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69
Keyexchange in real quadratic congruence function fields
 Designs, Codes and Cryptography 7
, 1996
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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 26 (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.
Parallel scalar multiplication on general elliptic curves over F_p hedged against NonDifferential SideChannel Attacks
, 2002
"... For speeding up elliptic curve scalar multiplication and making it secure against sidechannel attacks such as timing or power analysis, various methods have been proposed using specifically chosen elliptic curves. We show that both goals can be achieved simultaneously even for conventional elliptic ..."
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Cited by 22 (0 self)
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For speeding up elliptic curve scalar multiplication and making it secure against sidechannel attacks such as timing or power analysis, various methods have been proposed using specifically chosen elliptic curves. We show that both goals can be achieved simultaneously even for conventional elliptic curves over Fp . This result is shown via two facts. First, we recall the known fact that every elliptic curve over Fp admits a scalar multiplication via a (Montgomery ladder) Lucas chain.
A Scalable GF(p) Elliptic Curve Processor Architecture for Programmable Hardware
"... This work proposes a new elliptic curve processor architecture for the computation of point multiplication for curves defined over fields GF (p). This is a scalable architecture in terms of area and speed specially suited for memoryrich hardware platforms such a field programmable gate arrays ( ..."
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Cited by 20 (2 self)
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This work proposes a new elliptic curve processor architecture for the computation of point multiplication for curves defined over fields GF (p). This is a scalable architecture in terms of area and speed specially suited for memoryrich hardware platforms such a field programmable gate arrays (FPGAs). This processor uses a new type of highradix Montgomery multiplier that relies on the precomputation of frequently used values and on the use of multiple processing engines.
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 ..."
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Cited by 20 (3 self)
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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
Elliptic curve cryptosystems on reconfigurable hardware
 MASTER’S THESIS, WORCESTER POLYTECHNIC INST
, 1998
"... Security issues will play an important role in the majority of communication and computer networks of the future. As the Internet becomes more and more accessible to the public, security measures will have to be strengthened. Elliptic curve cryptosystems allow for shorter operand lengths than other ..."
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Cited by 19 (0 self)
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Security issues will play an important role in the majority of communication and computer networks of the future. As the Internet becomes more and more accessible to the public, security measures will have to be strengthened. Elliptic curve cryptosystems allow for shorter operand lengths than other publickey schemes based on the discrete logarithm in finite fields and the integer factorization problem and are thus attractive for many applications. This thesis describes an implementation of a crypto engine based on elliptic curves. The underlying algebraic structures are composite Galois fields GF((2 n) m) in a standard base representation. As a major new feature, the system is developed for a reconfigurable platform based on Field Programmable Gate Arrays (FPGAs). FPGAs combine the flexibility of software solutions with the security of traditional hardware implementations. In particular, it is possible to easily change all algorithm parameters such as curve coefficients, field order, or field representation. The thesis deals with the design and implementation of elliptic curve point multiplicationarchitectures. The architectures are described in VHDL and mapped to Xilinx FPGA devices. Architectures over Galois fields of different order and representation were implemented and compared. Area and timing measurements are provided for all architectures. It is shown that a full point multiplication on elliptic curves of realworld size can be implemented on commercially available FPGAs.
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
An elliptic curve cryptography based authentication and key agreement protocol for wireless communication
 In 2nd International Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications Symposium on Information Theory
, 1998
"... We propose an authentication and key agreement protocol for wireless communication based on elliptic curve cryptographic techniques. The proposed protocol requires signi cantly less bandwidth than the AzizDi e and BellerChangYacobi protocols, and furthermore, it has lower computational burden and ..."
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Cited by 17 (4 self)
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We propose an authentication and key agreement protocol for wireless communication based on elliptic curve cryptographic techniques. The proposed protocol requires signi cantly less bandwidth than the AzizDi e and BellerChangYacobi protocols, and furthermore, it has lower computational burden and storage requirements on the user side. The use of elliptic curve cryptographic techniques provide greater security using fewer bits, resulting in a protocol which requires low computational overhead, and thus, making it suitable for wireless and mobile communication systems, including smartcards and handheld devices. 1
A microcoded elliptic curve processor using FPGA technology
 IEEE Transactions on VLSI Systems
, 2002
"... Abstract—The implementation of a microcoded elliptic curve processor using fieldprogrammable gate array technology is described. This processor implements optimal normal basis field operations in P. The design is synthesized by a parameterized module generator, which can accommodate arbitrary and a ..."
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Cited by 17 (0 self)
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Abstract—The implementation of a microcoded elliptic curve processor using fieldprogrammable gate array technology is described. This processor implements optimal normal basis field operations in P. The design is synthesized by a parameterized module generator, which can accommodate arbitrary and also produce field multipliers with different speed/area tradeoffs. The control part of the processor is microcoded, enabling curve operations to be incorporated into the processor and hence reducing the chip’s I/O requirements. The microcoded approach also facilitates rapid development and algorithmic optimization: for example, projective and affine coordinates were supported using different microcode. The design was successfully tested on a Xilinx Virtex XCV10006 device and could perform an elliptic curve multiplication over the field P using affine and projective coordinates for aIIQISS and IUQ. Index Terms—Arithmetic, cryptography, Galois fields, microprogramming, public key cryptography, reconfigurable architectures. I.