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24
Redundant trinomials for finite fields of characteristic 2
 Proceedings of ACISP 05, LNCS 3574
, 2005
"... Abstract. In this paper we introduce socalled redundant trinomials to represent elements of nite elds of characteristic 2. The concept is in fact similar to almost irreducible trinomials introduced by Brent and Zimmermann in the context of random numbers generators in [BZ 2003]. See also [BZ]. In f ..."
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Abstract. In this paper we introduce socalled redundant trinomials to represent elements of nite elds of characteristic 2. The concept is in fact similar to almost irreducible trinomials introduced by Brent and Zimmermann in the context of random numbers generators in [BZ 2003]. See also [BZ]. In fact, Blake et al. [BGL 1994, BGL 1996] and Tromp et al. [TZZ 1997] explored also similar ideas some years ago. However redundant trinomials have been discovered independently and this paper develops applications to cryptography, especially based on elliptic curves. After recalling well known techniques to perform e cient arithmetic in extensions of F2, we describe redundant trinomial bases and discuss how to implement them e ciently. They are well suited to build F2n when no irreducible trinomial of degree n exists. Depending on n ∈ [2, 10, 000] tests with NTL show that improvements for squaring and exponentiation are respectively up to 45 % and 25%. More attention is given to relevant extension degrees for doing elliptic and hyperelliptic curve cryptography. For this range, a scalar multiplication can be speeded up by a factor up to 15%. 1.
Software Multiplication Using Gaussian Normal Bases
 IEEE Transactions on Computers
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Irreducible polynomials of maximum weight
 Util. Math
"... Abstract. We establish some necessary conditions for the existence of irreducible polynomials of degree n and weight n over F2. Such polynomials can be used to efficiently implement multiplication in F2n. We also provide a simple proof of a result of Bluher concerning the reducibility of a certain f ..."
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Abstract. We establish some necessary conditions for the existence of irreducible polynomials of degree n and weight n over F2. Such polynomials can be used to efficiently implement multiplication in F2n. We also provide a simple proof of a result of Bluher concerning the reducibility of a certain family of polynomials. 1.
Montgomery reduction algorithm for modular multiplication using lowweight polynomial form integers
 In Kornerup and Muller [870
"... Abstract. We extend lowweight polynomial form integers (LWPFIs) presented in [5]. An LWPFI p is an integer expressed as a degreel, monic polynomial such that p = t l + fl−1t l + · · · + f1t + f0, where t can be any positive integer. In [5], fi’s are limited to 0 and ±1, but here we let fi  ≤ ..."
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Abstract. We extend lowweight polynomial form integers (LWPFIs) presented in [5]. An LWPFI p is an integer expressed as a degreel, monic polynomial such that p = t l + fl−1t l + · · · + f1t + f0, where t can be any positive integer. In [5], fi’s are limited to 0 and ±1, but here we let fi  ≤ ξ for some small positive integer ξ. In modular multiplication based on LWPFI, elements in Zp are expressed in polynomial in t and multiplication is performed in Z[t]/f(t). The coefficients must be reduced for subsequent modular multiplications. In [5], a coefficient reduction algorithm based on a division algorithm derived from the Barrett reduction algorithm is presented. In this report, we present a coefficient reduction algorithm based on the Montgomery reduction algorithm and its detailed analysis results. Bounds on the input and output of our coefficient reduction algorithm is carefully analyzed. We give conditions for eliminating the final subtractions at the end of the Montgomery reduction algorithm. In addition, we present efficient modular addition and subtraction methods using LWPFI moduli. 1
Software multiplication using normal bases
 Dept. of Combinatorics and Optimization, Univ. of
, 2004
"... Fast algorithms for multiplication in finite fields are required for several cryptographic applications, in particular for implementing elliptic curve operations over the NIST recommended binary fields. In this paper we present new software algorithms for efficient multiplication over the binary fie ..."
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Fast algorithms for multiplication in finite fields are required for several cryptographic applications, in particular for implementing elliptic curve operations over the NIST recommended binary fields. In this paper we present new software algorithms for efficient multiplication over the binary field F2m that use a Gaussian normal basis representation. Two approaches are presented, direct normal basis multiplication, and a method that exploits a mapping to a ring where fast polynomialbased techniques can be employed. Our analysis including experimental results on an Intel Pentium family processor shows that the new algorithms are faster and can use memory more efficiently than previous methods. Despite significant improvements, we conclude that the penalty in multiplication is still sufficiently large to discourage the use of normal bases in software implementations of elliptic curve systems. Key words Multiplication in F2 m, normal basis, Gaussian normal basis, elliptic curve cryptography. 1
Implementation and Analysis of Elliptic Curve Cryptosystems over Polynomial basis and ONB
"... Abstract — Polynomial bases and normal bases are both used for elliptic curve cryptosystems, but field arithmetic operations such as multiplication, inversion and doubling for each basis are implemented by different methods. In general, it is said that normal bases, especially optimal normal bases ( ..."
