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More sophisticated algorithms exist, usually inspired by similar algorithms for integer factorization.These algorithms run faster than the naïve algorithm, some of them linear in the square root of the size of the group, and thus exponential in half the number of digits in the size of the group.
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.
A general algorithm for computing log a in finite groups G is to raise b to larger and larger powers k until the desired a is found.
This algorithm is sometimes called trial multiplication.
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.
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).
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