Dr. A. Betten
Fall 2009
MATH 360 Mathematics of Information Security
Midterm # 3 – Practice
Problem # 1
Bob chooses the elliptic curve y 2 = x3 + 2x + 4 mod 31. He also chooses the base point
P = (0, 2, 1) and the secret integer s = 11. Can you help him compute s · P ? Hint: you can
get by with 5 additions.
Problem # 2
Bob choses the elliptic curve E given by y 2 = x3 + 4x + 3 over F137 . Alice is trying to encode
her message as points on E. She uses the alphabet A = 01, B = 02, . . . , Z = 26. If m is
the message, can you help her encode m as point (x, y) on E? Use the Kobliz-encoding as
p
x = km + j where k = ! 27
" = 5 and 0 ≤ j < 5 to encode
SEE YOU SOON
Problem # 3
Suppose you know that 2 is a primitive root modulo 419. If 398 = 2x mod 419, what is x?
Use the Baby-Step-Giant-Step algorithm.
Problem # 4
In the finite field F256 generated by the irreducible polynomial X 8 + X 4 + X 3 + X + 1,
compute:
a) The product 100111 times 101111 (your answer must be reduced).
b) The inverse of 10100101.
Problem # 5
Can you compute the entry (10, 5) in the AES S-box? Hint: the previous problem can help.
Problem # 6
Create a finite field with 8 elements. In it, find a primitive element (i.e., a generator for the
multiplicative group).
Problem # 7
Consider the elliptic curve y 2 + xy = x3 + 1 over F4 .
a) List all the points.
b) Let Φ be the map that takes (x, y, z) to (x2 , y 2 , z 2 ). For each point P , compute Φ(P ).
c) Dwaw the points in a 4 × 4 grid (with ∞ on top) and show the map Φ using arrows.
Problem # 8
Consider the elliptic curve E given by y 2 = x3 − 10x + 21 over F557 . Suppose you know that
P = (2, 3, 1) has order 189, so the number of points N557 is a multiple of 189. Use the Hasse
bound
√
|q + 1 − Nq | ≤ 2 q
to determine N557 .
Problem # 9
Consider the elliptic curve E given by y 2 = x3 + 2x + 4 over F17 . The addition table is
10
15
13
6
12
4
8
2
14
7
1
5
11
9
3
0
15
10
7
14
5
11
3
9
6
13
0
12
4
2
8
1
13
7
12
15
8
6
0
4
10
5
9
3
14
11
1
2
6
14
15
11
7
9
5
1
4
10
8
13
2
0
12
3
12
5
8
7
0
15
2
6
13
3
11
1
10
14
9
4
4
11
6
9
15
1
7
3
2
14
12
10
0
8
13
5
8
3
0
5
2
7
4
15
12
1
14
9
13
10
11
6
2
9
4
1
6
3
15
5
0
11
13
14
8
12
10
7
14
6
10
4
13
2
12
0
11
15
3
7
9
1
5
8
7
13
5
10
3
14
1
11
15
12
2
8
6
4
0
9
1
0
9
8
11
12
14
13
3
2
15
4
5
7
6
10
5
12
3
13
1
10
9
14
7
8
4
0
15
6
2
11
11
4
14
2
10
0
13
8
9
6
5
15
1
3
7
12
9
2
11
0
14
8
10
12
1
4
7
6
3
5
15
13
3
8
1
12
9
13
11
10
5
0
6
2
7
15
4
14
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
You are trying to show that the group is cyclic. To do this, trace through the multiples of
a suitable point by drawing arrows in the following diagram: