Picking A, C and M for Linear congruential generator

Question

I am looking to implement a simple pseudorandom number generator (PRNG) that has a specified period and guaranteed no collisions for the duration of that period. After doing some research I came across the very famous LCG which is perfect. The problem is, I am having trouble understanding how to properly configure it. Here is my current implementation:

    function LCG (state)
    {
        var a = ?;
        var c = ?;
        var m = ?;

        return (a * state + c) % m;
    }

It says that in order to have a full period for all seed values the following conditions must be met:

1. c and m are relatively prime
2. a-1 is divisible by all prime factors of m
3. a-1 is a multiple of 4 if m is a multiple of 4

1 and 3 are simple to understand and test for. However what about 2, I don't quite understand what that means or how to check for it. And what about C, can it be zero? what if it's non-zero?

Overall I need to select A, C and M in such a way that I have a period of 48^5 - 1. M is equal to the period, I am not sure about A and C.

Answer 1

From Wikipedia:

Provided that c is nonzero, the LCG will have a full period for all seed values if and only if:

1. c and m are relatively prime,
2. a-1 is divisible by all prime factors of m,
3. a-1 is a multiple of 4 if m is a multiple of 4.

You said you want a period of 48⁵-1, so you must choose m≥48⁵-1. Let's try choosing m=48⁵-1 and see where that takes us. The conditions from the Wikipedia article prohibit you from choosing c=0 if you want the period to be m.

Note that 11, 47, 541, and 911 are the prime factors of 48⁵-1, since they're all prime and 11*47*541*911 = 48⁵-1.

Let's go through each of those conditions:

1. For c and m to be relatively prime, c and m must have no common prime factors. So, pick any prime numbers other than 11, 47, 541, and 911, then multiply them together to choose your c.
2. You'll need to choose a such that a-1 is divisible by all the prime factors of m, i.e., a = x*11*47*541*911 + 1 for any x of your choosing.
3. Your m is not a multiple of 4, so you can ignore the third condition.

In summary:

• m = 48⁵-1,
• c = any product of primes other than 11, 47, 541, and 911 (also, c must be less than m),
• a = x*11*47*541*911 + 1, for any nonnegative x of your choice (also, a must be less than m).

---

Here's a smaller test case (in Python) using a period of 48²-1 (which has prime factors 7 and 47):

def lcg(state):
    x = 1
    a = x*7*47 + 1
    c = 100
    m = 48**2 - 1
    return (a * state + c) % m

expected_period = 48**2 - 1
seeds = [5]
for i in range(expected_period):
    seeds.append(lcg(seeds[-1]))
print(len(set(seeds)) == expected_period)

It outputs True, as it should. (If you have any trouble reading Python, let me know and I can translate it to JavaScript.)