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Abstract — Polynomial bases and normal bases are both used for elliptic curve cryptosystems, but field arithmetic operations such as multiplication, inversion and doubling for each basis are implemented by different methods. In general, it is said that normal bases, especially optimal normal bases (ONB) which are special cases on normal bases, are efficient for the implementation in hardware in comparison with polynomial bases. However there seems to be more examined by implementing and analyzing these systems under similar condition. In this paper, we designed field arithmetic operators for each basis over GF(2 233), which field has a polynomial basis recommended by SEC2 and a typeII ONB both, and analyzed these implementation results. And, in addition, we predicted the efficiency of two elliptic curve cryptosystems using these field arithmetic operators.
Software implementation of arithmetic in F3 m
 International Workshop on the Arithmetic of Finite Fields (WAIFI 2007), volume 4547 of Lecture Notes in Computer Science
, 2007
"... Abstract. Fast arithmetic for characteristic three finite fields F3 m is desirable in pairingbased cryptography because there is a suitable family of elliptic curves over F3 m having embedding degree 6. In this paper we present some structure results for Gaussian normal bases of F3 m, and use the r ..."
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Abstract. Fast arithmetic for characteristic three finite fields F3 m is desirable in pairingbased cryptography because there is a suitable family of elliptic curves over F3 m having embedding degree 6. In this paper we present some structure results for Gaussian normal bases of F3 m, and use the results to devise faster multiplication algorithms. We carefully compare multiplication in F3 m using polynomial bases and Gaussian normal bases. Finally, we compare the speed of encryption and decryption for the BonehFranklin and SakaiKasahara identitybased encryption schemes at the 128bit security level, in the case where supersingular elliptic curves with embedding degrees 2, 4 and 6 are employed. 1.
GENERALISED MERSENNE NUMBERS REVISITED
"... Abstract. Generalised Mersenne Numbers (GMNs) were defined by Solinas in 1999 and feature in the NIST (FIPS 1862) and SECG standards for use in elliptic curve cryptography. Their form is such that modular reduction is extremely efficient, thus making them an attractive choice for modular multiplica ..."
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Abstract. Generalised Mersenne Numbers (GMNs) were defined by Solinas in 1999 and feature in the NIST (FIPS 1862) and SECG standards for use in elliptic curve cryptography. Their form is such that modular reduction is extremely efficient, thus making them an attractive choice for modular multiplication implementation. However, the issue of residue multiplication efficiency seems to have been overlooked. Asymptotically, using a cyclic rather than a linear convolution, residue multiplication modulo a Mersenne number is twice as fast as integer multiplication; this property does not hold for prime GMNs, unless they are of Mersenne’s form. In this work we exploit an alternative generalisation of Mersenne numbers for which an analogue of the above property — and hence the same efficiency ratio — holds, even at bitlengths for which schoolbook multiplication is optimal, while also maintaining very efficient reduction. Moreover, our proposed primes are abundant at any bitlength, whereas GMNs are extremely rare. Our multiplication and reduction algorithms can also be easily parallelised, making our arithmetic particularly suitable for hardware implementation. Furthermore, the field representation we propose also naturally protects against sidechannel attacks, including timing attacks, simple power analysis and differential power analysis, which is essential in many cryptographic scenarios, in constrast to GMNs. 1.
�2007 SWPS COMPLEXITY ANALYSIS FOR 4INPUT/1OUTPUT FPGAS APPLIED TO MULTIPLIER DESIGNS
"... Abstract. Some algorithms are more efficient than others. The complexity of an algorithm is a function describing the efficiency of the algorithm which has two measures: Space Complexity and Time Complexity. In this paper, we present complexity analysis for FPGA based designs which is based on 4inp ..."
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Abstract. Some algorithms are more efficient than others. The complexity of an algorithm is a function describing the efficiency of the algorithm which has two measures: Space Complexity and Time Complexity. In this paper, we present complexity analysis for FPGA based designs which is based on 4input and 1output LUT structure followed by the majority of FPGA manufacturers. The same procedure is then applied to KaratsubaOffman Multiplier (KOM) because of two reasons. Firstly, due to the increased use of FPGAs especially for security applications, it seems logical to compare various architectures for their efficiencies in FPGAs. Secondly, for diverse security applications, it provides a prior estimation to hardware resources and achievable timing. We consider a 4input and 1output structure as a basic building block available in majority of FPGAs by different FPGA manufacturers. We then compare our theoretical and experimental results for KOM in FPGAs which are fairly convincible. Key words. complexity analysis, field programmable gate arrays (FPGAs), KaratsubaOfman multiplier, cryptography, hardware implementations
ABSTRACT ON THE REPRESENTATION OF FINITE FIELDS
, 2010
"... Date: I hereby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this wo ..."
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Date: I hereby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work.