Efficient Elliptic Curve Exponentiation Using Mixed Coordinates

Cohen, Henrí; Miyaji, Atsuko; Ono, Takatoshi

doi:10.1007/3-540-49649-1_6

Cited by 298 publications

(323 citation statements)

References 23 publications

Supporting

Mentioning

318

Contrasting

Unclassified

Order By: Relevance

“…Note however that k repeated doublings can save 2 squarings in k − 1 doublings by using one more chaining variable for aZ 4 0 (i.e., compute W = aZ 4 0 at the first doubling and then update W by W ← CW in each iteration except for the final doubling) [7]. Also note that if a = −3, we can compute A in doubling as A = 3(X 0 + Z 2 0 )(X 0 − Z 2 0 ).…”

Section: Elliptic Addition/doubling In Projective Coordinatesmentioning

confidence: 99%

See 1 more Smart Citation

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

Section: Elliptic Addition/doubling In Projective Coordinatesmentioning

confidence: 99%

“…In particular, much research has been conducted on fast algorithms and implementation techniques of elliptic curve arithmetic over various finite fields [12,22,24,23,8,7,2,9].…”

Section: Introductionmentioning

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

show abstract

“…We optimize the EC exponentiation by combining mixed coordinate system from [5] and the Lim-Hwang's method [10].…”

Section: Elliptic Curve Arithmeticmentioning

confidence: 99%

“…The first k v doublings, in case W [v] < 2 w−1 , can be more efficiently computed [5]. There have been an analysis on this in [5], however it is incorrect and we want to correct it here.…”

Section: Signed Window Algorithm For Ec Exponentiationmentioning

confidence: 99%

See 1 more Smart Citation

Fast Implementation of Elliptic Curve Defined over GF(pm) on CalmRISC with MAC2424 Coprocessor

Chung

Sim

Lee

2000

Cryptographic Hardware and Embedded Systems — CHES 2000

View full text Add to dashboard Cite

Abstract. In this paper, we propose fast finite field and elliptic curve (EC) algorithms useful for embedding cryptographic functions on high performance device such that most instructions take just one cycle. In such case, the integer multiplications and additions have the same computational cost so that the computational cost analyses that were previously done in traditional manner may be invalid and in some cases the new algorithms should be introduced for fast computation. In our implementation, column major method for field multiplication and BP inversion algorithm are used for fast field arithmetic, and mixed coordinates method is used for efficient EC exponentiation. We give here analyses on various algorithms that are useful for implementing EC exponentiation on CalmRISC microcontroller with MAC2424 coprocessor, as well as new exact analyses on BP (Bailey-Paar) inversion algorithm and EC exponentiation. Using techniques shown in this paper, we implemented EC exponentiation for various coordinate systems and the best result took 122ms, assuming 50ns clock cycle.

show abstract

A high‐speed RSD‐based flexible ECC processor for arbitrary curves over general prime field

Shah

Javeed

Azmat

et al. 2018

Circuit Theory & Apps

View full text Add to dashboard Cite

SummaryThis workpresents a novel high‐speed redundant‐signed‐digit (RSD)‐based elliptic curve cryptographic (ECC) processor for arbitrary curves over a general prime field. The proposed ECC processor works for any value of the prime number and curve parameters. It is based on a new high speed Montgomery multiplier architecture which uses different parallel computation techniques at both circuit level and architectural level. At the circuit level, RSD and carry save techniques are adopted while pre‐computation logic is incorporated at the architectural level. As a result of these optimization strategies, the proposed Montgomery multiplier offers a significant reduction in computation time over the state‐of‐the‐art. At the system level, to further enhance the overall performance of the proposed ECC processor, Montgomery ladder algorithm with (X,Y)‐only common Z coordinate (co‐Z) arithmetic is adopted. The proposed ECC processor is synthesized and implemented on different Xilinx Virtex (V) FPGA families for field sizes of 256 to 521 bits. On V‐6 platform, it computes a single 256 to 521 bits scalar point multiplication operation in 0.65 to 2.6 ms which is up to 9 times speed‐up over the state‐of‐the‐art.

show abstract

Efficient Elliptic Curve Exponentiation Using Mixed Coordinates

Cited by 298 publications

References 23 publications

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

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

Fast Implementation of Elliptic Curve Defined over GF(pm) on CalmRISC with MAC2424 Coprocessor

A high‐speed RSD‐based flexible ECC processor for arbitrary curves over general prime field

Contact Info

Product

Resources

About