TY - CHAP

T1 - Improved algorithms for efficient arithmetic on elliptic curves using fast endomorphisms

AU - Ciet, Mathieu

AU - Lange, Tanja

AU - Sica, Francesco

AU - Quisquater, Jean Jacques

PY - 2003

Y1 - 2003

N2 - In most algorithms involving elliptic curves, the most expensive part consists in computing multiples of points. This paper investigates how to extend the τ-adic expansion from Koblitz curves to a larger class of curves defined over a prime field having an efficiently-computable endomorphism φ in order to perform an efficient point multiplication with efficiency similar to Solinas' approach presented at CRYPTO '97. Furthermore, many elliptic curve cryptosystems require the computation of k0P + k1Q. Following the work of Solinas on the Joint Sparse Form, we introduce the notion of φ-Joint Sparse Form which combines the advantages of a φ-expansion with the additional speedup of the Joint Sparse Form. We also present an efficient algorithm to obtain the φ-Joint Sparse Form. Then, the double exponentiation can be done using the φ endomorphism instead of doubling, resulting in an average of l applications of φ and l/2 additions, where l is the size of the ki's. This results in an important speed-up when the computation of φ is particularly effective, as in the case of Koblitz curves.

AB - In most algorithms involving elliptic curves, the most expensive part consists in computing multiples of points. This paper investigates how to extend the τ-adic expansion from Koblitz curves to a larger class of curves defined over a prime field having an efficiently-computable endomorphism φ in order to perform an efficient point multiplication with efficiency similar to Solinas' approach presented at CRYPTO '97. Furthermore, many elliptic curve cryptosystems require the computation of k0P + k1Q. Following the work of Solinas on the Joint Sparse Form, we introduce the notion of φ-Joint Sparse Form which combines the advantages of a φ-expansion with the additional speedup of the Joint Sparse Form. We also present an efficient algorithm to obtain the φ-Joint Sparse Form. Then, the double exponentiation can be done using the φ endomorphism instead of doubling, resulting in an average of l applications of φ and l/2 additions, where l is the size of the ki's. This results in an important speed-up when the computation of φ is particularly effective, as in the case of Koblitz curves.

KW - Elliptic curves

KW - Fast endomorphisms

KW - Joint Sparse Form

UR - http://www.scopus.com/inward/record.url?scp=35248868300&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=35248868300&partnerID=8YFLogxK

U2 - 10.1007/3-540-39200-9_24

DO - 10.1007/3-540-39200-9_24

M3 - Chapter

AN - SCOPUS:35248868300

SN - 3540140395

SN - 9783540140399

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 388

EP - 400

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

A2 - Biham, Eli

PB - Springer Verlag

ER -