lib.rs

#![no_std]
#![forbid(unsafe_code)]

extern crate num;
extern crate rand;
extern crate num_bigint as bigint;



use core::ops::Sub;
use num::Integer;
pub use bigint::{BigUint,RandBigInt};
use num_traits::{Zero, One};
use num_traits::*;

use log::debug;
use log::error;
use log::info;

// Settings
// NIST recomends 5 rounds for miller rabin. This implementation does 8. Apple uses 16. Three iterations has a probability of 2^80 of failing
const MILLER_RABIN_ROUNDS: usize = 8usize;


/// # Generator
/// This is the most commonly used struct. It is used to generate:
/// - Large Unsigned Integers
/// - Composite Numbers
/// - Prime Numbers
/// - Safe Primes
/// 
/// ```
/// 
/// use num_primes::{Generator,Verification};
/// 
/// fn main(){
///     let prime = Generator::new_prime(512);
///     let _uint = Generator::new_uint(1024);
/// 
///     // p = 2q + 1 || where p is safe prime
///     let _safe_prime = Generator::safe_prime(64);
/// 
///     let _ver: bool = Verification::is_prime(&prime);
/// }
/// ```
pub struct Generator;
/// # Prime Verification
/// This struct is used to verify whether integers are prime, safe prime, or composite.
/// 
/// ```
/// use num_primes::{Generator,Verification};
/// 
/// fn main(){
///     let prime = Generator::new_prime(1024);
///     let safe = Generator::safe_prime(128);
/// 
///     let is_prime: bool = Verification::is_prime(&prime);
///     let is_safe_prime: bool = Verification::is_safe_prime(&safe);
/// 
///     assert_eq!(is_prime, true);
///     assert_eq!(is_safe_prime, true);
/// }
/// ```
pub struct Verification;
/// # Prime Factorization
/// This struct is used to factor large numbers and return their largest prime factor
/// ```
/// use num_primes::{Generator,Factorization};
/// 
/// fn main() {
///     // Generates New Unsighed Integer of 64 bits
///     let uint = Generator::new_uint(64);
///     // Prime Factorization    
///     let prime_factor = Factorization::prime_factor(uint);
/// 
///     match prime_factor {
///         Some(x) => println!("Largest Prime Factor: {}",x),
///         None => println!("No Prime Factors Found"),
///     }
/// }
/// ```
pub struct Factorization;

impl Generator {
    /// # Generate Large Composite Numbers
    /// This function guarantees the returned value is a composite number and is not prime.
    /// ```
    /// use num_primes::Generator;
    /// 
    /// fn main(){
    ///     // Generate Composite Number of 1024 bits
    ///     let composite = Generator::new_composite(1024);
    /// 
    ///     // Print Out Composite Number
    ///     println!("Composite Number: {}",composite);
    /// }
    /// ```
    pub fn new_composite(n: usize) -> BigUint {
        let mut rng = rand::thread_rng();
        loop {
            // Make mutable and set LSB and MSB
            let candidate: BigUint = rng.gen_biguint(n);
            //candidate.set_bit(0, true);
            //candidate.set_bit((n-1) as u32, true);
            if is_prime(&candidate) == false { 
                return candidate;
            }
        }
    }
    /// # Generate Large Unsigned Integer
    /// This function takes an input (n) for the number of bits the unsigned    integer should be
    /// 
    /// ```
    /// use num_primes::Generator;
    ///
    /// fn main() {
    ///     // Generate Large Random Unsigned Integer of 1024 bits
    ///     let x = Generator::new_uint(1024);
    /// 
    ///     // Print Integer
    ///     println!("Large Unsigned Integer: {}",x);
    /// }
    /// ```
    pub fn new_uint(n: usize) -> BigUint {
        let mut rng = rand::thread_rng();
        return rng.gen_biguint(n);
    }

    /// # Generate Prime Number
    /// This function generates a prime number of n-bits using three primality  tests.
    /// 
    /// ```
    /// use num_primes::Generator;
    /// 
    /// fn main(){
    ///     // Generate Two Primes (p,q) of 512 bits
    ///     let p = Generator::new_prime(512);
    ///     let q = Generator::new_prime(512);
    /// 
    ///     // Multiply p and q and return n
    ///     let n = p * q;
    /// 
    ///     // Print n
    ///     println!("n: {}",n);
    /// }
    /// ```
    pub fn new_prime(n: usize) -> BigUint {
        let mut rng = rand::thread_rng();
        
        loop {
            // Make mutable and set LSB and MSB
            let candidate: BigUint = rng.gen_biguint(n);
            
