Discrete Logarithm Problem

Discrete logarithms are logarithms defined with regard to multiplicative cyclic groups. If G is a multiplicative cyclic group and g is a generator of G, then from the definition of cyclic groups, we know every element h in G can be written as gx for some x. The discrete logarithm to the base g of h in the group G is defined to be x . For example, if the group is Z5*, and the generator is 2, then the discrete logarithm of 1 is 4 because 24 ≡ 1 mod 5.

The discrete logarithm problem is defined as: given a group G, a generator g of the group and an element h of G, to find the discrete logarithm to the base g of h in the group G. Discrete logarithm problem is not always hard. The hardness of finding discrete logarithms depends on the groups. For example, a popular choice of groups for discrete logarithm based crypto-systems is Zp* where p is a prime number. However, if p−1 is a product of small primes, then the Pohlig–Hellman algorithm can solve the discrete logarithm problem in this group very efficiently. That’s why we always want p to be a safe prime when using Zp* as the basis of discrete logarithm based crypto-systems. A safe prime is a prime number which equals 2q+1 where q is a large prime number. This guarantees that p-1 = 2q has a large prime factor so that the Pohlig–Hellman algorithm cannot solve the discrete logarithm problem easily. Even p is a safe prime, there is a sub-exponential algorithm which is called the index calculus. That means p must be very large (usually at least 1024-bit) to make the crypto-systems safe.

Source: https://www.doc.ic.ac.uk/~mrh/330tutor/ch06s02.html


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