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22
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
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
Finite Field Multiplier Using Redundant Representation
 IEEE Transactions on Computers
, 2002
"... This article presents simple and highly regular architectures for finite field multipliers using a redundant representation. The basic idea is to embed a finite field into a cyclotomic ring which has a basis with the elegant multiplicative structure of a cyclic group. One important feature of our ar ..."
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Cited by 21 (1 self)
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This article presents simple and highly regular architectures for finite field multipliers using a redundant representation. The basic idea is to embed a finite field into a cyclotomic ring which has a basis with the elegant multiplicative structure of a cyclic group. One important feature of our architectures is that they provide areatime tradeoffs which enable us to implement the multipliers in a partialparallel/hybrid fashion. This hybrid architecture has great significance in its VLSI implementation in very large fields. The squaring operation using the redundant representation is simply a permutation of the coordinates. It is shown that when there is an optimal normal basis, the proposed bitserial and hybrid multiplier architectures have very low space complexity. Constant multiplication is also considered and is shown to have advantage in using the redundant representation. Index terms: Finite field arithmetic, cyclotomic ring, redundant set, normal basis, multiplier, squaring.
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.
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
C.: Itoh–Tsujii inversion in standard basis and its application in cryptography and codes
 Des. Codes Cryptogr
, 2002
"... Abstract. This contribution is concerned with a generalization of Itoh and Tsujii’s algorithm for inversion in extension fields GF (q m). Unlike the original algorithm, the method introduced here uses a standard (or polynomial) basis representation. The inversion method is generalized for standard b ..."
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Cited by 15 (2 self)
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Abstract. This contribution is concerned with a generalization of Itoh and Tsujii’s algorithm for inversion in extension fields GF (q m). Unlike the original algorithm, the method introduced here uses a standard (or polynomial) basis representation. The inversion method is generalized for standard basis representation and relevant complexity expressions are established, consisting of the number of extension field multiplications and exponentiations. As the main contribution, for three important classes of fields we show that the Frobenius map can be explored to perform the exponentiations required for the inversion algorithm efficiently. As an important consequence, Itoh and Tsujii’s inversion method shows almost the same practical complexity for standard basis as for normal basis representation for the field classes considered.
Efficient Multiplier Architectures for Galois Fields GF(2 4n )
 IEEE Transactions on Computers
, 1998
"... This contribution introduces a new class of multipliers for finite fields GF ((2 n ) 4 ). The architecture is based on a modified version of the KaratsubaOfman algorithm (KOA). By determining optimized field polynomials of degree four, the last stage of the KOA and the modulo reduction can b ..."
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Cited by 13 (0 self)
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This contribution introduces a new class of multipliers for finite fields GF ((2 n ) 4 ). The architecture is based on a modified version of the KaratsubaOfman algorithm (KOA). By determining optimized field polynomials of degree four, the last stage of the KOA and the modulo reduction can be combined. This saves computation and area in VLSI implementations. The new algorithm leads to architectures which show a considerably improved gate complexity compared to traditional approaches and reduced delay if compared with KOAbased architectures with separate modulo reduction. The new multipliers lead to highly modular architectures an are thus well suited for VLSI implementations. Three types of field polynomials are introduced and conditions for their existence are established. For the small fields where n = 2; 3; : : : ; 8, which are of primary technical interest, optimized field polynomials were determined by an exhaustive search. For each field order, exact space and ti...
Comparison of Arithmetic Architectures for ReedSolomon Decoders in Reconfigurable Hardware
 IEEE Transactions on Computers
, 1997
"... ReedSolomon (RS) error correction codes are being widely used in modern communication systems such as compact disk players or satellite communication links. RS codes rely on arithmetic in finite, or Galois fields. The specific field GF (2 8 ) is of central importance for many practical systems. T ..."
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Cited by 13 (2 self)
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ReedSolomon (RS) error correction codes are being widely used in modern communication systems such as compact disk players or satellite communication links. RS codes rely on arithmetic in finite, or Galois fields. The specific field GF (2 8 ) is of central importance for many practical systems. The most costly, and thus most critical, elementary operations in RS decoders are multiplication and inversion in Galois fields. Although there have been considerable efforts in the area of Galois field arithmetic architectures, there appears to be very little reported work for Galois field arithmetic for reconfigurable hardware. This contribution provides a systematic comparison of two promising arithmetic architecture classes. The first one is based on a standard base representation, and the second one is based on composite fields. For both classes a multiplier and an inverter for GF (2 8 ) are described and theoretical gate counts are provided. Using a design entry based on a VHDL descr...
Subquadratic Computational Complexity Schemes for Extended Binary Field Multiplication Using Optimal Normal Bases
, 2007
"... Based on a recently proposed Toeplitz matrixvector product approach, a subquadratic computational complexity scheme is presented for multiplications in binary extended finite fields using Type I and II optimal normal bases. basis. Index Terms Finite field, subquadratic computational complexity mult ..."
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Cited by 7 (3 self)
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Based on a recently proposed Toeplitz matrixvector product approach, a subquadratic computational complexity scheme is presented for multiplications in binary extended finite fields using Type I and II optimal normal bases. basis. Index Terms Finite field, subquadratic computational complexity multiplication, normal basis, optimal normal