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A new class of irreducible pentanomials for polynomial based multipliers in binary fields [article]

Gustavo Banegas and Ricardo Custodio and Daniel Panario
2018 arXiv   pre-print
We introduce a new class of irreducible pentanomials over F_2 of the form f(x) = x^2b+c + x^b+c + x^b + x^c + 1. Let m=2b+c and use f to define the finite field extension of degree m.  ...  We give the exact number of operations required for computing the reduction modulo f. We also provide a multiplier based on Karatsuba algorithm in F_2[x] combined with our reduction process.  ...  In this paper, we present a new class of pentanomials over F 2 , defined by x 2b+c + x b+c + x b + x c + 1.  ... 
arXiv:1806.00432v1 fatcat:r5w3vglx6bex3cleyrmswcvk5a

On Low Complexity Bit Parallel Polynomial Basis Multipliers [chapter]

Arash Reyhani-Masoleh, M. Anwar Hasan
2003 Lecture Notes in Computer Science  
Based on a number of important classes of irreducible polynomials, we give exact complexity analyses of the multiplier gate count and time delay.  ...  Time and space complexities of such a multiplier heavily depend on the field defining irreducible polynomials.  ...  This work has been supported in part by an NSERC postdoctoral fellowship awarded to A. Reyhani-Masoleh and in part by an NSERC grant awarded to M. A. Hasan.  ... 
doi:10.1007/978-3-540-45238-6_16 fatcat:o7k7xpn2krh5tj6qtaeb7gor4q

Low complexity bit parallel architectures for polynomial basis multiplication over GF(2m)

A. Reyhani-Masoleh, M.A. Hasan
2004 IEEE transactions on computers  
In this effect, first we derive a new formulation for polynomial basis multiplication in terms of the reduction matrix Q.  ...  The main advantage of this new formulation is that it can be used with any field defining irreducible polynomial.  ...  ACKNOWLEDGMENTS The authors would like to thank the reviewers for their comments. The work has been supported in part by an NSERC postdoctoral fellowship to A. Reyhani-Masoleh.  ... 
doi:10.1109/tc.2004.47 fatcat:clt3j6zcmrcrhm6e4suqdvkha4

Bit-Serial and Bit-Parallel Montgomery Multiplication and Squaring over GF(2^m)

Arash Hariri, Arash Reyhani-Masoleh
2009 IEEE transactions on computers  
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 paper, we consider the Montgomery multiplication in the binary extension fields and study different structures of bit-serial and bit-parallel multipliers.  ...  ACKNOWLEDGMENTS The authors would like to thank the reviewers for their constructive comments. This work has been supported in part by an NSERC Discovery grant awarded to Arash Reyhani-Masoleh.  ... 
doi:10.1109/tc.2009.70 fatcat:huco55p7tfcttmadd6rpckfq5y

Fast Bit-Parallel Binary Multipliers Based on Type-I Pentanomials

Jose L. Imana
2018 IEEE transactions on computers  
Imaña 4 Abstract-In this paper, a fast implementation of bit-parallel polynomial basis (PB) 5 multipliers over the binary extension field GF ð2 m Þ generated by type-I irreducible 6 pentanomials is presented  ...  The complexity 26 of the multiplier mainly depends on the irreducible polynomial 27 fðyÞ selected for the finite field.  ...  The elements of the binary extension field GF ð2 m Þ can 88 be represented in the polynomial basis f1; x; . . . ; x mÀ1 g, where x is a 89 root of the irreducible generating polynomial fðyÞ.  ... 
doi:10.1109/tc.2017.2778730 fatcat:jtpsie7c3nfr7of3t2jwlncfwq

Space Efficient GF(2m) Multiplier for Special Pentanomials based on n-Term Karatsuba Algorithm

Sun-Mi Park, Ku-Young Chang, Dowon Hong, Changho Seo
2020 IEEE Access  
Recently, new multiplication schemes over the binary extension field GF(2 m ) based on an n-term Karatsuba algorithm have been proposed for irreducible trinomials.  ...  In this paper, we extend these schemes for trinomials to any irreducible polynomials.  ...  MULTIPLIER FOR TYPE P.1 PENTANOMIALS In this section, we propose a new multiplier for type P.1 pentanomials.  ... 
doi:10.1109/access.2020.2971702 fatcat:fha5ntecdredbetgeck7mqj5gi

18 Years of Redundant Basis Multipliers over Galois Field

Roma Chourasia, Kavita Khare
2016 International Journal of Computer Applications  
Based on review the Word Level Redundant Basis multiplier is the most efficient among all multipliers in terms of hardware utilization.  ...  Different techniques used so far for the implementation of redundant basis multipliers over Galois Field are explored here.  ...  Multiplication in the rings is faster than multiplication in a field defined by pentanomials because the new rings are based on 3 and 4-term polynomials.  ... 
doi:10.5120/ijca2016908878 fatcat:xdidogzd6beddas64nds6maoze

A New Approach to Subquadratic Space Complexity Parallel Multipliers for Extended Binary Fields

Haining Fan, M. Anwar Hasan
2007 IEEE transactions on computers  
Based on Toeplitz matrix-vector products and coordinate transformation techniques, we present a new scheme for subquadratic space complexity parallel multiplication in GF ð2 n Þ using the shifted polynomial  ...  This design algorithm can be modified for the construction of most of the subquadratic space complexity multipliers previously reported in the literature.  ...  ACKNOWLEDGMENTS The authors thank the reviewers for their careful reading and useful comments. The work was supported in part by NSERC through grants awarded to Dr. Hasan.  ... 
doi:10.1109/tc.2007.19 fatcat:tzzv5soclzdwlanxbxgtfrpvua

