Dr. A. Betten
Fall 2009
MATH 360 Mathematics of Information Security
Assignment # 5
Problem # 22
On the elliptic curve over F29 defined by y 2 = x3 + 4x + 20, compute
a) (5, 22) + (16, 27)
b) 2 · (5, 22)
Problem # 23
a) How many points over F5 does the elliptic curve defined by y 2 = x3 + 3x have?
b) How many points over F5 does the elliptic curve defined by y 2 = x3 + 4x have?
c) Is the group of points in a) cyclic?
c) Is the group of points in b) cyclic?
Problem # 24
Bob chooses the elliptic curve over F5 defined by y 2 = x3 + x + 1. The list of points and the
addition table is
i
Pi


7 8 5 6 3 4 1 2 0
0 (0, 1, 1)
 8 6 7 4 5 2 3 0 1 


1 (0, 4, 1)
 5 7 3 8 1 6 0 4 2 


2 (2, 1, 1)
 6 4 8 2 7 0 5 1 3 


3 (2, 4, 1)
 3 5 1 7 0 8 2 6 4 
 4 2 6 0 8 1 7 3 5 
4 (3, 1, 1)


5 (3, 4, 1)
 1 3 0 5 2 7 4 8 6 


6 (4, 3, 1)
 2 0 4 1 6 3 8 5 7 
7 (4, 2, 1)
0 1 2 3 4 5 6 7 8
8 (0, 1, 0)
This means that Pi + Pj = Pk where k is the entry in row i and column j of the table.
Suppose Bob publishes G = P0 = (0, 1, 1) and B = P6 = (4, 3, 1). Alice sends (P2 , P6 ).
Which point is the message M that Alice wants to send?