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Guide to Elliptic Curve Cryptography
, 2004
"... Elliptic curves have been intensively studied in number theory and algebraic geometry for over 100 years and there is an enormous amount of literature on the subject. To quote the mathematician Serge Lang: It is possible to write endlessly on elliptic curves. (This is not a threat.) Elliptic curves ..."
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Cited by 369 (17 self)
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Elliptic curves have been intensively studied in number theory and algebraic geometry for over 100 years and there is an enormous amount of literature on the subject. To quote the mathematician Serge Lang: It is possible to write endlessly on elliptic curves. (This is not a threat.) Elliptic curves also figured prominently in the recent proof of Fermat's Last Theorem by Andrew Wiles. Originally pursued for purely aesthetic reasons, elliptic curves have recently been utilized in devising algorithms for factoring integers, primality proving, and in publickey cryptography. In this article, we aim to give the reader an introduction to elliptic curve cryptosystems, and to demonstrate why these systems provide relatively small block sizes, highspeed software and hardware implementations, and offer the highest strengthperkeybit of any known publickey scheme.
Optimal Extension Fields for Fast Arithmetic in PublicKey Algorithms
, 1998
"... Abstract. This contribution introduces a class of Galois field used to achieve fast finite field arithmetic which we call an Optimal Extension Field (OEF). This approach is well suited for implementation of publickey cryptosystems based on elliptic and hyperelliptic curves. Whereas previous reported ..."
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Cited by 65 (14 self)
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Abstract. This contribution introduces a class of Galois field used to achieve fast finite field arithmetic which we call an Optimal Extension Field (OEF). This approach is well suited for implementation of publickey cryptosystems based on elliptic and hyperelliptic curves. Whereas previous reported optimizations focus on finite fields of the form GF (p) and GF (2 m), an OEF is the class of fields GF (p m), for p a prime of special form and m a positive integer. Modern RISC workstation processors are optimized to perform integer arithmetic on integers of size up to the word size of the processor. Our construction employs wellknown techniques for fast finite field arithmetic which fully exploit the fast integer arithmetic found on these processors. In this paper, we describe our methods to perform the arithmetic in an OEF and the methods to construct OEFs. We provide a list of OEFs tailored for processors with 8, 16, 32, and 64 bit word sizes. We report on our application of this approach to construction of elliptic curve cryptosystems and demonstrate a substantial performance improvement over all previous reported software implementations of Galois field arithmetic for elliptic curves.
LowWeight Binary Representations for Pairs of Integers
, 2001
"... . Shamir's method speeds up the computation of the product of powers of two elements of a group, a common object in publickey algorithms. Shamir's method is based on binary expansions and was designed for modular and nite eld arithmetic. Elliptic curve arithmetic uses signed binary expansions r ..."
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Cited by 65 (0 self)
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. Shamir's method speeds up the computation of the product of powers of two elements of a group, a common object in publickey algorithms. Shamir's method is based on binary expansions and was designed for modular and nite eld arithmetic. Elliptic curve arithmetic uses signed binary expansions rather than the ordinary binary expansions of modular arithmetic. This note extends Shamir's method to the elliptic curve setting by specifying an optimal signed binary representation for a pair of positive integers. 1 Shamir Methods Shamir suggested [4] a simple but powerful trick for speeding up an operation that is common in publickey cryptography. Let G be a subgroup of the multiplicative group of nonzero elements of a nite eld F q . 1 The basic publickey operation in G is exponentiation: computing g a for a given element g 2 G and a positive integer a. This is typically accomplished [6] by the binary method, based on the binary expansion e of a. The method requires a squa...
Generalized Mersenne Numbers
, 1999
"... . There is a well known shortcut for modular multiplication modulo a Mersenne number, performing modular reduction without integer division. We generalize this technique to a larger class of primes, and discuss parameter choices which are particularly well suited for machine implementation. Keywords ..."
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Cited by 46 (0 self)
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. There is a well known shortcut for modular multiplication modulo a Mersenne number, performing modular reduction without integer division. We generalize this technique to a larger class of primes, and discuss parameter choices which are particularly well suited for machine implementation. Keywords: modular arithmetic, elliptic curves. Introduction It has long been known that certain integers are particularly well suited for modular reduction. The best known examples (e.g., [1]) are the Mersenne numbers m = 2 k \Gamma 1. In this case, the integers (mod m) are represented as kbit integers. When performing modular multiplication, one carries out an integer multiplication followed by a modular reduction. One thus has the problem of reducing modulo m a 2kbit number. Modular reduction is usually done by integer division, but this is unnecessary in the Mersenne case. Let n ! m 2 be the integer to be reduced (mod m). Let T be the integer represented by the k most significant bits o...
Toward Acceleration of RSA Using 3D Graphics Hardware
"... Abstract. Demand in the consumer market for graphics hardware that accelerates rendering of 3D images has resulted in commodity devices capable of astonishing levels of performance. These results were achieved by specifically tailoring the hardware for the target domain. As graphics accelerators bec ..."
