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65
The Relationship Between Breaking the DiffieHellman Protocol and Computing Discrete Logarithms
, 1998
"... Both uniform and nonuniform results concerning the security of the DiffieHellman keyexchange protocol are proved. First, it is shown that in a cyclic group G of order jGj = Q p e i i , where all the multiple prime factors of jGj are polynomial in log jGj, there exists an algorithm that re ..."
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Cited by 49 (3 self)
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Both uniform and nonuniform results concerning the security of the DiffieHellman keyexchange protocol are proved. First, it is shown that in a cyclic group G of order jGj = Q p e i i , where all the multiple prime factors of jGj are polynomial in log jGj, there exists an algorithm that reduces the computation of discrete logarithms in G to breaking the DiffieHellman protocol in G and has complexity p maxf(p i )g \Delta (log jGj) O(1) , where (p) stands for the minimum of the set of largest prime factors of all the numbers d in the interval [p \Gamma 2 p p+1; p+2 p p+ 1]. Under the unproven but plausible assumption that (p) is polynomial in log p, this reduction implies that the DiffieHellman problem and the discrete logarithm problem are polynomialtime equivalent in G. Second, it is proved that the DiffieHellman problem and the discrete logarithm problem are equivalent in a uniform sense for groups whose orders belong to certain classes: there exists a p...
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 47 (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 46 (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.
Low Complexity Bit Parallel Architectures for Polynomial Basis Multiplication over GF(2 m)
 IEEE TRANSACTIONS ON COMPUTERS
, 2004
"... 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 ..."
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Cited by 34 (3 self)
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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.
HighRadix Design of a Scalable Modular Multiplier
 in Cryptographic Hardware and Embedded Systems — CHES 2001, Ç. K. Koç and C. Paar, Eds. 2001, Lecture Notes in Computer Science
, 2001
"... This paper describes an algorithm and architecture based on an extension of a scalable radix2 architecture proposed in a previous work. The algorithm is proven to be correct and the hardware design is discussed in detail. Experimental results are shown to compare a radix8 implementation with a ..."
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Cited by 30 (8 self)
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This paper describes an algorithm and architecture based on an extension of a scalable radix2 architecture proposed in a previous work. The algorithm is proven to be correct and the hardware design is discussed in detail. Experimental results are shown to compare a radix8 implementation with a radix2 design. The scalable Montgomery multiplier is adjustable to constrained areas yet being able to work on any given precision of the operands. Similar to some systolic implementations, this design avoid the high load on signals that broadcast to several components, making the delay independent of operand's precision.
The DiffieHellman Protocol
 DESIGNS, CODES, AND CRYPTOGRAPHY
, 1999
"... The 1976 seminal paper of Diffie and Hellman is a landmark in the history of cryptography. They introduced the fundamental concepts of a trapdoor oneway function, a publickey cryptosystem, and a digital signature scheme. Moreover, they presented a protocol, the socalled DiffieHellman protoco ..."
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Cited by 29 (0 self)
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The 1976 seminal paper of Diffie and Hellman is a landmark in the history of cryptography. They introduced the fundamental concepts of a trapdoor oneway function, a publickey cryptosystem, and a digital signature scheme. Moreover, they presented a protocol, the socalled DiffieHellman protocol, allowing two parties who share no secret information initially, to generate a mutual secret key. This paper summarizes the present knowledge on the security of this protocol.
A new construction of MasseyOmura parallel multiplier over GF(2m
 IEEE Transactions on Computers
, 2001
"... AbstractÐThe MasseyOmura multiplier of GF
2m uses a normal basis and its bit parallel version is usually implemented using m identical combinational logic blocks whose inputs are cyclically shifted from one another. In the past, it was shown that, for a class of finite fields defined by irreducib ..."
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Cited by 28 (5 self)
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AbstractÐThe MasseyOmura multiplier of GF
2m uses a normal basis and its bit parallel version is usually implemented using m identical combinational logic blocks whose inputs are cyclically shifted from one another. In the past, it was shown that, for a class of finite fields defined by irreducible allone polynomials, the parallel MasseyOmura multiplier had redundancy and a modified architecture of lower circuit complexity was proposed. In this article, it is shown that, not only does this type of multipliers contain redundancy in that special class of finite fields, but it also has redundancy in fields GF
2m defined by any irreducible polynomial. By removing the redundancy, we propose a new architecture for the normal basis parallel multiplier, which is applicable to any arbitrary finite field and has significantly lower circuit complexity compared to the original MasseyOmura normal basis parallel multiplier. The proposed multiplier structure is also modular and, hence, suitable for VLSI realization. When applied to fields defined by the irreducible allone polynomials, the multiplier's circuit complexity matches the best result available in the open literature. Index TermsÐFinite field, MasseyOmura multiplier, allone polynomial, optimal normal bases. æ 1
Mastrovito Multiplier for General Irreducible Polynomials
 IEEE Transactions on Computers
, 2000
"... ..."
Parallel Multipliers Based on Special Irreducible Pentanomials
 IEEE Trans on Computers
, 2003
"... Abstract—The stateoftheart Galois field GFð2 m Þ 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 equal ..."
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Cited by 21 (0 self)
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Abstract—The stateoftheart Galois field GFð2 m Þ 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 equally spaced 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 nor an irreducible ESP exists, the use of irreducible pentanomials has been suggested. Irreducible pentanomials are abundant, and there are several eligible candidates for a given m. In this 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 GFð2 m Þ. æ
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 19 (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.