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Efficient Algorithms for Elliptic Curve Cryptosystems
, 1997
"... Elliptic curves are the basis for a relative new class of publickey schemes. It is predicted that elliptic curves will replace many existing schemes in the near future. It is thus of great interest to develop algorithms which allow efficient implementations of elliptic curve crypto systems. This th ..."
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Cited by 66 (9 self)
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Elliptic curves are the basis for a relative new class of publickey schemes. It is predicted that elliptic curves will replace many existing schemes in the near future. It is thus of great interest to develop algorithms which allow efficient implementations of elliptic curve crypto systems. This thesis deals with such algorithms. Efficient algorithms for elliptic curves can be classified into lowlevel algorithms, which deal with arithmetic in the underlying finite field and highlevel algorithms, which operate with the group operation. This thesis describes three new algorithms for efficient implementations of elliptic curve cryptosystems. The first algorithm describes the application of the KaratsubaOfman Algorithm to multiplication in composite fields GF ((2 n ) m ). The second algorithm deals with efficient inversion in composite Galois fields of the form GF ((2 n ) m ). The third algorithm is an entirely new approach which accelerates the multiplication of points which i...
ªLowComplexity Bitparallel Canonical and Normal Basis Multipliers for a Class of Finite Fields,º
 IEEE Trans. Computers
, 1998
"... Abstract—We present a new lowcomplexity bitparallel canonical basis multiplier for the field GF(2 m) generated by an allonepolynomial. The proposed canonical basis multiplier requires m 2 1 XOR gates and m 2 AND gates. We also extend this canonical basis multiplier to obtain a new bitparallel n ..."
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Cited by 37 (8 self)
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Abstract—We present a new lowcomplexity bitparallel canonical basis multiplier for the field GF(2 m) generated by an allonepolynomial. The proposed canonical basis multiplier requires m 2 1 XOR gates and m 2 AND gates. We also extend this canonical basis multiplier to obtain a new bitparallel normal basis multiplier. Index Terms—Finite fields, multiplication, normal basis, canonical basis, allonepolynomial. 1
Mastrovito Multiplier for All Trinomials
 IEEE Trans. Computers
, 1999
"... An e cient algorithm for the multiplication in GF (2m)was introduced by Mastrovito. The space complexity of the Mastrovito multiplier for the irreducible trinomial x m + x +1was given as m 2, 1 XOR and m 2 AND gates. In this paper, we describe an architecture based on a new formulation of the multip ..."
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Cited by 36 (3 self)
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An e cient algorithm for the multiplication in GF (2m)was introduced by Mastrovito. The space complexity of the Mastrovito multiplier for the irreducible trinomial x m + x +1was given as m 2, 1 XOR and m 2 AND gates. In this paper, we describe an architecture based on a new formulation of the multiplication matrix, and show that the Mastrovito multiplier for the generating trinomial x m + x n +1, where m 6 = 2n, also requires m 2, 1 XOR and m 2 AND gates. However, m 2, m=2 XOR gates are su cient when the generating trinomial is of the form x m + x m=2 +1 for an even m. We also calculate the time complexity of the proposed Mastrovito multiplier, and give design examples for the irreducible trinomials x 7 + x 4 + 1 and x 6 + x 3 +1.
A Very Compact SBox for AES
 in Proceedings of CHES 2005, ser. LNCS
, 2005
"... Abstract. A key step in the Advanced Encryption Standard (AES) algorithm is the “Sbox. ” Many implementations of AES have been proposed, for various goals, that effect the Sbox in various ways. In particular, the most compact implementations to date of Satoh et al.[1] and Mentens et al.[2] perform ..."
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Cited by 34 (1 self)
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Abstract. A key step in the Advanced Encryption Standard (AES) algorithm is the “Sbox. ” Many implementations of AES have been proposed, for various goals, that effect the Sbox in various ways. In particular, the most compact implementations to date of Satoh et al.[1] and Mentens et al.[2] perform the 8bit Galois field inversion of the Sbox using subfields of 4 bits and of 2 bits. Our work refines this approach to achieve a more compact Sbox. We examined many choices of basis for each subfield, not only polynomial bases as in previous work, but also normal bases, giving 432 cases. The isomorphism bit matrices are fully optimized, improving on the “greedy algorithm. ” Introducing some NOR gates gives further savings. The best case improves on [1] by 20%. This decreased size could help for arealimited hardware implementations, e.g., smart cards, and to allow more copies of the Sbox for parallelism and/or pipelining of AES. 1
A SideChannel Analysis Resistant Description of the AES Sbox
 In Fast Software Encryption, 12th International Workshop, FSE 2005
"... Abstract. So far, efficient algorithmic countermeasures to secure the AES algorithm against (firstorder) differential sidechannel attacks have been very expensive to implement. In this article, we introduce a new masking countermeasure which is not only secure against firstorder sidechannel attac ..."
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Cited by 25 (2 self)
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Abstract. So far, efficient algorithmic countermeasures to secure the AES algorithm against (firstorder) differential sidechannel attacks have been very expensive to implement. In this article, we introduce a new masking countermeasure which is not only secure against firstorder sidechannel attacks, but which also leads to relatively small implementations compared to other masking schemes implemented in dedicated hardware. Our approach is based on shifting the computation of the finite field inversion in the AES Sbox down to GF (4). In this field, the inversion is a linear operation and therefore it is easy to mask. Summarizing, the new masking scheme combines the concepts of multiplicative and additive masking in such a way that security against firstorder sidechannel attacks is maintained, and that small implementations in dedicated hardware can be achieved.
