Simulation of a Hall-Irwin distribution applied on Z_p

simul.png

simulate.py

from random import randint
import matplotlib.pyplot as plt

P = 29
ITERATIONS = 100000
SUMNUMS = 100

a = [0]*P
for k in range(ITERATIONS):
  b = sum([randint(0,P) for i in range(SUMNUMS)])%P
  a[b]+=1

plt.bar(range(P), a)
plt.show()

theory.md

While the simulation graph seems to indicate that a sum of 100 random numbers over Z_29 has a uniform distribution, I don't see any specific reason that would lead me to believe that this is actually true, especially for any prime field and any number of summands.

This could be proved or refuted by taking the probability distribution given in theorem 2.1 of this paper and checking that \sum^{n-1}_{i=0} P(i*p + j) is the same \forall j in {0, ... ,p-1} where n is the number of summands and p is the prime that defines the field. Note that the k used in the definition of P would need to be replaced by p-1.