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25
Resistance against Differential Power Analysis for Elliptic Curve Cryptosystems
, 1999
"... Differential Power Analysis, first introduced by Kocher et al. in [14], is a powerful technique allowing to recover secret smart card information by monitoring power signals. In [14] a specific DPA attack against smartcards running the DES algorithm was described. As few as 1000 encryptions were su ..."
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Cited by 162 (2 self)
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Differential Power Analysis, first introduced by Kocher et al. in [14], is a powerful technique allowing to recover secret smart card information by monitoring power signals. In [14] a specific DPA attack against smartcards running the DES algorithm was described. As few as 1000 encryptions were sufficient to recover the secret key. In this paper we generalize DPA attack to elliptic curve (EC) cryptosystems and describe a DPA on EC DiffieHellman key exchange and EC ElGamal type encryption. Those attacks enable to recover the private key stored inside the smartcard. Moreover, we suggest countermeasures that thwart our attack.
A Survey of Fast Exponentiation Methods
 Journal of Algorithms
, 1998
"... Publickey cryptographic systems often involve raising elements of some group (e.g. GF(2 n), Z/NZ, or elliptic curves) to large powers. An important question is how fast this exponentiation can be done, which often determines whether a given system is practical. The best method for exponentiation de ..."
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Cited by 155 (0 self)
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Publickey cryptographic systems often involve raising elements of some group (e.g. GF(2 n), Z/NZ, or elliptic curves) to large powers. An important question is how fast this exponentiation can be done, which often determines whether a given system is practical. The best method for exponentiation depends strongly on the group being used, the hardware the system is implemented on, and whether one element is being raised repeatedly to different powers, different elements are raised to a fixed power, or both powers and group elements vary. This problem has received much attention, but the results are scattered through the literature. In this paper we survey the known methods for fast exponentiation, examining their relative strengths and weaknesses. 1
Software Implementation of Elliptic Curve Cryptography Over Binary Fields
, 2000
"... This paper presents an extensive and careful study of the software implementation on workstations of the NISTrecommended elliptic curves over binary fields. We also present the results of our implementation in C on a Pentium II 400 MHz workstation. ..."
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Cited by 147 (9 self)
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This paper presents an extensive and careful study of the software implementation on workstations of the NISTrecommended elliptic curves over binary fields. We also present the results of our implementation in C on a Pentium II 400 MHz workstation.
Efficient arithmetic on Koblitz curves
 Designs, Codes, and Cryptography
, 2000
"... Abstract. It has become increasingly common to implement discretelogarithm based publickey protocols on elliptic curves over finite fields. The basic operation is scalar multiplication: taking a given integer multiple of a given point on the curve. The cost of the protocols depends on that of the ..."
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Cited by 79 (0 self)
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Abstract. It has become increasingly common to implement discretelogarithm based publickey protocols on elliptic curves over finite fields. The basic operation is scalar multiplication: taking a given integer multiple of a given point on the curve. The cost of the protocols depends on that of the elliptic scalar multiplication operation. Koblitz introduced a family of curves which admit especially fast elliptic scalar multiplication. His algorithm was later modified by Meier and Staffelbach. We give an improved version of the algorithm which runs 50 % faster than any previous version. It is based on a new kind of representation of an integer, analogous to certain kinds of binary expansions. We also outline further speedups using precomputation and storage.
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.
Software Implementation of the NIST Elliptic Curves Over Prime Fields
 TOPICS IN CRYPTOLOGY – CTRSA 2001, VOLUME 2020 OF LNCS
, 2001
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Improved Algorithms for Elliptic Curve Arithmetic in GF(2^n)
, 1998
"... This paper describes three contributions for efficient implementation of elliptic curve cryptosystems in GF (2^n). The first is a new method for doubling an elliptic curve point, which is simpler to implement than the fastest known method, due to Schroeppel, and which favors sparse elliptic curve co ..."
