Walking on the Hypercube

Jelly, 9 bytes

%⁴2*^µÐḶL

Takes two command-line arguments.

%⁴2*^µÐḶL        A monadic link. Inputs: a_0. b also taken from command line.
%⁴2*^              Variadic link. Input: a
%⁴                   a modulo b. ⁴ is second input, b.
  2*                 Get 2 to that power
    ^                and bitwise xor with a.
     µ             Start a new, monadic link (input: a_0)
      ÐḶ             All elements of the cycle created when the preceding link
                     is applied repeatedly, starting with a_0.
        L            Length.

Try it here.

Haskell, 124

import Data.Bits
(y:z:w)%(x:s)|x==y||x==z=[i|(i,r)<-zip[1..]s,r==x]!!0|0<1=w%s
g n=(tail>>=(%)).iterate(\a->xor a$2^mod a n)

This finds the circle by the two-pointers-going-around-in-different-speeds algorithm, and heavily uses/abuses Haskell's approach to lists (for example, the two pointers are actually lists).

g is the function that computes the answer. give it n and then a[0] and it will return the number to you (note that n should be defined to be of type Int to avoid type ambiguity).