BSGS template

# Use

Given an integer a, b, P, where a, b are coprime, obtaining a positive integer x, such that A ^X ≡ B (P MOD)

# Derivation

The remaining cases are generated based mold cycle, i.e. A ⁰ , A ^{. 1} , ......, A ^{n-. 1-} remaining under the category modulus n (prime number) Significance

And A ^n- , A ^{n-+. 1} , ......, A ^{2N. 1-} same residue class, so the answer must be in [0, n-1] within.

Values ​​of powers of p sense die to obtain a part, and store them up, and then the remaining portion of such evaluation is not possible to find a way of using a direct look-up table has been evaluated.

Order evaluation length m = ⌈√ P ⌉.

No evaluation section how to find, such as A ^m , ......  ,  A ^{2 m - . 1} this section, if a solution, it must be A ^I  ⋅  A ^m  ≡  B  ( m O D  P ) of the case,

The A ^m to the right, into  A ^I  ≡  B '  ( m O D  P ),  B' = B · A ^{- m} . The Direct Access b ' have no corresponding answer i.

Thus, no portion of each segment can be evaluated to determine whether only one query solvable within the section.
A hash table to hold, x _i is a number from I to i, A ^x minimum value of x, the insert query are O ( log n- ),

Achieve overall complexity O (√ n- log n- ).