January 2007
Beginner
544 pages
14h 21m
English
Content preview from Computer Security and Cryptography15.4 THE ELLIPTIC GROUP OVER
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15.4 THE ELLIPTIC GROUP OVER 
The elliptic group described in Proposition 15.2 consists of points in the plane. It can be generalized to obtain a discrete group consisting of pairs of points in the set 
when p > 3 as follows.
The elliptic curve 
p(a, b) over consists of the abstract point at infinity 
and all pairs of points (x, y) with 
that satisfy
The elements of p(a, b) are found as follows:
- Compute z = x3 + Ax + B (modulo p).
- Check if z is a quadratic residue, by verifying
; in this case, there are two points (x, ±y) in p(a, b) with y2 = z (modulo p).
Proposition 15.4: If p = 3 (modulo ...
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