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16
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
Customizable elliptic curve cryptosystems
 IEEE Transactions on Very Large Scale Integration (VLSI) Systems
, 2005
"... Abstract—This paper presents a method for producing hardware designs for elliptic curve cryptography (ECC) systems over the finite field qp@P A, using the optimal normal basis for the representation of numbers. Our field multiplier design is based on a parallel architecture containing multiplebit s ..."
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Cited by 10 (1 self)
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Abstract—This paper presents a method for producing hardware designs for elliptic curve cryptography (ECC) systems over the finite field qp@P A, using the optimal normal basis for the representation of numbers. Our field multiplier design is based on a parallel architecture containing multiplebit serial multipliers; by changing the number of such serial multipliers, designers can obtain implementations with different tradeoffs in speed, size and level of security. A design generator has been developed which can automatically produce a customised ECC hardware design that meets userdefined requirements. To facilitate performance characterization, we have developed a parametric model for estimating the number of cycles for our generic ECC architecture. The resulting hardware implementations are among the fastest reported: for a key size of 270 bits, a point multiplication in a Xilinx XC2V6000 FPGA at 35 MHz can run over 1000 times faster
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).
Fault detection architectures for field multiplication using polynomial bases
 Issue on Fault Diagnosis and Tolerance in Cryptography
, 2006
"... In many cryptographic schemes, the most time consuming basic arithmetic operation is the finite field multiplication and its hardware implementation for bit parallel operation may require millions of logic gates. Some of these gates may become faulty in the field due to natural causes or malicious a ..."
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Cited by 3 (3 self)
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In many cryptographic schemes, the most time consuming basic arithmetic operation is the finite field multiplication and its hardware implementation for bit parallel operation may require millions of logic gates. Some of these gates may become faulty in the field due to natural causes or malicious attacks, which may lead to the generation of erroneous outputs by the multiplier. In this paper, we propose new architectures to detect erroneous outputs caused by certain types of faults in bitparallel and bitserial polynomial basis multipliers over finite fields of characteristic two. In particular, parity prediction schemes are developed for detecting errors due to single and certain multiple stuckat faults. Although the issue of detecting soft errors in registers is not considered, the proposed schemes have the advantage that they can be used with any irreducible binary polynomial chosen to define the finite field. Key words: Finite fields, polynomial basis multiplier, error detection.
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.
Quadrinomial Modular Arithmetic Using Modified Polynomial Basis
 In Proceedings of the International Conference on Information Technology: Coding and Computing ( ITCC’2005
, 2005
"... Finite field arithmetic has advantageous space and time complexity when the field is built with a sparse polynomial. Katti and Brennan in their paper [3] introduced a new type of polynomial, wich we will call here the Nearly All One Polynomial (NAOP), and they show that the NAOP modular arithmetic i ..."
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Cited by 3 (0 self)
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Finite field arithmetic has advantageous space and time complexity when the field is built with a sparse polynomial. Katti and Brennan in their paper [3] introduced a new type of polynomial, wich we will call here the Nearly All One Polynomial (NAOP), and they show that the NAOP modular arithmetic is roughly equivalent to quadrinomial arithmetic. In this paper we will introduce a new representation: the modified polynomial basis, to compute modulo quadrinomials. We obtain a faster bitparallel multiplier in F2n with time complexity equal to TA + (2 + ⌈log2(n + 1)⌉)TX and a time complexity equal to (n + 1) 2 AND and ((n + 1) 2 + m − k − 1) XOR. For fields F2n of degree n ranging between 160 and 500, which cannot be constructed with an irreducible trinomial or an optimal normal basis, our multiplier improve the time complexity by a factor of 8 % the previous multipliers proposed of [6, 3, 8], in compensation the space complexity increase by a factor
Montgomery Multiplier for a Class of Special Irreducible Pentanomials
"... In this paper we describe a Montgomery multiplier for elements of GF(2 m) defined by a type II pentanomial. The multiplier described is similar to Wu’s multiplier in [2, 3] which was defined for trinomials. In [2, 3] Wu showed that the Montgomery multiplier requires the same number of XOR and AND ga ..."
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Cited by 1 (0 self)
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In this paper we describe a Montgomery multiplier for elements of GF(2 m) defined by a type II pentanomial. The multiplier described is similar to Wu’s multiplier in [2, 3] which was defined for trinomials. In [2, 3] Wu showed that the Montgomery multiplier requires the same number of XOR and AND gates as other multipliers and that the time taken for the multiplication is also about the same as other multipliers defined for similar fields. In this paper we show that for fields defined by pentanomials we can implement a Montgomery multiplier that results in better time and gate complexity compared to similar multipliers in [1]. Gains in both space and time, was not possible for fields defined by trinomials. 1
The Parity of the Number of Irreducible Factors for Some Pentanomials
, 2008
"... It is well known that StickelbergerSwan theorem is very important for determining reducibility of polynomials over a binary field. Using this theorem it was determined the parity of the number of irreducible factors for some kinds of polynomials over a binary field, for instance, trinomials, tetran ..."
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It is well known that StickelbergerSwan theorem is very important for determining reducibility of polynomials over a binary field. Using this theorem it was determined the parity of the number of irreducible factors for some kinds of polynomials over a binary field, for instance, trinomials, tetranomials, selfreciprocal polynomials and so on. We discuss this problem