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### Faster polynomial multiplication over finite fields [article]

David Harvey, Joris van der Hoeven, Grégoire Lecerf
2014 arXiv   pre-print
Let p be a prime, and let M_p(n) denote the bit complexity of multiplying two polynomials in F_p[X] of degree less than n.  ...  The cost is thus M p 0 (2 n ) + O(n lg p), where M p 0 (d) denotes the cost of a multiplication in F p [X]/(X d ¡ 1). Arithmetic in finite fields Let p be a prime and let > 1.  ...  Our techniques also give rise to new strategies for polynomial evaluationinterpolation over F q . This may for instance be applied to the ecient multiplication of polynomial matrices over F q .  ...

### Faster Polynomial Multiplication over Finite Fields

David Harvey, Joris Van Der Hoeven, Grégoire Lecerf
2017 Journal of the ACM
In section 3, we recall basic complexity results for arithmetic in finite fields.  ...  In this paper we are mainly interested in the case that R is the finite field F p = Z / p Z for some prime p.  ...

### Fast Three-Input Multipliers over Small Composite Fields for Multivariate Public Key Cryptography

Haibo Yi, Weijian Li
2015 International Journal of Security and Its Applications
Third, our multipliers adapt table look-up and polynomial basis, since they are faster over specific fields, respectively. We demonstrate the improvement of our design mathematically.  ...  Finite field multiplication is playing a crucial role in the implementations of multivariate cryptography and most of them use two-input multipliers.  ...  Since addition is much faster than constant multiplication over finite fields, we have C MA  . Therefore,  is proved.  ...

### Polynomial Multiplication over Binary Fields Using Charlier Polynomial Representation with Low Space Complexity [chapter]

Sedat Akleylek, Murat Cenk, Ferruh Özbudak
2010 Lecture Notes in Computer Science
One can obtain binomial or trinomial irreducible polynomials in Charlier polynomial representation which allows us faster modular reduction over binary fields when there is no desirable such low weight  ...  In this paper, we give a new way to represent certain finite fields GF (2 n ). This representation is based on Charlier polynomials.  ...  The binary extension field multiplication can be performed in two steps: polynomial multiplication over GF (2) and modular reduction over GF (2 n ).  ...

### Page 418 of IEEE Transactions on Computers Vol. 52, Issue 4 [page]

2003 IEEE Transactions on Computers
We consider rings modulo trinomials and 4-term polynomials. In each case, we show that our multiplier is faster than multipliers over elements in a finite field defined by irreducible pentanomials.  ...  However, this also results in faste multipliers as the ring elements are defined using simpk polynomials over the finite field GF(2).  ...

### EFFICIENT OPERATIONS IN LARGE FINITE FIELDS FOR ELLIPTIC CURVE CRYPTOGRAPHIC

Yan-Haw Chen, Chien-Hsing Huang
2020 International Journal of Engineering Technologies and Management Research
An efficient method to compute the finite field multiplication for Elliptic Curve point multiplication at high speed encryption of the message is presented.  ...  The modified Horner rule method is not only to finite field operations but also to Elliptic curve scalar multiplication in the encryption and decryption.  ...  The industry uses Elliptic curve groups over the large finite fields of GF(2m) and GF(p), Koblitz EC groups in GF(2m) (Koblitz, 1987) faster than GF(p).  ...

### On square-free factorization of multivariate polynomials over a finite field

Laurent Bernardin
1997 Theoretical Computer Science
In this paper we present a new deterministic algorithm for computing the square-free decomposition of multivariate polynomials with coefficients from a finite field.  ...  We will show that the modular approach presented by Yun has no significant performance advantage over our algorithm.  ...  Therefore, we can compute the square-free decompositions of bl and b2 over a finite field of characteristic 7 much faster than over other fields.  ...

