Multiplicative order

Multiplicative order
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The multiplicative order of x relative to y is the least integer n such that x^n is 1 modulo y . For example, 37 mo 1000 is 100 because 37^100 is 1 modulo 1000 (and no number smaller than 100 would do). Note that the multiplicative order is undefined if x and y are not relatively prime.

There is an algorithm in Bach & Shallit, Algorithmic Number Theory I, exercise 5.8, page 118. A naive algorithm can require more than the age of the universe to produce an answer for large arguments.

J

mo=: 4 : 0
 a=. x: x
 m=. x: y
 assert. 1=a+.m
 *./ a mopk"1 |: __ q: m
)

mopk=: 4 : 0
 a=. x: x
 'p k'=. x: y
 pm=. (p^k)&|@^
 t=. (p-1)*p^k-1  NB. totient
 'q e'=. __ q: t
 x=. a pm t%q^e
 d=. (1<x)+x (pm i. 1:)&> (e-1) */\@$&.> q
 */q^d
)

For example:

   37 mo 1000
100
   2 mo _1+10^80x
190174169488577769580266953193403101748804183400400

Java

public BigInteger mo(BigInteger x, BigInteger y){
  BigInteger retVal = BigInteger.ZERO;
  for(;x.modPow(retVal, y) != BigInteger.ONE;retVal = retVal.add(BigInteger.ONE));
  return retVal;
}