            //candidate.set_bit(0, true);
            //candidate.set_bit((n-1) as u32, true);
            
            if is_prime(&candidate) == true { 
                return candidate;
            }
        }
    }

    /// # Generate Safe Primes
    /// This function will generate safe prime numbers, or numbers of the form p = 2q + 1 where p is the safe prime.
    /// ```
    /// use num_primes::Generator;
    /// 
    /// fn main(){
    ///     // p = 2q + 1 where p is the safe prime. This generates a number of 64 bits.
    ///     let safe_prime = Generator::safe_prime(64);
    /// }
    /// ```
    pub fn safe_prime(n: usize) -> BigUint {
        let mut rng = rand::thread_rng();
        loop {
            // Make mutable and set LSB and MSB
            let candidate: BigUint = rng.gen_biguint(n);
            //candidate.set_bit(0, true);
            //candidate.set_bit((n-1) as u32, true);
            if is_prime(&candidate) == true {
                if is_safe_prime(&candidate) == true {
                    // checks with (p-1/n)
                    return candidate
                }
            }
        }
    }
}

impl Verification {
    pub fn is_prime(n: &BigUint) -> bool {
        return is_prime(n);
    }
    pub fn is_composite(n: &BigUint) -> bool {
        let x: bool = is_prime(n);
        
        if x == true {
            return false
        }
        else if x == false {
            return true
        }
        else {
            panic!("An Error Has Occured On Checking Composite Number");
        }
    }
    pub fn is_safe_prime(n: &BigUint) -> bool {
        return is_safe_prime(n);
    }
    /// # Very Smooth Number
    /// This Function Is Deprecated And Should Rarely Be Used
    /// ```
    /// use num_traits::FromPrimitive;
    /// use num_bigint::BigUint;
    /// use num_primes::Verification;
    /// 
    /// fn main(){
    ///     // Set BigUint To 7
    ///     let x: BigUint = BigUint::from_u64(7u64).unwrap();
    /// 
    ///     // Verify Its A Smooth Number
    ///     let result: bool = Verification::is_smooth_number(&x,31.0,5);
    /// 
    ///     println!("Is A {} Smooth Number: {}",x,result);
    /// }
    /// ```
    pub fn is_very_smooth_number(m: &BigUint, n: f64, c: u32) -> bool {
        return vsn(m,n,c);
    }
}

impl Factorization {
    /// # Prime Factorization
    /// This is a method of factoring the largest prime factors of large numbers. It is slow and should not be relied on. It can easily factor 32-bit numbers and sometimes 64-bit.
    /// 
    /// It returns as a `Option<BigUint>` type
    /// 
    /// ```
    /// use num_primes::{Generator,Factorization};
    /// 
    /// fn main() {
    ///     // Generates New Unsighed Integer of 32 bits
    ///     let uint = Generator::new_uint(32);
    ///     // Prime Factorization    
    ///     let factor = Factorization::prime_factor(uint);
    /// 
    ///     match factor {
    ///         Some(factor) => println!("Largest Prime Factor: {}",factor),
    ///         None => println!("No Prime Factors Found"),
    ///     }
    /// }
    /// ```
    pub fn prime_factor(mut n: BigUint) -> Option<BigUint> {
        // Check Primality and if prime, returns prime
        if is_prime(&n) {
            return Some(n)
        }

        let one = BigUint::one();
        let two = &one + &one;
        
        // STEP 1 | n divided by 2
        while n.is_even() {
            n = n / &two;
        }
        
        // STEP 2 | 3..sqrt(n) | Divide i by n. On failure, add 2 to i
        let n_sqrt = n.sqrt().to_usize().unwrap();
        
        for mut i in 3..n_sqrt {
            while n.divides(&BigUint::from(i)) {
                n = n / BigUint::from(i);
            }
            i = i + 2usize;
        }

        // Step 3
        if n > two {
            return Some(n)
        }
        else {
            return None
        }


