Abstract
Let y 2 = x 3 + ax + b be an elliptic curve over F; p, p being a prime number greater than 3, and consider a; b ∈ [1, p]. In this paper, we study elliptic curve isomorphisms, with a view towards reduction in the size of elliptic curves coefficients. We first consider reducing the ratio a/b. We then apply these considerations to determine the number of elliptic curve isomorphism classes. Later we work on both coefficients. We introduce the number M.p/as the lower bound of all M ∈ ℕ such that each isomorphism class has a representative with max (a,b) <M. Using results from the theory of uniform distributions, we prove upper and lower bounds of the form c 1p1/2 <M(p) <c 2p 3/4 with explicit constants c 1, c 2 > 0.
| Original language | English |
|---|---|
| Pages (from-to) | 89-100 |
| Number of pages | 12 |
| Journal | Journal of Mathematical Cryptology |
| Volume | 5 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - Jun 2011 |
| Externally published | Yes |
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Keywords
- Cryptography
- Elliptic curves
- Exponential sums
- Uniform distribution
ASJC Scopus subject areas
- Computer Science Applications
- Applied Mathematics
- Computational Mathematics
Cite this
Compact elliptic curve representations. / Ciet, Mathieu; Quisquater, Jean Jacques; Sica, Francesco.
In: Journal of Mathematical Cryptology, Vol. 5, No. 1, 06.2011, p. 89-100.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Compact elliptic curve representations
AU - Ciet, Mathieu
AU - Quisquater, Jean Jacques
AU - Sica, Francesco
PY - 2011/6
Y1 - 2011/6
N2 - Let y 2 = x 3 + ax + b be an elliptic curve over F; p, p being a prime number greater than 3, and consider a; b ∈ [1, p]. In this paper, we study elliptic curve isomorphisms, with a view towards reduction in the size of elliptic curves coefficients. We first consider reducing the ratio a/b. We then apply these considerations to determine the number of elliptic curve isomorphism classes. Later we work on both coefficients. We introduce the number M.p/as the lower bound of all M ∈ ℕ such that each isomorphism class has a representative with max (a,b) <M. Using results from the theory of uniform distributions, we prove upper and lower bounds of the form c 1p1/2 <M(p) 2p 3/4 with explicit constants c 1, c 2 > 0.
AB - Let y 2 = x 3 + ax + b be an elliptic curve over F; p, p being a prime number greater than 3, and consider a; b ∈ [1, p]. In this paper, we study elliptic curve isomorphisms, with a view towards reduction in the size of elliptic curves coefficients. We first consider reducing the ratio a/b. We then apply these considerations to determine the number of elliptic curve isomorphism classes. Later we work on both coefficients. We introduce the number M.p/as the lower bound of all M ∈ ℕ such that each isomorphism class has a representative with max (a,b) <M. Using results from the theory of uniform distributions, we prove upper and lower bounds of the form c 1p1/2 <M(p) 2p 3/4 with explicit constants c 1, c 2 > 0.
KW - Cryptography
KW - Elliptic curves
KW - Exponential sums
KW - Uniform distribution
UR - http://www.scopus.com/inward/record.url?scp=84858422907&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84858422907&partnerID=8YFLogxK
U2 - 10.1515/JMC.2011.007
DO - 10.1515/JMC.2011.007
M3 - Article
AN - SCOPUS:84858422907
VL - 5
SP - 89
EP - 100
JO - Journal of Mathematical Cryptology
JF - Journal of Mathematical Cryptology
SN - 1862-2976
IS - 1
ER -