1. Lecture 3.2: Public Key Cryptography II
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Lecture 3.2: Public Key Cryptography II
CS 436/636/736
Spring 2013
Nitesh Saxena

2. Today’s Informative/Fun Bit – Acoustic Emanations
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Today’s Informative/Fun Bit – Acoustic Emanations
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Public Key Cryptography -- II

3. Course Administration
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Course Administration
HW1 – due at 11am on Feb 08
Any questions, or help needed?
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Public Key Cryptography -- II

4. Outline of Today’s Lecture
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Outline of Today’s Lecture
Number Theory
Modular Arithmetic
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Public Key Cryptography -- II

5. Public Key Cryptography -- II
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Modular Arithmetic
Definition: x is congruent to y mod m, if m divides (x-y). Equivalently, x and y have the same remainder when divided by m.
Notation:
Example:
We work in Zm = {0, 1, 2, …, m-1}, the group of integers modulo m
Example: Z9 ={0,1,2,3,4,5,6,7,8}
We abuse notation and often write = instead of
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Public Key Cryptography -- II

6. Public Key Cryptography -- II
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Addition in Zm :
Addition is well-defined:
3 + 4 = 7 mod 9.
3 + 8 = 2 mod 9.
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Public Key Cryptography -- II

7. Additive inverses in Zm
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Additive inverses in Zm
0 is the additive identity in Zm
Additive inverse of a is -a mod m = (m-a)
Every element has unique additive inverse.
4 + 5= 0 mod 9.
4 is additive inverse of 5.
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8. Public Key Cryptography -- II
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Multiplication in Zm :
Multiplication is well-defined:
3 * 4 = 3 mod 9.
3 * 8 = 6 mod 9.
3 * 3 = 0 mod 9.
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Public Key Cryptography -- II

9. Multiplicative inverses in Zm
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Multiplicative inverses in Zm
1 is the multiplicative identity in Zm
Multiplicative inverse (x*x-1=1 mod m)
SOME, but not ALL elements have unique multiplicative inverse.
In Z9 : 3*0=0, 3*1=3, 3*2=6, 3*3=0, 3*4=3, 3*5=6, …, so 3 does not have a multiplicative inverse (mod 9)
On the other hand, 4*2=8, 4*3=3, 4*4=7, 4*5=2, 4*6=6, 4*7=1, so 4-1=7, (mod 9)
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Public Key Cryptography -- II

10. Which numbers have inverses?
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Which numbers have inverses?
In Zm, x has a multiplicative inverse if and only if x and m are relatively prime or gcd(x,m)=1
E.g., 4 in Z9
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Public Key Cryptography -- II

11. Extended Euclidian: a-1 mod n
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Extended Euclidian: a-1 mod n
Main Idea: Looking for inverse of a mod n means looking for x such that x*a – y*n = 1.
To compute inverse of a mod n, do the following:
Compute gcd(a, n) using Euclidean algorithm.
Since a is relatively prime to m (else there will be no inverse) gcd(a, n) = 1.
So you can obtain linear combination of rm and rm-1 that yields 1.
Work backwards getting linear combination of ri and ri-1 that yields 1.
When you get to linear combination of r0 and r1 you are done as r0=n and r1= a.
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Public Key Cryptography -- II

12. Public Key Cryptography -- II
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Example – 15-1 mod 37
37 = 2 *
15 = 2 * 7 + 1
7 = 7 * 1 + 0
Now,
15 – 2 * 7 = 1
15 – 2 (37 – 2 * 15) = 1
5 * 15 – 2 * 37 = 1
So, 15-1 mod 37 is 5.
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Public Key Cryptography -- II

13. Modular Exponentiation: Square and Multiply method
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Modular Exponentiation: Square and Multiply method
Usual approach to computing xc mod n is inefficient when c is large.
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
Show an example: x^64 will require 6 squarings (or 6 multiplications).
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Public Key Cryptography -- II

14. Public Key Cryptography -- II
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Example: 3037 mod 77
z = z2 mod n
if bi = 1 then z = z* x mod n i b z 5 1
=1*1*30 mod 77 4
=30*30 mod 77 3
=53*53 mod 77 2
=37*37*30 mod 77
=29*29 mod 77
=71*71*30 mod 77
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Public Key Cryptography -- II

15. Public Key Cryptography -- II
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Other Definitions
An element g in G is said to be a generator of a group if a = gi for every a in G, for a certain integer i
A group which has a generator is called a cyclic group
The number of elements in a group is called the order of the group
Order of an element a is the lowest i (>0) such that ai = e (identity)
A subgroup is a subset of a group that itself is a group
Public Key Cryptography -- II

16. Public Key Cryptography -- II
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Lagrange’s Theorem
Order of an element in a group divides the order of the group
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17. Euler’s totient function
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Euler’s totient function
Given positive integer n, Euler’s totient function is the number of positive numbers less than n that are relatively prime to n
Fact: If p is prime then
{1,2,3,…,p-1} are relatively prime to p.
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Public Key Cryptography -- II

18. Euler’s totient function
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Euler’s totient function
Fact: If p and q are prime and n=pq then
Each number that is not divisible by p or by q is relatively prime to pq.
E.g. p=5, q=7: {1,2,3,4,-,6,-,8,9,-,11,12,13,-,-,16,17,18,19,-,-,22,23,24,-,26,27,-,29,-,31,32,33,34,-}
pq-p-(q-1) = (p-1)(q-1)
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Public Key Cryptography -- II

19. Euler’s Theorem and Fermat’s Theorem
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Euler’s Theorem and Fermat’s Theorem
If a is relatively prime to n then
If a is relatively prime to p then ap-1 = 1 mod p
Proof : follows from Lagrange’s Theorem
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Public Key Cryptography -- II

20. Euler’s Theorem and Fermat’s Theorem
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Euler’s Theorem and Fermat’s Theorem
EG: Compute 9100 mod 17: p =17, so p-1 = = 6·16+4. Therefore, 9100=96·16+4=(916)6(9)4 . So mod 17 we have 9100  (916)6(9)4 (mod 17)  (1)6(9)4 (mod 17)  (81)2 (mod 17)  16
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Public Key Cryptography -- II

21. Public Key Cryptography -- II
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Some questions
2-1 mod 4 =?
What is the complexity of
(a+b) mod m
(a*b) mod m
a-1 mod (m)
xc mod (n)
Order of a group is 5. What can be the order of an element in this group?
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Public Key Cryptography -- II

22. Public Key Cryptography -- II
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Further Reading
Chapter 4 of Stallings
Chapter 2.4 of HAC
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