Results 1  10
of
15
An EndtoEnd Systems Approach to Elliptic Curve Cryptography
 In Cryptographic Hardware and Embedded Systems (CHES
, 2002
"... Since its proposal by Victor Miller [17] and Neal Koblitz [15] in the mid 1980s, Elliptic Curve Cryptography (ECC) has evolved into a mature publickey cryptosystem. Offering the smallest key size and the highest strength per bit, its computational efficiency can benefit both client devices and serv ..."
Abstract

Cited by 26 (3 self)
 Add to MetaCart
Since its proposal by Victor Miller [17] and Neal Koblitz [15] in the mid 1980s, Elliptic Curve Cryptography (ECC) has evolved into a mature publickey cryptosystem. Offering the smallest key size and the highest strength per bit, its computational efficiency can benefit both client devices and server machines. We have designed a programmable hardware accelerator to speed up point multiplication for elliptic curves over binary polynomial fields GF (2^m). The accelerator is based on a scalable architecture capable of handling curves of arbitrary field degrees up to m = 255. In addition, it delivers optimized performance for a set of commonly used curves through hardwired reduction logic. A prototype implementation running in a Xilinx XCV2000E FPGA at 66.4 MHz shows a performance of 6987 point multiplications per second for GF(2^163). We have integrated ECC into OpenSSL, today's dominant implementation of the secure Internet protocol SSL, and tested it with the Apache web server and opensource web browsers.
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 ..."
Abstract

Cited by 21 (14 self)
 Add to MetaCart
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.
A Cryptographic Processor for Arbitrary Elliptic Curves over GF(2 m
, 2003
"... We describe a cryptographic processor for Elliptic Curve Cryptography (ECC). ECC is evolving as an attractive alternative to other publickey cryptosystems such as the RivestShamirAdleman algorithm (RSA) by offering the smallest key size and the highest strength per bit. The cryptographic processo ..."
Abstract

Cited by 20 (3 self)
 Add to MetaCart
We describe a cryptographic processor for Elliptic Curve Cryptography (ECC). ECC is evolving as an attractive alternative to other publickey cryptosystems such as the RivestShamirAdleman algorithm (RSA) by offering the smallest key size and the highest strength per bit. The cryptographic processor performs point multiplication for elliptic curves over binary polynomial fields GF(2 m). In contrast to other designs that only support one curve at a time, our processor is capable of handling arbitrary curves without requiring reconfiguration. More specifically, it can handle both named curves as standardized by the National Institute for Standards and Technology (NIST) as well as any other generic curves up to a field degree of 255. Efficient support for arbitrary curves is particularly important for the targeted server applications that need to handle requests for secure connections generated by a multitude of heterogeneous client devices. Such requests may specify curves which are infrequently used or not even known at implementation time. We have implemented the cryptographic processor in a fieldprogrammable gate array (FPGA) running at a clock frequency of 66.4 MHz. Its performance is 6955 point multiplications per
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 ..."
Abstract

Cited by 17 (0 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 17 (2 self)
 Add to MetaCart
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
Low Complexity Multiplication in a Finite Field Using Ring Representation
 IEEE Transactions on Computers
, 2003
"... Abstract—Elements of a finite field, GFð2 m Þ, are represented as elements in a ring in which multiplication is more time efficient. This leads to faster multipliers with a modest increase in the number of XOR and AND gates needed to construct the multiplier. Such multipliers are used in error contr ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
Abstract—Elements of a finite field, GFð2 m Þ, are represented as elements in a ring in which multiplication is more time efficient. This leads to faster multipliers with a modest increase in the number of XOR and AND gates needed to construct the multiplier. Such multipliers are used in error control coding and cryptography. We consider rings modulo trinomials and 4term polynomials. In each case, we show that our multiplier is faster than multipliers over elements in a finite field defined by irreducible pentanomials. These results are especially significant in the field of elliptic curve cryptography, where pentanomials are used to define finite fields. Finally, an efficient systolic implementation of a multiplier for elements in a ring defined by x n þ x þ 1 is presented. Index Terms—Finite field multiplication, ring representation, systolic arrays. æ 1
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 ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
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.
Relationship between GF(2 m) Montgomery and Shifted Polynomial Basis Multiplication Algorithms
"... Abstract Applying the matrixvector product idea of the Mastrovito multiplier to the GF(2 m) Montgomery multiplication algorithm, we present a new multiplier for irreducible trinomials. This multiplier and the corresponding shifted polynomial basis (SPB) multiplier have the same circuit structure f ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
Abstract Applying the matrixvector product idea of the Mastrovito multiplier to the GF(2 m) Montgomery multiplication algorithm, we present a new multiplier for irreducible trinomials. This multiplier and the corresponding shifted polynomial basis (SPB) multiplier have the same circuit structure for the same set of parameters. Furthermore, by establishing isomorphisms between the Montgomery and the SPB constructions of GF(2 m), we show that the Montgomery algorithm can be used to perform the SPB multiplication without any changes, and vice versa.
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 ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
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