SOBER: A Stream Cipher based on Linear Feedback over GF(2 g )
, 1999
"... This paper introduces a mechanism for creating a family of stream ciphers based on Linear Feedback Shift Registers over the Galois Finite Field of order 2 ..."
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Cited by 23 (7 self)
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This paper introduces a mechanism for creating a family of stream ciphers based on Linear Feedback Shift Registers over the Galois Finite Field of order 2
A generalized method for constructing subquadratic complexity GF(2 k ) multipliers
 IEEE Transactions on Computers
, 2004
"... We introduce a generalized method for constructing subquadratic complexity multipliers for even characteristic field extensions. The construction is obtained by recursively extending short convolution algorithms and nesting them. To obtain the short convolution algorithms the Winograd short convolu ..."
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Cited by 22 (0 self)
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We introduce a generalized method for constructing subquadratic complexity multipliers for even characteristic field extensions. The construction is obtained by recursively extending short convolution algorithms and nesting them. To obtain the short convolution algorithms the Winograd short convolution algorithm is reintroduced and analyzed in the context of polynomial multiplication. We present a recursive construction technique that extends any d point multiplier into an n = d k point multiplier with area that is subquadratic and delay that is logarithmic in the bitlength n. We present a thorough analysis that establishes the exact space and time complexities of these multipliers. Using the recursive construction method we obtain six new constructions, among which one turns out to be identical to the Karatsuba multiplier. All six algorithms have subquadratic space complexities and two of the algorithms have significantly better time complexities than the Karatsuba algorithm. Keywords: Bitparallel multipliers, finite fields, Winograd convolution 1
Mastrovito multiplier for general irreducible polynomials
 IEEE Transactions on Computers
, 2000
"... We present a new formulation of the Mastrovito multiplication matrix for the field GF(2 m) generated by an arbitrary irreducible polynomial. We study in detail several specific types of irreducible polynomials, e.g., trinomials, allonepolynomials, and equallyspacedpolynomials, and obtain the tim ..."
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Cited by 20 (0 self)
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We present a new formulation of the Mastrovito multiplication matrix for the field GF(2 m) generated by an arbitrary irreducible polynomial. We study in detail several specific types of irreducible polynomials, e.g., trinomials, allonepolynomials, and equallyspacedpolynomials, and obtain the time and space complexity of these designs. Particular examples, illustrating the properties of the proposed architecture, are also given. The complexity results established in this paper match the best complexity results known to date. The most important new result is the space complexity of the Mastrovito multiplier for an equallyspacedpolynomial, which is found as (m 2 − ∆) XOR gates and m 2 AND gates, where ∆ is the spacing factor.
Elliptic curve cryptosystems on reconfigurable hardware
 MASTER’S THESIS, WORCESTER POLYTECHNIC INST
, 1998
"... Security issues will play an important role in the majority of communication and computer networks of the future. As the Internet becomes more and more accessible to the public, security measures will have to be strengthened. Elliptic curve cryptosystems allow for shorter operand lengths than other ..."
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Cited by 19 (0 self)
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Security issues will play an important role in the majority of communication and computer networks of the future. As the Internet becomes more and more accessible to the public, security measures will have to be strengthened. Elliptic curve cryptosystems allow for shorter operand lengths than other publickey schemes based on the discrete logarithm in finite fields and the integer factorization problem and are thus attractive for many applications. This thesis describes an implementation of a crypto engine based on elliptic curves. The underlying algebraic structures are composite Galois fields GF((2 n) m) in a standard base representation. As a major new feature, the system is developed for a reconfigurable platform based on Field Programmable Gate Arrays (FPGAs). FPGAs combine the flexibility of software solutions with the security of traditional hardware implementations. In particular, it is possible to easily change all algorithm parameters such as curve coefficients, field order, or field representation. The thesis deals with the design and implementation of elliptic curve point multiplicationarchitectures. The architectures are described in VHDL and mapped to Xilinx FPGA devices. Architectures over Galois fields of different order and representation were implemented and compared. Area and timing measurements are provided for all architectures. It is shown that a full point multiplication on elliptic curves of realworld size can be implemented on commercially available FPGAs.
Parallel Multipliers Based on Special Irreducible Pentanomials
 IEEE Transactions on Computers
, 2003
"... The stateoftheart Galois field GF(2m)multipliers offer advantageous space and time complexities when the field is generated by some special irreducible polynomial. To date, the best complexity results have been obtained when the irreducible polynomial is either a trinomial or an equallyspace pol ..."
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Cited by 17 (0 self)
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The stateoftheart Galois field GF(2m)multipliers offer advantageous space and time complexities when the field is generated by some special irreducible polynomial. To date, the best complexity results have been obtained when the irreducible polynomial is either a trinomial or an equallyspace polynomial (ESP). Unfortunately, there exist only a few irreducible ESPs in the range of interest for most of the applications, e.g., errorcorrecting codes, computer algebra, and elliptic curve cryptography. Furthermore, it is not always possible to find an irreducible trinomial of degree m in this range. For those cases, where neither an irreducible trinomial or an irreducible ESP exists, the use of irreducible pentanomials has been suggested. Irreducible pentanomials are abundant, 2and there are several eligible candidates for a given m. Inthis paper, we promote the use of two special types of irreducible pentanomials. We propose new Mastrovito and dual basis multiplier architectures based on these special irreducible pentanomials, and give rigorous analyses of their space and time complexity. Index Terms: Finite fields arithmetic, parallel multipliers, pentanomials, multipliers for GF(2m). 1