Low-Complexity Parallel Systolic Montgomery Multipliers over GF(2m) Using Toeplitz Matrix-Vector Representation

C.-Y. LEE
2008 IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences  
The results reveal that our proposed multipliers reduce the space complexity of approximately 15% compared with an existing systolic Montgomery multiplier for trinomials.  ...  In this paper, a generalized Montgomery multiplication algorithm in GF(2 m ) using the Toeplitz matrix-vector representation is presented.  ...  The work was supported in part by the National Science Council of the Republic of China under grant number NSC 96-2221-E-262-008-MY2.  ... 
doi:10.1093/ietfec/e91-a.6.1470 fatcat:75kzhk35frhovpch6qryzyshcm

Fast Asymptotic Square Root for Two Types of Special Pentanomials

Yu Zhang, Yin Li, Qing Chen
2019 IEEE Access  
Specifically, for the field GF(2 m ) is defined by two type of specific irreducible pentanomials, i.e., Type C.1 and Type C.2 pentanomials, we derived explicit formulae for space and time complexities  ...  Inspired by the Montgomery and generalized polynomial basis (GPB) squaring operation, we introduce and study the notion of asymptotic square root over binary extension fields GF(2 m ).  ...  Particularly, we have proposed a new type of asymptotic square root operation for two classes of irreducible pentanomials.  ... 
doi:10.1109/access.2019.2911012 fatcat:xbachi4ym5dohnxo63slvgvo5e

Fault Detection Architectures for Field Multiplication Using Polynomial Bases

A. Reyhani-Masoleh, M.A. Hasan
2006 IEEE transactions on computers  
In this paper, we propose new architectures to detect erroneous outputs caused by certain types of faults in bit-parallel and bit-serial polynomial basis multipliers over finite fields of characteristic  ...  Although the issue of detecting soft errors in registers is not considered, the proposed schemes have the advantage that they can be used with any irreducible binary polynomial chosen to define the finite  ...  This work has been supported in part by an NSERC Discovery grant awarded to A. Reyhani-Masoleh and in part by NSERC Discovery and Strategic grants awarded to M. A. Hasan.  ... 
doi:10.1109/tc.2006.147 fatcat:exrsfrdxsfae5m7pcb6qrbyhj4

Design and Implementation of a Sequential Polynomial Basis Multiplier over GF(2m)

2017 KSII Transactions on Internet and Information Systems  
The proposed sequential multiplier supports multiplication of any two arbitrary finite field elements over GF(2 m ) for generic irreducible polynomials, therefore made versatile.  ...  In this paper, a modified serial multiplication algorithm with interleaved modular reduction is proposed, which allows for an efficient realization of a sequential polynomial basis multiplier.  ...  The multipliers based on trinomials and pentanomials have relatively lower hardware complexity when compared to other classes of irreducible polynomials due to less number of terms and consequently lower  ... 
doi:10.3837/tiis.2017.05.021 fatcat:lstwstb4avabxjnw7zfd643fzi

Area- Efficient VLSI Implementation of Serial-In Parallel-Out Multiplier Using Polynomial Representation in Finite Field GF(2m) [article]

Saeideh Nabipour, Gholamreza Zare Fatin, Javad Javidan
2022 arXiv   pre-print
The reduced finite field multiplier complexity is achieved by means of utilizing logic NAND gate in a particular architecture.  ...  Finite field multiplier is regarded as the bottleneck arithmetic unit for such applications and it is the most complicated operation over finite field GF(2m) which requires a huge amount of logic resources  ...  In [26] presented three small classes of irreducible polynomials for lowcomplexity bit-parallel multipliers.  ... 
arXiv:2007.08284v2 fatcat:othrtcbrr5a3ddy2zvond2l7pu

Low-Space Complexity Digit-Serial Multiplier Based on Modified Polynomial Basis Over GF(2m)

Jeng-Shyang Pan, Shu-Xia Dong, Chun-Sheng Yang
2017 Journal of Information Hiding and Multimedia Signal Processing  
In this paper, in order to reduce the complexities of multiplication, a new polynomial basis is proposed, which is generated by the irreducible trinomial and called modified polynomial basis (MPB).  ...  The multiplication is one of the most time-consuming and hardware-consuming operations in finite field for the applications of elliptic curve cryptography.  ...  In addition to the trinomial and pentanomial, there are many variants. Many multipliers based on polynomial basis or it's variants have been studied in [3, 4, 5] .  ... 
dblp:journals/jihmsp/0001DY17 fatcat:5hlimclfqvdhvn3p2cq72hci6i

Bit-serial and digit-serial GF(2m) Montgomery multipliers using linear feedback shift registers

M. Morales-Sandoval, C. Feregrino-Uribe, P. Kitsos
2011 IET Computers & Digital Techniques  
The proposed multipliers are for different classes of irreducible polynomials: general, all one polynomials (AOP), pentanomials and trinomials.  ...  This work presents novel multipliers for Montgomery multiplication defined on binary fields GF(2 m ).  ...  multiplier discussed in previous sections using four kinds of irreducible polynomials: general, AOP, trinomials and pentanomials.  ... 
doi:10.1049/iet-cdt.2010.0021 fatcat:tpkee25zubdj7le7nm7zpdgy3u
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