CSCI381 Fall 2005

1
Efficient Exponentiation in Groups mod m and in
elliptic curve groups

• CSCI381 Fall 2005
• GWU

2
Efficient exponentiation(from Memon notes)

• Usual approach to computing xc mod n is
  inefficient when c is large.
• Example 551 involves 51 multiplications mod n
• Instead, represent c as bit string bk-1 b0 and
  use the following algorithm
• z 1
• For i k-1 downto 0 do
• z z2 mod n
• if bi 1 then z z x mod n
• How many multiplications? k 2ceiling(log2c)

3
Example

• Calculate 551 mod 7 efficiently
• 51 110011 25 24 21 20
• 551 ((((52)2)2)2)2 ? (((52)2)2)2 ? 52 ? 51
• How many multiplications did you need?

4
551 mod 7
i bi z
5 1 1 5
4 1 25 mod 7 4 20 mod 7 6
3 0 36 mod 7 1 1
2 0 1 1
1 1 1 5
0 1 25 mod 7 4 20 mod 7 6
5
Efficient Exponentiation Elliptic Curves

• E.g. y2 x3 6x 4 mod 7
• Curve O, (0, 2), (0, 5), (1, 2), (1, 5), (3,
  0), (4, 1), (4, 6), (6, 2), (6, 5)
• (3, 0) (3, 0) ?
• (3, 0) (4, 1) ?
• (1, 2) (1, 2) ?
• Is there a generator?

6
Efficient Exponentiation Elliptic Curves

• E.g. y2 x3 6x 4 mod 7
• Curve O, (0, 2), (0, 5), (1, 2), (1, 5), (3,
  0), (4, 1), (4, 6), (6, 2), (6, 5)
• 5 (1, 2) ?

7
Elliptic Curves Efficient Exponentiation

• Represent c as bit string bk-1 b0
• Then
• z 1
• For i k-1 downto 0 do
• z 2z
• if bi 1 then z z x

8
Or use algorithm from handout

• f(P, m)
• If m 0
• return (O)
• else
• If m (mod 2) 0
• return ( f(PP, m?2) )
• else
• return (Pf(PP, m?2))

9
5(1, 2) ?