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27
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
A New Approach to Subquadratic Space Complexity Parallel Multipliers for Extended Binary Fields
 IEEE Transactions on Computers
, 2007
"... Based on Toeplitz matrixvector products and coordinate transformation techniques, we present a new scheme for subquadratic space complexity parallel multiplication in GF(2 n) using the shifted polynomial basis. Both the space complexity and the asymptotic gate delay of the proposed multiplier are b ..."
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Cited by 21 (14 self)
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Based on Toeplitz matrixvector products and coordinate transformation techniques, we present a new scheme for subquadratic space complexity parallel multiplication in GF(2 n) using the shifted polynomial basis. Both the space complexity and the asymptotic gate delay of the proposed multiplier are better than those of the best existing subquadratic space complexity parallel multipliers. For example, with n being a power of 2 and 3, the space complexity is about 8 % and 10 % better, while the asymptotic gate delay is about 33 % and 25 % better, respectively. Another advantage of the proposed matrixvector product approach is that it can also be used to design subquadratic space complexity polynomial, dual, weakly dual and triangular basis parallel multipliers. To the best of our knowledge, this is the first time that subquadratic space complexity parallel multipliers are proposed for dual, weakly dual and triangular bases. A recursive design algorithm is also proposed for efficient construction of the proposed subquadratic space complexity multipliers. This design algorithm can be modified for the construction of most of the subquadratic space complexity multipliers previously reported in the literature.
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.
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
Low Complexity Bit Parallel Architectures for Polynomial Basis Multiplication over GF(2 m
 IEEE Transactions on Computers
, 2004
"... Abstract—Representing the field elements with respect to the polynomial (or standard) basis, we consider bit parallel architectures for multiplication over the finite field GFð2 m Þ. In this effect, first we derive a new formulation for polynomial basis multiplication in terms of the reduction matri ..."
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Cited by 17 (2 self)
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Abstract—Representing the field elements with respect to the polynomial (or standard) basis, we consider bit parallel architectures for multiplication over the finite field GFð2 m Þ. 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. Using this formulation, we then develop a generalized architecture for the multiplier and analyze the time and gate complexities of the proposed multiplier as a function of degree m and the reduction matrix Q. To the best of our knowledge, this is the first time that these complexities are given in terms of Q. Unlike most other articles on bit parallel finite field multipliers, here we also consider the number of signals to be routed in hardware implementation and we show that, compared to the wellknown Mastrovito’s multiplier, the proposed architecture has fewer routed signals. In this article, the proposed generalized architecture is further optimized for three special types of polynomials, namely, equally spaced polynomials, trinomials, and pentanomials. We have obtained explicit formulas and complexities of the multipliers for these three special irreducible polynomials. This makes it very easy for a designer to implement the proposed multipliers using hardware description languages like VHDL and Verilog with minimum knowledge of finite field arithmetic. Index Terms—Finite or Galois field, Mastrovito multiplier, allone polynomial, polynomial basis, trinomial, pentanomial and equallyspaced polynomial. 1
Reconfigurable Implementation of Elliptic Curve Crypto Algorithms
 RECONFIGURABLE ARCHITECTURES WORKSHOP, 16TH INTERNATIONAL PARALLEL AND DISTRIBUTED PROCESSING SYMPOSIUM
, 2002
"... For FPGA based coprocessors for elliptic curve cryptography, a significant performance gain can be achieved when hybrid coordinates are used to represent points on the elliptic curve. We provide a new area/performance tradeoff analysis of different hybrid representations over fields of characteristi ..."
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Cited by 13 (0 self)
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For FPGA based coprocessors for elliptic curve cryptography, a significant performance gain can be achieved when hybrid coordinates are used to represent points on the elliptic curve. We provide a new area/performance tradeoff analysis of different hybrid representations over fields of characteristic two. Moreover, we present a new generic cryptoprocessor architecture that can be adapted to various area/performance constraints and finite field sizes, and show how to apply high level synthesis techniques to the controller design.
A new combinational logic minimization technique with applications to cryptology
 of Lecture Notes in Computer Science
, 2010
"... to cryptology. ..."
Parallel Montgomery Multiplication in GF(2 k ) Using Trinomial Residue Arithmetic
 In 17th IEEE Symposium on Computer Arithmetic (ARITH05
, 2005
"... We propose the first general multiplication algorithm in GF(2 k) with a subquadratic area complexity of O(k 8/5) = O(k 1.6). Using the Chinese Remainder Theorem, we represent the elements of GF(2 k); i.e. the polynomials in GF(2)[X] of degree at most k − 1, by their remainder modulo a set of n pair ..."
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Cited by 7 (0 self)
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We propose the first general multiplication algorithm in GF(2 k) with a subquadratic area complexity of O(k 8/5) = O(k 1.6). Using the Chinese Remainder Theorem, we represent the elements of GF(2 k); i.e. the polynomials in GF(2)[X] of degree at most k − 1, by their remainder modulo a set of n pairwise prime trinomials, T1,..., Tn, of degree d and such that nd ≥ k. Our algorithm is based on Montgomery’s multiplication applied to the ring formed by the direct product of the trinomials.
Evaluating Instruction Set Extensions for Fast Arithmetic on Binary Finite Fields
 PROC. INT. CONF. APPLICATIONSPECIFIC SYSTEMS, ARCHITECTURES, AND PROCESSORS (ASAP
, 2004
"... Binary finite fields GF(2^n) are very commonly used in cryptography, particularly in publickey algorithms such as Elliptic Curve Cryptography (ECC). On wordoriented programmable processors, field elements are generally represented as polynomials with coefficients from {0, 1}. Key arithmetic operati ..."
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Cited by 5 (1 self)
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Binary finite fields GF(2^n) are very commonly used in cryptography, particularly in publickey algorithms such as Elliptic Curve Cryptography (ECC). On wordoriented programmable processors, field elements are generally represented as polynomials with coefficients from {0, 1}. Key arithmetic operations on these polynomials, such as squaring and multiplication, are not supported by integeroriented processor architectures. Instead, these are implemented in software, causing a very large fraction of the cryptography execution time to be dominated by a few elementary operations. For example, more than 90% of the execution time of 163bit ECC may be consumed by two simple field operations: squaring and multiplication. A few
1 Block Recombination Approach for Subquadratic Space Complexity Binary Field Multiplication based on Toeplitz MatrixVector Product
"... In this paper, we present a new method for parallel binary finite field multiplication which results in subquadratic space complexity. The method is based on decomposing the building blocks of FanHasan subquadratic Toeplitz matrixvector multiplier. We reduce the space complexity of their architect ..."
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Cited by 5 (3 self)
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In this paper, we present a new method for parallel binary finite field multiplication which results in subquadratic space complexity. The method is based on decomposing the building blocks of FanHasan subquadratic Toeplitz matrixvector multiplier. We reduce the space complexity of their architecture by recombining the building blocks. In comparison to other similar schemes available in the literature, our proposal presents a better space complexity while having the same time complexity. We also show that block recombination can be used for efficient implementation of the GHASH function of Galois Counter Mode (GCM).