1. Diffie-Hellman Key Exchange
MATH 3396
Instructor: Alex Karassev

2. Discrete logarithm problem (DLP)
Тема №1 Серверное оборудование
a, b from Fp*
The smallest non-negative integer x such that ax = b is called the discrete logarithm of b to the base a
Notation: x = Log a b числа b по основанию а
Example: 42 = 5 in F11 and therefore 2 = log4 5
Knowing a and b it is hard to find x
Exhaustive search: O(p)
Improvement: O(p1/2)
УЦ Сетевая Академия ЛАНИТ, 2008

3. Sophie Germain primes A prime q such that 2q+1 is also prime
In this case p = 2q+1 is called safe prime
First few Sophie Germain primes: 2, 3, 5, 11, 23, 29, 41, 53
First few safe primes: 5,11,23,47,59
Largest known (as of 2016) Sophie Germain prime has digits
Conjecture: there are infinitely many Sophie Germain prime

4. Diffie-Hellman key exchange
Choose large prime p (preferably, a safe prime)
Choose g in Fp* such that ord g is a large prime
(if p = 2 q +1, ord g = q)
p and g are NOT secret (in fact, usually g = 2 or 3)

5. Diffie-Hellman Key exchange
Тема №1 Серверное оборудование
p, g
non-secure channel A
Secret а
A = ga
Secret b
B = gb B
Alice computes K=Ba
Shared secret key:
K = Ba = (gb)a = gab = (ga)b = Ab
Bob computes
K=Ab
A possible way to find K = gab : knowing p, g, А, find а, such that ga = A mod p which is the Discrete Logarithm Problem
УЦ Сетевая Академия ЛАНИТ, 2008

6. Тема №1 Серверное оборудование
Example
p = 11, g = 2
A = 8
a = 3
A = 23 = 8
B = 6
b = 9
B = 29 =512 = 46*11+6 = 6
Alice computes K = Ba = 63 = 216 = 19* = 7
Bob computes K = Ab = 89 = = = = *11+ 7 = 7
Shared secret key K = 7
УЦ Сетевая Академия ЛАНИТ, 2008

7. Man-in-the-middle attack
Diffie-Hellman key exchange protocol is not protected against man-in-the middle attack:
An authenticated version of Diffie-Hellman protocol can be obtained with the use of digital signature

8. Digital signature – brief overview
Bob needs to send Alice a document m
Alice needs to make sure that document has been sent by Bob and not by somebody else
Bob send c=e(m), and encrypted E(f(m)), where s=f(m) is a function of m (digital signature), and E is the encryption function using Bob’s private key for digital signature
Alice receives c and s, and computes m=d(c) and s=D(E(f(m))), using Bob’s public key for digital signature
Verification: if f(m) = s, then the document has been sent by Bob