    }
}

// if true, then is not prime
// if false, then maybe prime
fn div_small_primes(numb: &BigUint) -> bool {
    let zero: BigUint = Zero::zero();
    let one: BigUint = One::one();

    
    static SMALL_PRIMES: [u32; 2048] = [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293,307,311,313,317,331,337,347,349,353,359,367,373,379,383,389,397,401,409,419,421,431,433,439,443,449,457,461,463,467,479,487,491,499,503,509,521,523,541,547,557,563,569,571,577,587,593,599,601,607,613,617,619,631,641,643,647,653,659,661,673,677,683,691,701,709,719,727,733,739,743,751,757,761,769,773,787,797,809,811,821,823,827,829,839,853,857,859,863,877,881,883,887,907,911,919,929,937,941,947,953,967,971,977,983,991,997,1009,1013,1019,1021,1031,1033,1039,1049,1051,1061,1063,1069,1087,1091,1093,1097,1103,1109,1117,1123,1129,1151,1153,1163,1171,1181,1187,1193,1201,1213,1217,1223,1229,1231,1237,1249,1259,1277,1279,1283,1289,1291,1297,1301,1303,1307,1319,1321,1327,1361,1367,1373,1381,1399,1409,1423,1427,1429,1433,1439,1447,1451,1453,1459,1471,1481,1483,1487,1489,1493,1499,1511,1523,1531,1543,1549,1553,1559,1567,1571,1579,1583,1597,1601,1607,1609,1613,1619,1621,1627,1637,1657,1663,1667,1669,1693,1697,1699,1709,1721,1723,1733,1741,1747,1753,1759,1777,1783,1787,1789,1801,1811,1823,1831,1847,1861,1867,1871,1873,1877,1879,1889,1901,1907,1913,1931,1933,1949,1951,1973,1979,1987,1993,1997,1999,2003,2011,2017,2027,2029,2039,2053,2063,2069,2081,2083,2087,2089,2099,2111,2113,2129,2131,2137,2141,2143,2153,2161,2179,2203,2207,2213,2221,2237,2239,2243,2251,2267,2269,2273,2281,2287,2293,2297,2309,2311,2333,2339,2341,2347,2351,2357,2371,2377,2381,2383,2389,2393,2399,2411,2417,2423,2437,2441,2447,2459,2467,2473,2477,2503,2521,2531,2539,2543,2549,2551,2557,2579,2591,2593,2609,2617,2621,2633,2647,2657,2659,2663,2671,2677,2683,2687,2689,2693,2699,2707,2711,2713,2719,2729,2731,2741,2749,2753,2767,2777,2789,2791,2797,2801,2803,2819,2833,2837,2843,2851,2857,2861,2879,2887,2897,2903,2909,2917,2927,2939,2953,2957,2963,2969,2971,2999,3001,3011,3019,3023,3037,3041,3049,3061,3067,3079,3083,3089,3109,3119,3121,3137,3163,3167,3169,3181,3187,3191,3203,3209,3217,3221,3229,3251,3253,3257,3259,3271,3299,3301,3307,3313,3319,3323,3329,3331,3343,3347,3359,3361,3371,3373,3389,3391,3407,3413,3433,3449,3457,3461,3463,3467,3469,3491,3499,3511,3517,3527,3529,3533,3539,3541,3547,3557,3559,3571,3581,3583,3593,3607,3613,3617,3623,3631,3637,3643,3659,3671,3673,3677,3691,3697,3701,3709,3719,3727,3733,3739,3761,3767,3769,3779,3793,3797,3803,3821,3823,3833,3847,3851,3853,3863,3877,3881,3889,3907,3911,3917,3919,3923,3929,3931,3943,3947,3967,3989,4001,4003,4007,4013,4019,4021,4027,4049,4051,4057,4073,4079,4091,4093,4099,4111,4127,4129,4133,4139,4153,4157,4159,4177,4201,4211,4217,4219,4229,4231,4241,4243,4253,4259,4261,4271,4273,4283,4289,4297,4327,4337,4339,4349,4357,4363,4373,4391,4397,4409,4421,4423,4441,4447,4451,4457,4463,4481,4483,4493,4507,4513,4517,4519,4523,4547,4549,4561,4567,4583,4591,4597,4603,4621,4637,4639,4643,4649,4651,4657,4663,4673,4679,4691,4703,4721,4723,4729,4733,4751,4759,4783,4787,4789,4793,4799,4801,4813,4817,4831,4861,4871,4877,4889,4903,4909,4919,4931,4933,4937,4943,4951,4957,4967,4969,4973,4987,4993,4999,5003,5009,5011,5021,5023,5039,5051,5059,5077,5081,5087,5099,5101,5107,5113,5119,5147,5153,5167,5171,5179,5189,5197,5209,5227,5231,5233,5237,5261,5273,5279,5281,5297,5303,5309,5323,5333,5347,5351,5381,5387,5393,5399,5407,5413,5417,5419,5431,5437,5441,5443,5449,5471,5477,5479,5483,5501,5503,5507,5519,5521,5527,5531,5557,5563,5569,5573,5581,5591,5623,5639,5641,5647,5651,5653,5657,5659,5669,5683,5689,5693,5701,5711,5717,5737,5741