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Cited by 19 (0 self)
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Abstract. Demand in the consumer market for graphics hardware that accelerates rendering of 3D images has resulted in commodity devices capable of astonishing levels of performance. These results were achieved by specifically tailoring the hardware for the target domain. As graphics accelerators become increasingly programmable however, this performance has made them an attractive target for other domains. Specifically, they have motivated the transformation of costly algorithms from a general purpose computational model into a form that executes on said graphics hardware. We investigate the implementation and performance of modular exponentiation using a graphics accelerator, with the view of using it to execute operations required in the RSA public key cryptosystem. 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
Implementation Options for Finite Field Arithmetic for Elliptic Curve Cryptosystems
, 1999
"... Contents 1. Motivation 2. Overview on Finite Field Arithmetic 3. Arithmetic in GF(p) 4. Arithmetic in GF(2 m ) 5. Arithmetic in GF(p m ) 6. Open Problems ECC '99 WPI Why PublicKey Algorithms? Traditional tool for data security: Privatekey (or symmetric) cryptography Main applications: ffl En ..."
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Cited by 11 (0 self)
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Contents 1. Motivation 2. Overview on Finite Field Arithmetic 3. Arithmetic in GF(p) 4. Arithmetic in GF(2 m ) 5. Arithmetic in GF(p m ) 6. Open Problems ECC '99 WPI Why PublicKey Algorithms? Traditional tool for data security: Privatekey (or symmetric) cryptography Main applications: ffl Encryption ffl Message Authentication Traditional shortcomings: 1. Key distribution, especially with large, dynamic user population (Internet) 2. How to assure sender authenticity and nonrepudiation? Solution: Publickey schemes, e.g., DiffieHellman key exchange or digital signatures. ECC '99 WPI Practical PublicKey Algorithms There are three families of PK algorithms of practical relevance: Integer Factorization Schemes Exp: RSA, Rabin, etc. required ope
LowPower Elliptic Curve Cryptography Using Scaled Modular Arithmetic
 Proceedings of 6th International Workshop on Cryptographic Hardware in Embedded Systems (CHES), volume 3156 of Lecture Notes in Computer Science
, 2004
"... Abstract. We introduce new modulus scaling techniques for transforming a class of primes into special forms which enables efficient arithmetic. The scaling technique may be used to improve multiplication and inversion in finite fields. We present an efficient inversion algorithm that utilizes the st ..."
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Cited by 11 (4 self)
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Abstract. We introduce new modulus scaling techniques for transforming a class of primes into special forms which enables efficient arithmetic. The scaling technique may be used to improve multiplication and inversion in finite fields. We present an efficient inversion algorithm that utilizes the structure of scaled modulus. Our inversion algorithm exhibits superior performance to the Euclidean algorithm and lends itself to efficient hardware implementation due to its simplicity. Using the scaled modulus technique and our specialized inversion algorithm we develop an elliptic curve processor architecture. The resulting architecture successfully utilizes redundant representation of elements in GF (p) and provides a lowpower, high speed, and small footprint specialized elliptic curve implementation. 1
Incomplete reduction in modular arithmetic
 IEE Proceedings: Computers and Digital Technique
, 2002
"... We describe a novel method for obtaining fast software implementations of the arithmetic operations in the finite field GF(p) with an arbitrary prime modulus p which is of arbitrary length. The most important feature of the method is that it avoids bitlevel operations which are slow on microprocess ..."
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Cited by 10 (1 self)
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We describe a novel method for obtaining fast software implementations of the arithmetic operations in the finite field GF(p) with an arbitrary prime modulus p which is of arbitrary length. The most important feature of the method is that it avoids bitlevel operations which are slow on microprocessors and performs wordlevel operations which are significantly faster. The proposed method has applications in publickey cryptographic algorithms defined over the finite field GF(p), most notably the elliptic curve digital signature algorithm. 1
Lowweight polynomial form integers for efficient modular multiplication
 IEEE Transactions on Computers
, 2006
"... In 1999, Jerome Solinas introduced families of moduli called the generalized Mersenne numbers (GMNs), which are expressed in lowweight polynomial form, p = f(t), where t is limited to a power of 2. GMNs are very useful in elliptic curve cryptosystems over prime fields, since only integer additions ..."
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Cited by 6 (2 self)
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In 1999, Jerome Solinas introduced families of moduli called the generalized Mersenne numbers (GMNs), which are expressed in lowweight polynomial form, p = f(t), where t is limited to a power of 2. GMNs are very useful in elliptic curve cryptosystems over prime fields, since only integer additions and subtractions are required in modular reductions. However, since there are not many GMNs and each GMN requires a dedicated implementation, GMNs are hardly useful for other cryptosystems. Here we modify GMN by removing restriction on the choice of t and restricting the coefficients of f(t) to 0 and ±1. We call such families of moduli lowweight polynomial form integers (LWPFIs). We show an efficient modular multiplication method using LWPFI moduli. LWPFIs allow general implementation and there exist many LWPFI moduli. One may consider LWPFIs as a tradeoff between general integers and GMNs.