What is discrete logarithm problem and explain it?

Discrete logarithms are logarithms defined with regard to multiplicative cyclic groups. 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.

How do you solve a discrete log?

Finding a discrete logarithm can be very easy. For example, say G = Z/mZ and g = 1. More specifically, say m = 100 and t = 17. Then logg t = 17 (or more precisely 17 mod 100).

What is log in discrete mathematics?

In mathematics, for given real numbers a and b, the logarithm logb a is a number x such that bx = a. Analogously, in any group G, powers bk can be defined for all integers k, and the discrete logarithm logb a is an integer k such that bk = a. Discrete logarithms are quickly computable in a few special cases.

What is the discrete logarithm assumption?

1. The discrete logarithm problem is a cryptography concept that underlies the basis of much of modern cryptography. In general the discrete logarithm problem states that for the equation bk (mod p) = a. where b , a , and p are known values and p is prime, finding k is hard.

What is the importance of discrete logarithms?

Aside from the intrinsic interest that the problem of computing discrete logarithms has, it is of considerable importance in cryptography. An efficient algorithm for discrete logarithms would make several authentication and key-exchange systems insecure.

What do you mean by discrete logarithm problem explain its importance in public key cryptography?

It enables two parties independently to compute a shared secret that cannot be obtained by an eavesdropper. In its most basic form, two parties agree on and make public a large prime p (the modulus) and a generator g that generates the group Z∗p or a large subgroup of it.

Is the discrete log problem NP complete?

NP-complete. The subset of NP to which all problems in NP can be reduced, i.e. the subset of NP that is NP-hard. All problems which are in NP but neither in P nor in NPC.

What is the use of discrete logarithm?

The term “discrete logarithm” is most commonly used in cryptography, although the term “generalized multiplicative order” is sometimes used as well (Schneier 1996, p. 501). In number theory, the term “index” is generally used instead (Gauss 1801; Nagell 1951, p.

Does RSA use discrete logarithm?

RSA labs makes a similar statement: The discrete logarithm problem bears the same relation to these systems as factoring does to the RSA system: the security of these systems rests on the assumption that discrete logarithms are difficult to compute.

What is AP problem?

A P problem is one that can be solved in “polynomial time,” which means that an algorithm exists for its solution such that the number of steps in the algorithm is bounded by a polynomial function of n, where n corresponds to the length of the input for the problem. Thus, P problems are said to be easy, or tractable.

Which is the definition of the discrete logarithm problem?

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.

When to use Diffie-Hellman for discrete logarithm?

With Diffie–Hellman a cyclic group modulus a prime p is used, allowing an efficient computation of the discrete logarithm with Pohlig–Hellman if the order of the group (being p −1) is sufficiently smooth, i.e. has no large prime factors . While computing discrete logarithms and factoring integers are distinct problems, they share some properties:

Is the discrete logarithm problem computationally intractable?

The discrete logarithm problem is considered to be computationally intractable. That is, no efficient classical algorithm is known for computing discrete logarithms in general. A general algorithm for computing logb a in finite groups G is to raise b to larger and larger powers k until the desired a is found.

Can a discrete logarithm be computed in polynomial time?

Can the discrete logarithm be computed in polynomial time on a classical computer? The discrete logarithm problem is considered to be computationally intractable. That is, no efficient classical algorithm is known for computing discrete logarithms in general.