,5743,5749,5779,5783,5791,5801,5807,5813,5821,5827,5839,5843,5849,5851,5857,5861,5867,5869,5879,5881,5897,5903,5923,5927,5939,5953,5981,5987,6007,6011,6029,6037,6043,6047,6053,6067,6073,6079,6089,6091,6101,6113,6121,6131,6133,6143,6151,6163,6173,6197,6199,6203,6211,6217,6221,6229,6247,6257,6263,6269,6271,6277,6287,6299,6301,6311,6317,6323,6329,6337,6343,6353,6359,6361,6367,6373,6379,6389,6397,6421,6427,6449,6451,6469,6473,6481,6491,6521,6529,6547,6551,6553,6563,6569,6571,6577,6581,6599,6607,6619,6637,6653,6659,6661,6673,6679,6689,6691,6701,6703,6709,6719,6733,6737,6761,6763,6779,6781,6791,6793,6803,6823,6827,6829,6833,6841,6857,6863,6869,6871,6883,6899,6907,6911,6917,6947,6949,6959,6961,6967,6971,6977,6983,6991,6997,7001,7013,7019,7027,7039,7043,7057,7069,7079,7103,7109,7121,7127,7129,7151,7159,7177,7187,7193,7207,7211,7213,7219,7229,7237,7243,7247,7253,7283,7297,7307,7309,7321,7331,7333,7349,7351,7369,7393,7411,7417,7433,7451,7457,7459,7477,7481,7487,7489,7499,7507,7517,7523,7529,7537,7541,7547,7549,7559,7561,7573,7577,7583,7589,7591,7603,7607,7621,7639,7643,7649,7669,7673,7681,7687,7691,7699,7703,7717,7723,7727,7741,7753,7757,7759,7789,7793,7817,7823,7829,7841,7853,7867,7873,7877,7879,7883,7901,7907,7919,7927,7933,7937,7949,7951,7963,7993,8009,8011,8017,8039,8053,8059,8069,8081,8087,8089,8093,8101,8111,8117,8123,8147,8161,8167,8171,8179,8191,8209,8219,8221,8231,8233,8237,8243,8263,8269,8273,8287,8291,8293,8297,8311,8317,8329,8353,8363,8369,8377,8387,8389,8419,8423,8429,8431,8443,8447,8461,8467,8501,8513,8521,8527,8537,8539,8543,8563,8573,8581,8597,8599,8609,8623,8627,8629,8641,8647,8663,8669,8677,8681,8689,8693,8699,8707,8713,8719,8731,8737,8741,8747,8753,8761,8779,8783,8803,8807,8819,8821,8831,8837,8839,8849,8861,8863,8867,8887,8893,8923,8929,8933,8941,8951,8963,8969,8971,8999,9001,9007,9011,9013,9029,9041,9043,9049,9059,9067,9091,9103,9109,9127,9133,9137,9151,9157,9161,9173,9181,9187,9199,9203,9209,9221,9227,9239,9241,9257,9277,9281,9283,9293,9311,9319,9323,9337,9341,9343,9349,9371,9377,9391,9397,9403,9413,9419,9421,9431,9433,9437,9439,9461,9463,9467,9473,9479,9491,9497,9511,9521,9533,9539,9547,9551,9587,9601,9613,9619,9623,9629,9631,9643,9649,9661,9677,9679,9689,9697,9719,9721,9733,9739,9743,9749,9767,9769,9781,9787,9791,9803,9811,9817,9829,9833,9839,9851,9857,9859,9871,9883,9887,9901,9907,9923,9929,9931,9941,9949,9967,9973,10007,10009,10037,10039,10061,10067,10069,10079,10091,10093,10099,10103,10111,10133,10139,10141,10151,10159,10163,10169,10177,10181,10193,10211,10223,10243,10247,10253,10259,10267,10271,10273,10289,10301,10303,10313,10321,10331,10333,10337,10343,10357,10369,10391,10399,10427,10429,10433,10453,10457,10459,10463,10477,10487,10499,10501,10513,10529,10531,10559,10567,10589,10597,10601,10607,10613,10627,10631,10639,10651,10657,10663,10667,10687,10691,10709,10711,10723,10729,10733,10739,10753,10771,10781,10789,10799,10831,10837,10847,10853,10859,10861,10867,10883,10889,10891,10903,10909,10937,10939,10949,10957,10973,10979,10987,10993,11003,11027,11047,11057,11059,11069,11071,11083,11087,11093,11113,11117,11119,11131,11149,11159,11161,11171,11173,11177,11197,11213,11239,11243,11251,11257,11261,11273,11279,11287,11299,11311,11317,11321,11329,11351,11353,11369,11383,11393,11399,11411,11423,11437,11443,11447,11467,11471,11483,11489,11491,11497,11503,11519,11527,11549,11551,11579,11587,11593,11597,11617,11621,11633,11657,11677,11681,11689,11699,11701,11717,11719,11731,11743,11777,11779,11783,11789,11801,11807,11813,11821,11827,11831,11833,11839,11863,11867,11887,11897,11903,11909,11923,11927,11933,11939,11941,11953,11959,11969,11971,11981,11987,12007,12011,12037,12