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22
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.
Asymmetric squaring formulae
, 2006
"... We present efficient squaring formulae based on the ToomCook multiplication algorithm. The latter always requires at least one nontrivial constant division in the interpolation step. We show such nontrivial divisions are not needed in the case two operands are equal for three, four and fiveway s ..."
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Cited by 11 (0 self)
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We present efficient squaring formulae based on the ToomCook multiplication algorithm. The latter always requires at least one nontrivial constant division in the interpolation step. We show such nontrivial divisions are not needed in the case two operands are equal for three, four and fiveway squarings. Our analysis shows that our 3way squaring algorithms have much less overhead than the best known 3way ToomCook algorithm. Our experimental results show that one of our new 3way squaring methods performs faster than mpz_mul() in GNU multiple precision library (GMP) for squaring integers of approximately 2400–6700 bits on Pentium IV Prescott 3.2GHz. For squaring in Z[x], our 3way squaring algorithms are much superior to other known squaring algorithms for small input size. In addition, we present 4way and 5way squaring formulae which do not require any constant divisions by integers other than a power of 2. Under some reasonable assumptions, our 5way squaring formula is faster than the recently proposed Montgomery’s 5way Karatsubalike formulae. Keywords: Squaring, Karatsuba algorithm, Toom
Overlapfree KaratsubaOfman Polynomial Multiplication Algorithm
"... We describe how a recently proposed way to split input operands allows for fast VLSI implementations of GF(2)[x] KaratsubaOfman multipliers. The XOR gate delay of the proposed multiplier is better than that of previous KaratsubaOfman multipliers. For example, it is reduced by about 33 % and 25 % ..."
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Cited by 11 (0 self)
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We describe how a recently proposed way to split input operands allows for fast VLSI implementations of GF(2)[x] KaratsubaOfman multipliers. The XOR gate delay of the proposed multiplier is better than that of previous KaratsubaOfman multipliers. For example, it is reduced by about 33 % and 25 % for n = 2 i and n = 3 i (i> 1), respectively. Index Terms Karatsuba algorithm, KaratsubaOfman algorithm, polynomial multiplication, subquadratic space complexity multiplier, finite field. I.
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.
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
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).
Comments on “five, Six, and SevenTerm KaratsubaLike Formulae
 IEEE Transactions on Computers
, 2007
"... We show that multiplication complexities of nterm KaratsubaLike formulae of GF (2)[x] (7 < n < 19) presented in the above paper can be further improved using the Chinese Remainder Theorem and the construction multiplication modulo (x − ∞) w. Index Terms Karatsuba algorithm, polynomial multiplicati ..."
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Cited by 5 (3 self)
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We show that multiplication complexities of nterm KaratsubaLike formulae of GF (2)[x] (7 < n < 19) presented in the above paper can be further improved using the Chinese Remainder Theorem and the construction multiplication modulo (x − ∞) w. Index Terms Karatsuba algorithm, polynomial multiplication, finite field.
Parallel Montgomery Multiplication in GF(2^k) using Trinomial Residue Arithmetic
 Proceedings 17th IEEE Symposium on computer Arithmetic
, 2005
"... Abstract We propose the first general multiplication algorithm in GF(2k) with a subquadratic area complexity of O(k8/5) = O(k1.6). Using the Chinese Remainder Theorem, we represent the elements of GF(2k); i.e. the polynomials in GF(2)[X] of degree at most k 1, by their remainder modulo a set of n ..."
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Cited by 3 (0 self)
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Abstract We propose the first general multiplication algorithm in GF(2k) with a subquadratic area complexity of O(k8/5) = O(k1.6). Using the Chinese Remainder Theorem, we represent the elements of GF(2k); 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.
Issues in Implementation of
"... I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ..."
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Cited by 1 (0 self)
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I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public.
Low Space Complexity Multiplication over Binary Fields with Dickson Polynomial Representation
, 2013
"... We study Dickson bases for binary field representation. Such a representation seems interesting when no optimal normal basis exists for the field. We express the product of two field elements as Toeplitz or Hankel matrixvector products. This provides a parallel multiplier which is subquadratic in s ..."
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We study Dickson bases for binary field representation. Such a representation seems interesting when no optimal normal basis exists for the field. We express the product of two field elements as Toeplitz or Hankel matrixvector products. This provides a parallel multiplier which is subquadratic in space and logarithmic in time. Using the matrixvector formulation of the field multiplication, we also present sequential multiplier structures with linear space complexity.