Get Discrete Logarithms

Additive DLog : The analogous definition for (Zp,+) is well-defined with given a,g,n find x>0 such that xg≡amodn. ... Multiplicative DLog :, the discrete Logarithm Problem (DLP) is given a,g,n∈Z+ find x∈Z>1 such that ax≡gmodn, if such x exists.

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). Discrete Logarithms - Dartmouth College Mathematics dartmouth.edu https://math.dartmouth.edu › ~carlp › dltalk09 dartmouth.edu https://math.dartmouth.edu › ~carlp › dltalk09

The discrete logarithmic problem is over a finite group (for example, Z∗p); in contrast to R∗, we don't have group elements arbitrarily close together; we call this type of group 'discrete'.

Discrete logarithm is just the inverse operation. For example, take the equation 3k≡12 (mod 23) for k. As shown above k=4 is a solution, but it is not the only solution. Since 322≡1 (mod 23), it also follows that if n is an integer then 34+22n≡12×1n≡12 (mod 23).

In its most standard form, the discrete logarithm problem in a finite group G can be stated as follows: Given α ∈ G and β ∈ hαi, find the least positive integer x such that αx = β. In additive notation (which we will often use), this means xα = β.

The discrete logarithm problem is a cornerstone of modern cryptography, providing the foundation for the security of the Diffie-Hellman key exchange protocol. Its computational difficulty ensures that securely exchanged keys remain confidential, enabling secure communication over potentially insecure channels.

A discrete logarithm is just the inverse operation. For example, take the equation 3k≡12 (mod 23) for k. As shown above k=4 is a solution, but it is not the only solution. Since 322≡1 (mod 23), it also follows that if n is an integer, then 34+22n≡12×1n≡12 (mod 23).