Fast Multiplication on Elliptic Curves Over GF(2m) without precomputation

López, Julio; Dahab, Ricardo

doi:10.1007/3-540-48059-5_27

Cited by 343 publications

(262 citation statements)

References 12 publications

Supporting

Mentioning

249

Contrasting

Unclassified

Order By: Relevance

“…Using projective coordinates can eliminate the use of Galois field inversion in point addition and point doubling. The point addition and point doubling in projective coordinates can be computed as following [28]:…”

Section: Projective Coordinatesmentioning

confidence: 99%

See 1 more Smart Citation

FPGA implementations of elliptic curve cryptography and Tate pairing over a binary field

Huang

Sweany

et al. 2008

Journal of Systems Architecture

View full text Add to dashboard Cite

Elliptic curve cryptography (ECC) is an alternative to traditional techniques for public key cryptography. It offers smaller key size without sacrificing security level. Tate pairing is a bilinear map used in identity based cryptography schemes. In a typical elliptic curve cryptosystem, elliptic curve point multiplication is the most computationally expensive component. Similarly, Tate pairing is also quite computationally expensive. Therefore, it is more attractive to implement the ECC and Tate pairing using hardware than using software. The Cryptography is the most standard and efficient way to protect the security of web transactions. It can be used to protect the confidentiality, integrity, authentication, and non-reputation of the web transactions. There are two major categories of cryptography 1 schemes, i.e., public-key cryptography and symmetric-key cryptography. In public-key cryptography, the receiver and sender have their own private key and share a common public key. In symmetric-key cryptography, the receiver and sender must have the same private key, which makes it difficult to manage the private key. Public-key cryptography is easy for key distribution and key management. But it is not as efficient as symmetric-key cryptography [19,39]. Thus, it is interesting to use dedicated hardware for public-key cryptography to improve the performance.A well-known public-key cryptography algorithm is RSA, which was first proposed by and it is projected that its size will increase to 2048 bits after 2010. The performance issues of RSA with such a large key size will then become a dominant force, which can severely affect the performance of RSA. So, we would like to use 283-bit ECC in place of the 2048-bit RSA since it can significantly reduce the key length and still provides the same security level.Despite ECC's advantages over RSA, software based ECC implementations usually require long computation time, hence makes it difficult to be effectively utilized in real-time 2 The main contribution of our FPGA based design is the resources sharing and parallel processing optimization. The simulation results show that our implementation is significantly faster than the software implementation as well as previous FPGA implementations with the same security level [12, 27]. Tate PairingIdentity based cryptography schemes have opened a new territory for public key cryptography [8,38,42]. Using identity based cryptography schemes, a sender can derive the public key of a receiver without receiving the certificate of the receiver issued by a certificate authority (CA). The public key can be derived from the identity of the receiver such as the email or IP address. Pairing over elliptic curve can be used to construct the identity based cryptography schemes. It is a map from two points on the elliptic curve to another multiplicative group. It has special properties of bilinearity. Currently, the most commonly used pairing methods are Tate pairing [13] and Weil pairing [31]. Originally, Weil pairing was used to attack publi...

show abstract

Section: Projective Coordinatesmentioning

confidence: 99%

“…In our design, we use Montgomery point multiplication algorithm for the implementation of point multiplication [19,28,43]. The pseudocode is shown in Algorithm 8.…”

Section: Montgomery Algorithmmentioning

confidence: 99%

FPGA implementations of elliptic curve cryptography and Tate pairing over a binary field

Huang

Sweany

et al. 2008

Journal of Systems Architecture

View full text Add to dashboard Cite

show abstract

“…In its simplest form, a scalar multiplication can be realized through a sequence of point additions and doublings, respectively. There exist a number of advanced algorithms for point multiplication; one of the most efficient was proposed by López and Dahab in [18]. Their algorithm requires to carry out 4 log 2 k + 6 additions, 2 log 2 k + 4 multiplications, 2 log 2 k + 2 squarings and 2 log 2 k + 1 inversions in the underlying finite field GF(2 m ) to obtain the result of k · P [18, Lemma 4].…”

Section: Elliptic Curve Cryptographymentioning

confidence: 99%

Hardware/Software Co-design of Elliptic Curve Cryptography on an 8051 Microcontroller

Koschuch

Lechner

Weitzer

et al. 2006

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Abstract. 8-bit microcontrollers like the 8051 still hold a considerable share of the embedded systems market and dominate in the smart card industry. The performance of 8-bit microcontrollers is often too poor for the implementation of public-key cryptography in software. In this paper we present a minimalist hardware accelerator for enabling elliptic curve cryptography (ECC) on an 8051 microcontroller. We demonstrate the importance of removing system-level performance bottlenecks caused by the transfer of operands between hardware accelerator and external RAM. The integration of a small direct memory access (DMA) unit proves vital to exploit the full potential of the hardware accelerator. Our design allows to perform a scalar multiplication over the binary extension field GF(2 191 ) in 118 msec at a clock frequency of 12 MHz. Considering performance and hardware cost, our system compares favorably with previous work on similar 8-bit platforms.

show abstract

“…A different approach for computing kP was introduced by Montgomery, based on the observation that the x-coordinate of the sum of two points can be computed only using the x-coordinates of the two points if their difference is known [20] (see also [1,17]). Let k = l−1 i=0 k i 2 i be the binary representation of the multiplier k (assume that k l−1 = 1).…”

Section: Elliptic Scalar Multiplicationmentioning

confidence: 99%

“…For a typical value of l = 160, we have T W = 4I +1445M and T M = 1463M , so Montgomery's method may be a little bit faster. However, Montgomery's method is not a general algorithm for elliptic scalar multiplication in GF(p n ), since it can't compute the y-coordinate of kP (note however that this is not the case in GF(2 n ); see [17]). Of course, this may not be a problem in many applications, since most elliptic curve variants of key exchange and digital signature schemes only make use of x coordinates (e.g., see [26,27,28]).…”

Section: Elliptic Scalar Multiplicationmentioning

confidence: 99%

Fast Implementation of Elliptic Curve Arithmetic in GF(p n )

Lim

Hwang

2000

Public Key Cryptography

View full text Add to dashboard Cite

Abstract. Elliptic curve cryptosystems have attracted much attention in recent years and one of major interests in ECC is to develop fast algorithms for field/elliptic curve arithmetic. In this paper we present various improvement techniques for field arithmetic in GF(p n )(p a prime), in particular, fast field multiplication and inversion algorithms, and provide our implementation results on Pentium II and Alpha 21164 microprocessors.

show abstract

Fast Multiplication on Elliptic Curves Over GF(2m) without precomputation

Cited by 343 publications

References 12 publications

FPGA implementations of elliptic curve cryptography and Tate pairing over a binary field

FPGA implementations of elliptic curve cryptography and Tate pairing over a binary field

Hardware/Software Co-design of Elliptic Curve Cryptography on an 8051 Microcontroller

Fast Implementation of Elliptic Curve Arithmetic in GF(p n )

Contact Info

Product

Resources

About