Diffiehellman

Diffie-Hellman Key Exchange Whittfield Diffie and Martin Hellman are called the inventors of Public Key Cryptography. Diffie-Hellman Key Exchange is the first Public Key Algorithm published in 1976.

What is Diffie-Hellman? A Public Key Algorithm Only for Key Exchange Does NOT Encrypt or Decrypt Based on Discrete Logarithms Widely used in Security Protocols and Commercial Products Williamson of Britain’s CESG claims to have discovered it several years prior to 1976

Discrete Logarithms What is a logarithm? log 10 100 = 2 because 10 2 = 100 In general if log m b = a then m a = b Where m is called the base of the logarithm A discrete logarithm can be defined for integers only In fact we can define discrete logarithms mod p also where p is any prime number

Discrete Logarithm Problem The security of the Diffie-Hellman algorithm depends on the difficulty of solving the discrete logarithm problem (DLP) in the multiplicative group of a finite field

Sets, Groups and Fields A set is any collection of objects called the elements of the set Examples of sets: R , Z , Q If we can define an operation on the elements of the set and certain rules are followed then we get other mathematical structures called groups and fields

Groups A group is a set G with a custom-defined binary operation + such that: The group is closed under +, i.e., for a , b  G: a + b  G The Associative Law holds i.e., for any a , b , c  G: a + (b + c) = (a + b) + c There exists an identity element 0 , such that a + 0 = a For each a  G there exists an inverse element –a such that a + (-a) = 0 If for all a , b  G: a + b = b + a then the group is called an Abelian or commutative group If a group G has a finite number of elements it is called a finite group

More About Group Operations + does not necessarily mean normal arithmetic addition + just indicates a binary operation which can be custom defined The group operation could be denoted as • The group notation with + is called the additive notation and the group notation with • is called the multiplicative notation

Fields A field is a set F with two custom-defined binary operations + and • such that: The Field is closed under + and • , i.e., for a , b  F: a + b  F and a • b  F The Associative Law holds i.e., for any a , b , c  F: a + (b + c) = (a + b) + c and a • (b • c) = (a • b) • c There exist identity elements 0 and 1 , such that a + 0 = a and a • 1 = a For each a  F there exist inverse elements –a and a -1 such that a + (-a) = 0 and a • a -1 = 1 If a field F has a finite number of elements it is called a finite field

Examples of Groups Groups Set of real numbers R under + Set of real numbers R under * Set of integers Z under + Set of integers Z under *? Set of integers modulo a prime number p under + Set of integers modulo a prime number p under * Set of 3 X 3 matrices under + meaning matrix addition Set of 3 X 3 matrices under * meaning matrix multiplication? Fields Set of real numbers R under + and * Set of integers Z under + and * Set of integers modulo a prime number p under + and *

Generator of Group If for a  G, all members of the group can be written in terms of a by applying the group operation * on a a number of times then a is called a generator of the group G Examples 2 is a generator of Z * 11 2 and 3 are generator of Z * 19 1 6 3 7 9 10 5 8 4 2 2 m mod 11 10 9 8 7 6 5 4 3 2 1 m 1 10 5 12 6 3 11 15 17 6 3 14 7 13 16 8 4 2 2 m mod 19 11 12 10 11 16 10 12 15 4 14 14 13 17 16 13 17 1 18 18 6 2 7 15 5 8 9 3 3 m mod 19 9 8 7 6 5 4 3 2 1 m

Primitive Roots If x n = a then a is called the n-th root of x For any prime number p , if we have a number a such that powers of a mod p generate all the numbers between 1 to p-1 then a is called a Primitive Root of p . In terms of the Group terminology a is the generator element of the multiplicative group of the finite field formed by mod p Then for any integer b and a primitive root a of prime number p we can find a unique exponent i such that b = a i mod p The exponent i is referred to as the discrete logarithm or index , of b for the base a .

Diffie-Hellman Algorithm Five Parts Global Public Elements User A Key Generation User B Key Generation Generation of Secret Key by User A Generation of Secret Key by User B

Global Public Elements q Prime number   < q and  is a primitive root of q The global public elements are also sometimes called the domain parameters

User A Key Generation Select private X A X A < q Calculate public Y A Y A =  X A mod q

User B Key Generation Select private X B X B < q Calculate public Y B Y B =  X B mod q

Generation of Secret Key by User A K = (Y B ) X A mod q

Generation of Secret Key by User B K = (Y A ) X B mod q

Diffie-Hellman Example q = 97  = 5 X A = 36 X B = 58 Y A = 5 36 = 50 mod 97 Y B = 5 58 = 44 mod 97 K = (Y B ) X A mod q = 44 36 mod 97 = 75 mod 97 K = (Y A ) X B mod q = 50 58 mod 97 = 75 mod 97

Why Diffie-Hellman is Secure? Opponent has q ,  , Y A and Y B To get X A or X B the opponent is forced to take a discrete logarithm The security of the Diffie-Hellman Key Exchange lies in the fact that, while it is relatively easy to calculate exponentials modulo a prime, it is very difficult to calculate discrete logarithms. For large primes, the latter task is considered infeasible.

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