Choose a, c, m in Linear Congruential Generator - excel

Question

I am looking to implement a linear congruential generator in Excel.
As we know, we must choose the parameter of LCG is a, c, m, and Z0.
Wikipedia says that
The period of a general LCG is at most m, and for some choices of
factor a much less than that. The LCG will have a full period for all
seed values if and only if:
m and the offset c are relatively prime,
a - 1 is divisible by all prime factors of m,
a - 1 is divisible by 4 if m is divisible by 4.
Also,
m, 0 < m – the "modulus"
a, 0 < a < m – the "multiplier"
c, 0 < c < m – the "increment"
Z0, 0 < Z0 < m – the "seed" or "start value"
I need to choose those values, I want Z0 initial value is 10113383, and the rest is random. Nah, what values that has a specified period and guaranteed no collisions for the duration of that period?
I've tried to put some values, a=13, c=911, m=11584577 and it looks no collisions. But I'm not sure if I break the rules or not.

Answer 1

I regularly teach a number theory and cryptography course so have built up a library of programs in VBA and Python. Using these, I only needed to write one more:
Function GCD(num1 As Long, num2 As Long) As Long
Dim a As Long
Dim b As Long
a = num1
b = num2
Dim R As Long
R = 1
Do Until R = 0
R = a Mod b
a = b
b = R
Loop
GCD = a
End Function
Sub Helper_Factor(ByVal n As Long, ByVal p As Long, factors As Collection)
'Takes a passed collection and adds to it an array of the form
'(q,k) where q >= p is the smallest prime divisor of n
'p is assumed to be odd
'The function is called in such a way that
'the first divisor found is automatically prime
Dim q As Long, k As Long
q = p
Do While q <= Sqr(n)
If n Mod q = 0 Then
k = 1
Do While n Mod q ^ k = 0
k = k + 1
Loop
k = k - 1 'went 1 step too far
factors.Add Array(q, k)
Helper_Factor n / q ^ k, q + 2, factors
Exit Sub
End If
q = q + 2
Loop
'if we get here then n is prime - add it as a factor
factors.Add Array(n, 1)
End Sub
Function Factor(ByVal n As Long) As Collection
Dim factors As New Collection
Dim k As Long
Do While n Mod 2 ^ k = 0
k = k + 1
Loop
k = k - 1
If k > 0 Then
n = n / 2 ^ k
factors.Add Array(2, k)
End If
If n > 1 Then Helper_Factor n, 3, factors
Set Factor = factors
End Function
Function DisplayFactors(n As Long) As String
Dim factors As Collection
Dim i As Long, p As Long, k As Long
Dim sfactors As Variant
Set factors = Factor(n)
ReDim sfactors(1 To factors.Count)
For i = 1 To factors.Count
p = factors(i)(0)
k = factors(i)(1)
sfactors(i) = p & IIf(k > 1, "^" & k, "")
Next i
DisplayFactors = Join(sfactors, "*")
End Function
Function MaxPeriod(a As Long, c As Long, m As Long) As Boolean
'assumes 0 < a,c < m
Dim factors As Collection
Dim p As Long, i As Long
If GCD(c, m) > 1 Then Exit Function 'with default value of False
If m Mod 4 = 0 And (a - 1) Mod 4 > 0 Then Exit Function
'else:
Set factors = Factor(m)
For i = 1 To factors.Count
p = factors(i)(0)
If p < m And (a - 1) Mod p > 0 Then Exit Function
Next i
'if you survive to here:
MaxPeriod = True
End Function
For example, in the Intermediate Window you can check:
?maxperiod(13,911,11584577)
True
so you seem to be in luck