### Practical fast algorithm for finite field arithmetics using group rings

Makoto Matsumoto, Shigehiro Tagami
2004 Hiroshima Mathematical Journal
This paper studies a fast algorithm for finite field arithmetics, by representing a finite field as a residue of a group ring of a finite cyclic group, where the frobenius (q-th power) operation is e‰ciently  ...  When the characteristic of the field is greater than 2, our algorithm is often much faster than a standard method (NTL) in computing inverse and power.  ...  Hellekalek for informing us of NTL and other fast algorithms for finite field arithmetics.  ...

### On the Virtues of Generic Programming for Symbolic Computation [chapter]

Xin Li, Marc Moreno Maza, Éric Schost
2007 Lecture Notes in Computer Science
For instance, the improved implementation of square-free factorization in AXIOM is 7 times faster than the one in Maple and very close to the one in MAGMA.  ...  The purpose of this study is to measure the impact of C level code polynomial arithmetic on the performances of AXIOM highlevel algorithms, such as polynomial factorization.  ...  To do so, we first observe that dense univariate and multivariate polynomials over finite fields play a central role in computer algebra, thanks to modular algorithms.  ...

### Improving the coding speed of erasure codes with polynomial ring transforms [article]

Jonathan Detchart, Jérôme Lacan
2017 arXiv   pre-print
Most of them use the finite field arithmetic.  ...  In this paper, we propose an implementation and a coding speed evaluation of an original method called PYRIT (PolYnomial RIng Transform) to perform operations between elements of a finite field into a  ...  in a finite field by the multiplication in a ring by using transforms between particular finite fields and polynomial rings.  ...

### Improving the Coding Speed of Erasure Codes with Polynomial Ring Transforms

Jonathan Detchart, Jerome Lacan
2017 GLOBECOM 2017 - 2017 IEEE Global Communications Conference
Most of them use the finite field arithmetic.  ...  In this paper, we propose an implementation and a coding speed evaluation of an original method called PYRIT (PolYnomial RIng Transform) to perform operations between elements of a finite field into a  ...  in a finite field by the multiplication in a ring by using transforms between particular finite fields and polynomial rings.  ...

### Low complexity multiplication in a finite field using ring representation

R. Katti, J. Brennan
2003 IEEE transactions on computers
We consider rings modulo trinomials and 4-term polynomials. In each case, we show that our multiplier is faster than multipliers over elements in a finite field defined by irreducible pentanomials.  ...  Elements of a finite field, GF ð2 m Þ, are represented as elements in a ring in which multiplication is more time efficient.  ...  However, this also results in faster multipliers as the ring elements are defined using simpler polynomials over the finite field GF (2) .  ...

### An Arithmetic over GF (2^5) Tt Implement in ECC

A. R.Rishivarman, M. Thiagarajan B. Parthasarathy
2012 International Journal of Computer Applications
In this paper an efficient arithmetic for operations over elements of GF(2 5 ) represented in normal basis is presented. The arithmetic is applicable in public-key cryptography.  ...  Although the discrete logarithm problem as first employed by them was defined explicitly as the problem of finding logarithms with respect to a generator in the multiplicative group of the integers module  ...  The DLP in such a group is very hard as opposed to the DLP in the multiplicative group over a finite field.  ...

### Design of Low Register all One Polynomial Multipliers Over GF (2m) on FPGA

2019 VOLUME-8 ISSUE-10, AUGUST 2019, REGULAR ISSUE
This paper presents All-one-polynomial (AOP)- based systolic multipliers over GF (2m) need aid as a rule not acknowledged for useful execution for cryptosystems for example, elliptic bend cryptography  ...  Also that, systolic AOP multipliers typically suffer from those issue from the secondary register-complexity, particularly alongside field programmable gate array (FPGA) platforms the place the register  ...  Polynomial basis multiplication over GF (2 m ) Assume there has field F, and the elements a n ,a n−1 ,a n−2 ...  ...

### Finding roots of polynomials over finite fields

S.V. Fedorenko, P.V. Trifonov
2002 IEEE Transactions on Communications
We propose an improved algorithm for finding roots of polynomials over finite fields.  ...  of one exponentiation over the finite field.  ...  Chien search algorithm solves it by evaluation of at all with the time complexity (1) where and are the time complexities of one addition and multiplication in the finite field, respectively.  ...
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