Logarithm Problem Solving

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Its elements are congruence classes modulo p, and the group product of two elements may be obtained by ordinary integer multiplication of the elements followed by reduction modulo p.

The kth power of one of the numbers in this group may be computed by finding its kth power as an integer and then finding the remainder after division by p.

Logarithms might be intimidating, but solving a logarithm is much simpler once you realize that logarithms are just another way to write out exponential equations.

≡ a (mod m) if r is a primitive root of m and gcd(a,m) = 1.

Discrete logarithms are quickly computable in a few special cases.

However, no efficient method is known for computing them in general.wiki How is a “wiki,” similar to Wikipedia, which means that many of our articles are co-written by multiple authors. Once you rewrite the logarithm into a more familiar form, you should be able to solve it as you would solve any standard exponential equation.To create this article, volunteer authors worked to edit and improve it over time. It requires running time linear in the size of the group G and thus exponential in the number of digits in the size of the group.Therefore, it is an exponential-time algorithm, practical only for small groups G.However none of them run in polynomial time (in the number of digits in the size of the group).becomes a product bk, and equality means congruence modulo p in the integers. While computing discrete logarithms and factoring integers are distinct problems, they share some properties: ) there is not only no efficient algorithm known for the worst case, but the average-case complexity can be shown to be about as hard as the worst case using random self-reducibility.For example, consider (Z ≡ 1 (mod 17), these are the only solutions.Equivalently, the set of all possible solutions can be expressed by the constraint that k ≡ 4 (mod 16).When the numbers involved are large, it is more efficient to reduce modulo p multiple times during the computation.Regardless of the specific algorithm used, this operation is called modular exponentiation.

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