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24
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 ..."
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Cited by 29 (3 self)
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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 cryptographic processor for arbitrary elliptic curves over GF(2m
 in: Proceedings of the IEEE International Conference ApplicationSpecific Systems, Architectures, and Processors, ASAP’03, The Hague, The
, 2003
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A new approach to subquadratic space complexity parallel multipliers for extended binary fields
 IEEE Trans. Computers
, 2007
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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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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
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 18 (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
Low complexity multiplication in a finite field using ring representation
 IEEE Trans. Comput
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Towards faulttolerant cryptographic computations over finite fields
 ACM Trans. Embedded Comput. Syst
"... Cryptographic schemes, such as authentication, confidentiality, and integrity, rely on computations in very large finite fields, whose hardware realization may require millions of logic gates. In a straightforward design, even a single fault in such a complex circuit is likely to yield an incorrect ..."
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Cited by 9 (3 self)
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Cryptographic schemes, such as authentication, confidentiality, and integrity, rely on computations in very large finite fields, whose hardware realization may require millions of logic gates. In a straightforward design, even a single fault in such a complex circuit is likely to yield an incorrect result and may be exploited by an attacker to break the cryptosystem. In this regard, we consider computing over finite fields in presence of certain faults in multiplier circuits. Our work reported here deals with errors caused by such faults in polynomial basis multipliers over finite fields of characteristic two and presents a scheme to correct single errors. Towards this, pertinent theoretical results are derived, and both bitparallel and bitserial fault tolerant multipliers are proposed.
Tradeoff Analysis of FPGA Based Elliptic Curve Cryptosystems
 In Proceedings of The IEEE International Symposium on Circuits and Systems (ISCAS
"... FPGAs are an attractive platform for elliptic curve cryptography hardware. Since field multiplication is the most critical operation in elliptic curve cryptography, we have studied how efficient several field multipliers can be mapped to lookup table based FPGAs. Furthermore we have compared differ ..."
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FPGAs are an attractive platform for elliptic curve cryptography hardware. Since field multiplication is the most critical operation in elliptic curve cryptography, we have studied how efficient several field multipliers can be mapped to lookup table based FPGAs. Furthermore we have compared different curve coordinate representations with respect to the number of required field operations, and show how an elliptic curve coprocessor based on the Montgomery algorithm for curve multiplication can be implemented using our generic coprocessor architecture. 1.
Fault detection architectures for field multiplication using polynomial bases
 Issue on Fault Diagnosis and Tolerance in Cryptography
, 2006
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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 ..."
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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.