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Authors:
  Derek Hollowood -- D-Rex
  Kevin Lui       -- K-Lui

Description:

  A Number Theoretic Bit Generator is a bit generator whose security relies
  on a computationally difficult Number Theoretic problem. Here we implement
  various Number Theoretic Bit Generators to demonstrate their efficiency
  and statistical soundness.

====================

Files Available:

  Random Bit Generators:
    bbs.py        | Blum Blum Shub Random Bit Generator
    bm.py         | Blum-Micali Random Bit Generator
    dec.py        | Dual Elliptic Curve Random Bit Generator
    rsa.py        | RSA Random Bit Generator

  Helpers:
    ecur.py       | Elliptic Curve Arithmetic
    fun.py        | Useful Number Theoretic Functions

  Statistics:
    stest.py      | Statistical Tests on each instance of RBG
    tester.py     | Calls on stest.py to run statistical tests on many instances

====================

Details of Files:

bbs.py:
  This file implements the Blum Blum Shub Random Bit Generator. The function
  takes as input two large primes p and q, both congruent to 3 modulo 4,
  as well as an integer k, the number of bits to be generated. In general,
  phi(pq) should be at least 5 times larger than k to avoid repeating a period.
  
bm.py:
  This file implements the Blum-Micali Random Bit Generator. The function takes
  as input a large prime p as well as an integer k, the number of bits to
  be generated. Alternatively, one could modify the function and allow g to
  be inputted; care should be taken to ensure g is a primitive root. 
  In general, phi(p)=p-1 should be larger than k.

dec.py:
  This file implements the Dual Elliptic Curve Deterministic Random Bit
  Generator. The function takes as input a large prime p as well as an integer k,
  the number of bits to be generated. Alternatively, one could modify the
  function to allow elliptic curve parameters to be inputted; care should be
  taken to ensure the curve is non-singular. In general, phi(p) should be
  larger than k. 

rsa.py:
  This file implements the RSA Random Bit Generator. Theh function takes as input
  two large primes p and q as well as an integer k, the number of bits to be
  generated. In general, p and q should be close so that phi(p) and phi(q)
  share few factors. Furthermore, phi(pq) should be at least 5 times larger
  than k to avoid repeating a period.

ecur.py
  This file implements adding and multiplying on Elliptic Curves.

fun.py
  This file implements modulus inverse and prime factorization.

stest.py
  This file implements the statistical tests that are to be run on each RBG.
  In this implementation, the array of randomly generated bits should be a
  multiple of 5 or it will be cut off. The frequency is defined by the number of 
  1s divided by the total number. The bit switch is the number of times a bit
  differs from the next bit divided by the total bit. The poker test counts the
  number of disjoint 5 sequence, compares the result with the expected
  result, and gives the Chi square.

tester.py
  This file implements the ability to run stest.py on many instances.
  Currently, it runs 100 tests with 10 different parameters each with 10^6
  bits. The number of bits generated for each run can be changed by changing the
  size variable. The number of tests and the number of parameters can be
  changed by changing the for loop and expanding the prime array accordingly.
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