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Cited by 45 (5 self)
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This paper describes three contributions for efficient implementation of elliptic curve cryptosystems in GF (2^n). The first is a new method for doubling an elliptic curve point, which is simpler to implement than the fastest known method, due to Schroeppel, and which favors sparse elliptic curve coefficients. The second is a generalized and improved version of the Guajardo and Paar's formulas for computing repeated doubling points. The third contribution consists of a new kind of projective coordinates that provides the fastest known arithmetic on elliptic curves. The algorithms resulting from this new formulation lead to a running time improvement for computing a scalar multiplication of about 17% over previous projective coordinate methods.
On the Performance of Signature Schemes based on Elliptic Curves
, 1998
"... . This paper describes a fast software implementation of the elliptic curve version of DSA, as specified in draft standard documents ANSI X9.62 and IEEE P1363. We did the implementations for the fields GF(2 n ), using a standard basis, and GF(p). We discuss various design decisions that have t ..."
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Cited by 39 (2 self)
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. This paper describes a fast software implementation of the elliptic curve version of DSA, as specified in draft standard documents ANSI X9.62 and IEEE P1363. We did the implementations for the fields GF(2 n ), using a standard basis, and GF(p). We discuss various design decisions that have to be made for the operations in the underlying field and the operations on elliptic curve points. In particular, we conclude that it is a good idea to use projective coordinates for GF(p), but not for GF(2 n ). We also extend a number of exponentiation algorithms, that result in considerable speed gains for DSA, to ECDSA, using a signed binary representation. Finally, we present timing results for both types of fields on a PPro200 based PC, for a C/C++ implementation with small assemblylanguage optimizations, and make comparisons to other signature algorithms, such as RSA and DSA. We conclude that for practical sizes of fields and moduli, GF(p) is roughly twice as fast as GF(2 ...
An Overview of Elliptic Curve Cryptography
, 2000
"... Elliptic curve cryptography (ECC) was introduced by Victor Miller and Neal Koblitz in 1985. ECC proposed as an alternative to established publickey systems such as DSA and RSA, have recently gained a lot attention in industry and academia. The main reason for the attractiveness of ECC is the fact t ..."
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Cited by 29 (2 self)
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Elliptic curve cryptography (ECC) was introduced by Victor Miller and Neal Koblitz in 1985. ECC proposed as an alternative to established publickey systems such as DSA and RSA, have recently gained a lot attention in industry and academia. The main reason for the attractiveness of ECC is the fact that there is no subexponential algorithm known to solve the discrete logarithm problem on a properly chosen elliptic curve. This means that significantly smaller parameters can be used in ECC than in other competitive systems such RSA and DSA, but with equivalent levels of security. Some benefits of having smaller key sizes include faster computations, and reductions in processing power, storage space and bandwidth. This makes ECC ideal for constrained environments such as pagers, PDAs, cellular phones and smart cards. The implementation of ECC, on the other hand, requires several choices such as the type of the underlying finite field, algorithms for implementing the finite field arithmetic and so on. In this paper we give we presen an selective overview of the main methods.
Modular Exponentiation on Reconfigurable Hardware
, 1999
"... It is widely recognized that security issues will play a crucial role in the majority of future computer and communication systems. A central tool for achieving system security are cryptographic algorithms. For performance as well as for physical security reasons, it is often advantageous to realize ..."
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Cited by 19 (1 self)
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It is widely recognized that security issues will play a crucial role in the majority of future computer and communication systems. A central tool for achieving system security are cryptographic algorithms. For performance as well as for physical security reasons, it is often advantageous to realize cryptographic algorithms in hardware. In order to overcome the wellknown drawback of reduced flexibility that is associated with traditional ASIC solutions, this contribution proposes arithmetic architectures which are optimized for modern field programmable gate arrays (FPGAs). The proposed architectures perform modular exponentiation with very long integers. This operation is at the heart of many practical publickey algorithms such as RSA and discrete logarithm schemes. We combine two versions of Montgomery modular multiplication algorithm with new systolic array designs which are well suited for FPGA realizations. The first one is based on a radix of two and is capable of processing a ...