041,12043,12049,12071,12073,12097,12101,12107,12109,12113,12119,12143,12149,12157,12161,12163,12197,12203,12211,12227,12239,12241,12251,12253,12263,12269,12277,12281,12289,12301,12323,12329,12343,12347,12373,12377,12379,12391,12401,12409,12413,12421,12433,12437,12451,12457,12473,12479,12487,12491,12497,12503,12511,12517,12527,12539,12541,12547,12553,12569,12577,12583,12589,12601,12611,12613,12619,12637,12641,12647,12653,12659,12671,12689,12697,12703,12713,12721,12739,12743,12757,12763,12781,12791,12799,12809,12821,12823,12829,12841,12853,12889,12893,12899,12907,12911,12917,12919,12923,12941,12953,12959,12967,12973,12979,12983,13001,13003,13007,13009,13033,13037,13043,13049,13063,13093,13099,13103,13109,13121,13127,13147,13151,13159,13163,13171,13177,13183,13187,13217,13219,13229,13241,13249,13259,13267,13291,13297,13309,13313,13327,13331,13337,13339,13367,13381,13397,13399,13411,13417,13421,13441,13451,13457,13463,13469,13477,13487,13499,13513,13523,13537,13553,13567,13577,13591,13597,13613,13619,13627,13633,13649,13669,13679,13681,13687,13691,13693,13697,13709,13711,13721,13723,13729,13751,13757,13759,13763,13781,13789,13799,13807,13829,13831,13841,13859,13873,13877,13879,13883,13901,13903,13907,13913,13921,13931,13933,13963,13967,13997,13999,14009,14011,14029,14033,14051,14057,14071,14081,14083,14087,14107,14143,14149,14153,14159,14173,14177,14197,14207,14221,14243,14249,14251,14281,14293,14303,14321,14323,14327,14341,14347,14369,14387,14389,14401,14407,14411,14419,14423,14431,14437,14447,14449,14461,14479,14489,14503,14519,14533,14537,14543,14549,14551,14557,14561,14563,14591,14593,14621,14627,14629,14633,14639,14653,14657,14669,14683,14699,14713,14717,14723,14731,14737,14741,14747,14753,14759,14767,14771,14779,14783,14797,14813,14821,14827,14831,14843,14851,14867,14869,14879,14887,14891,14897,14923,14929,14939,14947,14951,14957,14969,14983,15013,15017,15031,15053,15061,15073,15077,15083,15091,15101,15107,15121,15131,15137,15139,15149,15161,15173,15187,15193,15199,15217,15227,15233,15241,15259,15263,15269,15271,15277,15287,15289,15299,15307,15313,15319,15329,15331,15349,15359,15361,15373,15377,15383,15391,15401,15413,15427,15439,15443,15451,15461,15467,15473,15493,15497,15511,15527,15541,15551,15559,15569,15581,15583,15601,15607,15619,15629,15641,15643,15647,15649,15661,15667,15671,15679,15683,15727,15731,15733,15737,15739,15749,15761,15767,15773,15787,15791,15797,15803,15809,15817,15823,15859,15877,15881,15887,15889,15901,15907,15913,15919,15923,15937,15959,15971,15973,15991,16001,16007,16033,16057,16061,16063,16067,16069,16073,16087,16091,16097,16103,16111,16127,16139,16141,16183,16187,16189,16193,16217,16223,16229,16231,16249,16253,16267,16273,16301,16319,16333,16339,16349,16361,16363,16369,16381,16411,16417,16421,16427,16433,16447,16451,16453,16477,16481,16487,16493,16519,16529,16547,16553,16561,16567,16573,16603,16607,16619,16631,16633,16649,16651,16657,16661,16673,16691,16693,16699,16703,16729,16741,16747,16759,16763,16787,16811,16823,16829,16831,16843,16871,16879,16883,16889,16901,16903,16921,16927,16931,16937,16943,16963,16979,16981,16987,16993,17011,17021,17027,17029,17033,17041,17047,17053,17077,17093,17099,17107,17117,17123,17137,17159,17167,17183,17189,17191,17203,17207,17209,17231,17239,17257,17291,17293,17299,17317,17321,17327,17333,17341,17351,17359,17377,17383,17387,17389,17393,17401,17417,17419,17431,17443,17449,17467,17471,17477,17483,17489,17491,17497,17509,17519,17539,17551,17569,17573,17579,17581,17597,17599,17609,17623,17627,17657,17659,17669,17681,17683,17707,17713,17729,17737,17747,17749,17761,17783,17789,17791,17807,17827,17837,17839,17851,17863];
    
