Num-Primes: CSPRNG Large Composite, Prime, Safe Prime Generator

This crate provides a beautifully simplistic API for generating large, cryptographically-random, unsigned integers in rust, including but not limited to composite, prime, and safe prime numbers.

It takes full advantage of the num crate on stable rust.

• Read the About
• Read the License
• Read the Contribution

Notice

Please note there is a critical bug in this program that I cannot seem to fix where it marks some prime numbers as not prime. It is in the miller-rabin implementation and I cannot seem to fix it. If anyone is up to it, feel free to look through the issues tab for information about the bug and submit a PR if you find a fix.

Usage

Add this to your Cargo.toml:

[dependencies]
num-primes = "0.2.0"

Warning

There is currently a major bug in is_prime() and is_composite() that makes some values return wrong. For example, a prime can sometimes be marked as composite unless it was generated as they use the same tests to test for primality.

How To Use

There are three main structs that are included in this library

Generator

Generate Composite Number

This function will generate a composite number of n-bits.

use num_primes::Generator;

fn main(){
  // Generate Composite of 1024 bits
  let composite = Generator::new_composite(1024);
}

Generate Prime Number

This function will generate a prime number of n-bits.

use num_primes::Generator;

fn main(){
  // Generate two primes (p,q) of 512 bits each
  let p = Generator::new_prime(512);
  let q = Generator::new_prime(512);
  
  // Multiply to get the modulus (n)
  let n = p * q;
}

Generate Safe Prime

This function will generate a safe prime number of n-bits. This function uses the same tests openssl uses to generate safe primes, which is (n-1)/2.

This function is quite time consuming and should be avoided for large sizes.

use num_primes::Generator;

fn main(){
  // Generate Safe Prime of 64 bits | Uses (n-1)/2 to check
  let safe_prime = Generator::safe_prime(64);
}

Generate Large Unsigned Integer

This function will generate a large unsigned integer of n-bits. This function is faster than generating a composite or prime number due to no checks being done.

use num_primes::Generator;

fn main(){
  // Generate a Large Unsigned Integer of 1024 bits without running any checks
  let x = Generator::new_uint(1024);
}

Verification

WARNING: There is currently a bug that makes verification of certain prime numbers fail. Be careful when using this feature.

Verify Composite Number

This function will verify whether a BigUint type is a composite by returning a boolean value.

use num_primes::{Generator,Verification};

fn main(){
  // Generates Composite Number of 1024 bits
  let x = Generator::new_composite(1024);
  
  // Check if the number is a Composite Number
  let is_composite: bool = Verification::is_composite(&x);
  
  // Asserts that 'is_composite' is true
  assert_eq!(is_composite, true);
}

Verify Prime Number

This function will verify whether a BigUint type is a prime by returning a boolean value.

use num_primes::{Generator,Verification};

fn main(){
  // Generates Prime Number of 1024 bits
  let x = Generator::new_prime(1024);
  
  // Check if the number is a Prime Number
  let is_prime: bool = Verification::is_prime(&x);
  
  // Asserts that 'is_prime' is true
  assert_eq!(is_prime, true);
}

Verify Safe Prime Number

This function will verify whether a BigUint type is a safe prime by returning a boolean value.

use num_primes::{Generator,Verification};

fn main(){
  // Generates a Safe Prime Number of 128 bits
  let x = Generator::safe_prime(128);
  
  // Check if the number is a Safe Prime Number
  let is_safe_prime: bool = Verification::is_safe_prime(&x);
  
  // Asserts that `is_safe_prime` is true
  assert_eq!(is_safe_prime, true);
}

[Experimental] Verify VSN (Smooth Numbers)

EXPERIMENTAL: Please Avoid Using This Function As Of Now

Read Wolfram Alpha - Smooth Numbers

Read OEIS - P-Smooth Numbers

Read Wikipedia - Examples of VSN and VSSR

This function will verify whether a number is a very smooth number. It accepts three parameters as follows:

• m: &BigUint | prime
• n: f64 | constant
• c: u32 | constant

It follows the following equation:

1. Return m's Greatest Prime Factor as p
   1. if p <= log(n)c then its p-smooth
   2. if p > log(n)c then its not p-smooth

use num::traits::FromPrimitive;
use num_bigint::BigUint;
use num_primes::Verification;

fn main() {
    // Set BigUint To 7u64
    let x: BigUint = BigUint::from_u64(7u64).unwrap();

    // Verify Its A Smooth Number with parameters 
  		// m = 7 (&BigUint)
  		// n = 31.0 (f64)
  		// c = 5 (u32)
    let result: bool = Verification::is_very_smooth_number(&x,31.0,5);

  	// Print What Type of Smooth Number It Is And Whether Or Not It Is Smooth
    println!("Is A {} Smooth Number: {}",x,result);
  
  	// This problem should be 7-smooth
  	assert_eq!(result, true);
}

Factorization

NOTICE: Factorization is still in the works.

