome polynomials is reformulated in the decoding procedure so that the new transforms can be applied to the decoding procedure. A fast extended Euclidean algorithm is developed to determine the error locator polynomial. The computational complexity of the proposed decoding algorithm is O(n lg(n − k) + (n − k) lg 2 (n − k)), improving upon the best currently available decoding complexity O(n lg 2 (n) lg lg(n)), and reaching the best known complexity bound that was established by Justesen in 1976. However, Justesen's approach is only for the codes over some specific fields, which can apply Cooley-Tucky FFTs. As revealed by the computer simulations, the proposed decoding algorithm is 50 times faster than the conventional one for the (2 16 , 2 15) RS code over F 2 16 .