    for p in SMALL_PRIMES.iter() {
        if numb % &BigUint::from(*p) == zero {
            return false
        }
        // Fixes part of Issue 1 but may slow down generation | https://github.com/AtropineTearz/num-primes/issues/1
        if numb / &BigUint::from(*p) == one {
            return true
        }
    }
    return true
}



fn fermat(candidate: &BigUint) -> bool {
    let mut rng = rand::thread_rng();
    let random: BigUint = rng.gen_biguint_below(candidate);
    
    // p - 1
    let exponent: BigUint = candidate.sub(BigUint::one());
    
    let result = random.modpow(&(exponent), candidate);

    //let result = random.pow_mod(&(candidate - One::one()), candidate);
    result == One::one()
}

fn miller_rabin(candidate: &BigUint, limit: usize) -> bool {    
    // One and Two in ramp::Int form
    let zero: BigUint = Zero::zero();
    let one = BigUint::one();
    let two: BigUint = &one + &one;
    let two_2 = num_bigint::ToBigUint::to_biguint(&2).unwrap();

    // Check Whether Candidate Is 2 (which is prime)
    if candidate == &two_2 {
        return true
    }

    // Check Whether Candidate Is Even
    /*
    if candidate.mod(two) {
        return false
    }
    */
    
    let (d,s) = rewrite(&candidate);
    let step = s.sub(&one).to_usize().unwrap();

    let mut rng = rand::thread_rng();

    for _i in 0..limit {
        // Generate Random Number between [2,n-1) | Exclusive End Range; Uses (n-1), not (n-2)
        let a = rng.gen_biguint_range(&two, &(candidate-&one));
        
        // Reference Implementation
        // Pretty sure `sample_range()` has an inclusive end
        //let basis = Int::sample_range(&two, &(candidate-&two));
        
        // (a^d mod n)
        let mut x = a.modpow(&d, &candidate);