Prime Factorization

Read GeeksforGeeks - Efficient program to print all prime factors of a given number

This function lets you factorize composite numbers and prime numbers to find their Greatest Prime Factor.

use num_primes::{Generator,Factorization};

fn main() {
    // Generates New Unsighed Integer of 32 bits
    let uint = Generator::new_uint(32);
    
  	// Prime Factorization Returns Option<BigUint>    
    let factor = Factorization::prime_factor(uint);

  	// match the Option<BigUint>
    match factor {
        Some(factor) => println!("Largest Prime Factor: {}",factor),
        None => println!("No Prime Factors Found"),
    }
}

How Does It Work

The steps are listed below with n being the input number being factored:

A Primality Check is used first to determine whether the number is a prime or not.

1. while n is even, divide by 2
2. After Step 1, n must be odd.
   1. n_sqrt = Take the square root of n
3. Start a loop from i = 3 to n_sqrt
   1. While i / n
      1. Divide n by i
   2. On Failure of i dividing by n,
      1. increment i by 2 and continue
4. If n is a prime number and n > 2
   1. Return n

About

Prime Number Generation Design

The Prime Number Generation and parts of its code is based on Snips-AI's Medium Blog on Generating Prime Numbers.

A conservative attempt is made at deciding whether a number is prime or not. The number goes through the generation phase and 3 tests to determine if its prime:

Generation Phase

1. A single parameter is passed to the generator function that indicates the number of bits the prime should be.

2. The userspace CSPRNG is seeded by the operating system to generate the random numbers using the rand crate.

3. An unsigned integer is generated until it passes the prime test.

4. The number is sent to be processed by four tests

Primality Tests

The numbers go through multiple tests to determine whether they are composite or prime.

1. A check is done to see if the number is even.
2. An array of the first 2048 primes is used to check whether the number is divisble by any of the primes in the array.
3. Fermat's Little Theorem is performed
4. Miller-Rabin Primality Test, the gold standard recommended by the official RSA documentation and by NIST on generating primes, is performed with 16 iterations, the same used by Apple's cryptosystem.

If the number passes these tests, it is considered with high probability to be prime. Feel free to verify them yourselves on Wolfram Alpha by simply typing in the prime number.

Safe Primes

Safe Primes are generated simply by checking if (p-1)/2 is a prime with the tests listed above.

OpenSSL-Prime vs. Num-Primes

https://security.stackexchange.com/questions/176394/how-does-openssl-generate-a-big-prime-number-so-fast

OpenSSL LTS (1.1.1) has a doc page for prime generation, including how safe primes are generated.

OpenSSL should be prefered for serious cryptographic implementations due to the security of their code and their code having been audited.

Num-Primes may be useful in certain situations, like in embedded systems or where OpenSSLis overkill.

OpenSSL-prime:

• Generates Safe Primes using (n-1)/2 but is much more efficient
• Performs a default of 20 checks on prime numbers.

Num-Primes:

• Generates Safe Primes using (n-1)/2 but takes much longer for unknown reasons
• Performs a default of 16 checks on prime numbers through 4 different primality tests
• Supports #[no_std] and stable rust
• May be useful in situations where using OpenSSL would be overkill.

Differences Between num-primes and ramp-primes

ramp-primes is the original implementation of the prime number generator using the ramp library.

num-primes is the improved implementation using the num library and is available on stable release with no_std support.

num-primes (Stable)

Uses num, a pure-rust implementation for numeric types and traits.

• num is stable and can work on the Default, Beta, and Nightly Branch

• num is in Pure Rust

• num implements more features

• num has around ~6,000,000 downloads

• num is more developed than ramp.

num-primes has:

• Better API and Documentation
• More Functionality
• no_std support

ramp-primes (Unstable)

Uses ramp, a high-performance multiple-precision (aka "BigNum") library for working with numbers bigger than can normally be handled.

• ramp only works on unstable rust (the nightly branch).
• ramp is written in Rust and Inline Assembly (security concern)
• ramp is generally faster
• ramp is specifically focused on multiple-precision arithmetic
• ramp has around ~17,000 downloads
• ramp is not as developed as num

ramp-primes is:

• The First Implementation
• Generally Faster For Generation
• Less Features
• Only Works On Unstable Rust (Nightly Build)

License

Licensed under either of

• Apache License, Version 2.0

• MIT license

at your option.

Contribution

Unless you explicitly state otherwise, any contribution intentionally submitted for inclusion in the work by you, as defined in the Apache-2.0 license, shall be dual licensed as above, without any additional terms or conditions.