        // Reference Implementation
        //let mut y = Int::modpow(&basis, &d, candidate);

        if x == one || x == (candidate - &one) {
            continue
            // return true
        }
        else {
            // Convert To Usizes For Loop
            // step = (s - 1)
            let one_usize = one.to_usize().unwrap();
            let zero_usize = zero.to_usize().unwrap();
            
            let mut break_early = false;
            // Issue #1 | Changed one_usize to zero_usize; step (s-1) was equal to iterations-1 and therefore needed an extra iteration
            for _ in zero_usize..step {
                x = x.modpow(&two,candidate);

                // Reference Implementation
                //y = Int::modpow(&y, &two, candidate);
                
                if x == one {
                    return false
                } 
                else if x == (candidate - BigUint::one()) {
                    break_early = true;
                    break;
                }
                
            }

            if !break_early {
                return false
            }
        }
    }
    return true
}

// Rewrite for Miller-Rabin
fn rewrite(n: &BigUint) -> (BigUint,BigUint) {
    let one: BigUint = BigUint::one();
    let two: BigUint = BigUint::one() + BigUint::one();
    let mut s: BigUint = BigUint::zero();
    
    
    

    // (n-1) becomes even number
    let mut d: BigUint = n - &one;

    // The Main Loop That Checks Whether The Number is even and then divides by 2 and stores a counter 
    
    while d.is_even() == true {
        d = d.div_floor(&two);
        s += &one;
    }

    return (d.clone(),s)
}

// true = probably prime
// false = not prime (composite)
fn is_prime(candidate: &BigUint) -> bool {
    let zero: BigUint = Zero::zero();
    let one: BigUint = One::one();
    let two: BigUint = &one + &one;

    if candidate == &zero {
        return false
    }
    
    if candidate.is_even() && candidate != &two {
        return false
    }
    
    // First, simple trial divide
    if div_small_primes(candidate) == false {
        return false
    }

     // Second, Fermat's little theo test on the candidate
    if fermat(candidate) == false {
        return false;
    }

    // Finally, Miller-Rabin test
    if miller_rabin(candidate, MILLER_RABIN_ROUNDS) == false {
        return false
    }
    else {
        return true
    }
    return true
}

// p = 2q + 1
#[deprecated]
fn is_safe_prime_add(number: &BigUint) -> bool {    
    // number == q
    
    let one = BigUint::one();
    let two = &one + &one;

    let x = number * two;
    
    // p
    let p = x + one;
    
    if is_prime(&p) {
        return true;
    }
    else {
        return false;
    }
}

// (p - 1)/2
fn is_safe_prime(number: &BigUint) -> bool {
    let one = BigUint::one();
    let two = &one + &one;

    let result = (number - one) / two;

    if is_prime(&result) {
        return true
    }
    else {
        return false
    }
}

// TODO FIX ME
fn vsn(m: &BigUint,n: f64, c: u32) -> bool {
    // c: fixed constant

    // if m's greatest prime factor < log(n)^c
    let result = n.log10().powi(c as i32).ceil();

    let factor = Factorization::prime_factor(m.clone()).unwrap();

    if factor <= BigUint::from_f64(result).unwrap() {
        return true
    }
    else if factor > BigUint::from_f64(result).unwrap() {
        return false
    }
    else {
        panic!("The Very Smooth Function Is Deprecated And Should Not Be Used")
    }
}

fn pollard_rho(mut n: BigUint) {
    // Initialize Random Number Generator
    let mut rng = rand::thread_rng();
    
    // Set one and two
    let zero: BigUint = Zero::zero();
    let one: BigUint = One::one();
    let two: BigUint = &one + &one;

    // x
    let x = &two;
    let mut y = &zero;

    let mut i: usize = 0usize;
    let mut counter: usize = 10usize;
    //let x = rng.gen_biguint_range(&zero,&(&two % (n - two));

}


#[cfg(test)]
#[test]
fn generate(){
    let _x = Generator::new_prime(512);
}

#[cfg(test)]
#[test]
fn generate_safe_prime(){
    // p = 2q + 1 where p is safe prime
    let _p = Generator::safe_prime(64);
}

#[test]
fn prime_factor(){
    let x = Generator::new_uint(16);
    let _prime_factor = Factorization::prime_factor(x);
}