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//! Module for working with integer factorization. //! //! This module contains functions for factoring integers, //! computing the LCM and GCD of integers, and testing if //! integers are perfect squares and perfect cubes. use std::cmp::min; use std::mem; use super::prime; /// Find the GCD of `a` and `b` using the Euclidean algorithm. /// /// This function will return `0` if both arguments are zero. /// /// # Examples /// /// ``` /// use reikna::factor::gcd; /// assert_eq!(gcd(76, 54), 2); /// assert_eq!(gcd(18, 24), 6); /// assert_eq!(gcd(5, 2), 1); /// ``` pub fn gcd(mut a: u64, mut b: u64) -> u64 { if a < b { mem::swap(&mut a, &mut b); } while b != 0 { mem::swap(&mut a, &mut b); b %= a; } a } /// Return the GCD of the set of integers /// /// This function works by applying the fact that the /// GCD is both commutative and associative, and as such the GCD /// of a set can be found by computing the GCD of each of its /// members with a running GCD total. /// /// If an empty set is given, `0` will be returned. /// /// # Examples /// /// ``` /// use reikna::factor::gcd_all; /// assert_eq!(gcd_all(&vec![16, 4, 32]), 4); /// assert_eq!(gcd_all(&vec![3, 10, 18]), 1); /// ``` pub fn gcd_all(set: &[u64]) -> u64 { let mut gcd_: u64 = 0; for n in set { gcd_ = gcd(*n, gcd_); } gcd_ } /// Return `true` if `a` and `b` are coprime. /// /// This is a helper function that calls `gcd(a, b)` and checks /// if the result is `1`. /// /// # Examples /// /// ``` /// use reikna::factor::coprime; /// assert_eq!(coprime(8, 4), false); /// assert_eq!(coprime(9, 8), true); /// ``` pub fn coprime(a: u64, b: u64) -> bool { gcd(a, b) == 1 } /// Return the LCM of `a` and `b`. /// /// This function works by computing the GCD of `a` and `b` /// using `gcd()`, then applying the fact that /// /// ```text /// a * b /// lcm(a, b) = --------- /// gcm(a, b) /// ``` /// /// If both `a` and `b` are zero, `0` is returned. /// /// # Examples /// /// ``` /// ``` pub fn lcm(a: u64, b: u64) -> u64 { if a == 0 && b == 0 { return 0; } a * b / gcd(a, b) } /// Return the LCM of the set of integers /// /// This function works by applying the fact that the /// LCM is both commutative and associative, and as such the LCM /// of a set can be found by computing the LCM of each of its /// members with a running LCM total. /// /// If an empty set is given, `1` will be returned. /// /// # Examples /// /// ``` /// use reikna::factor::lcm_all; /// assert_eq!(lcm_all(&vec![8, 9, 21]), 504); /// assert_eq!(lcm_all(&vec![4, 7, 12, 21, 42]), 84); /// ``` pub fn lcm_all(set: &[u64]) -> u64 { let mut lcm_ = 1; for n in set { lcm_ = lcm(*n, lcm_); } lcm_ } /// List of least significant bytes for values /// that could be perfect squares. pub const GOOD_BYTES: [bool; 256] = [true , true , false, false, true , false, false, false, false, true , false, false, false, false, false, false, true , true , false, false, false, false, false, false, false, true , false, false, false, false, false, false, false, true , false, false, true , false, false, false, false, true , false, false, false, false, false, false, false, true , false, false, false, false, false, false, false, true , false, false, false, false, false, false, true , true , false, false, true , false, false, false, false, true , false, false, false, false, false, false, false, true , false, false, false, false, false, false, false, true , false, false, false, false, false, false, false, true , false, false, true , false, false, false, false, true , false, false, false, false, false, false, false, true , false, false, false, false, false, false, false, true , false, false, false, false, false, false, false, true , false, false, true , false, false, false, false, true , false, false, false, false, false, false, true , true , false, false, false, false, false, false, false, true , false, false, false, false, false, false, false, true , false, false, true , false, false, false, false, true , false, false, false, false, false, false, false, true , false, false, false, false, false, false, false, true , false, false, false, false, false, false, false, true , false, false, true , false, false, false, false, true , false, false, false, false, false, false, false, true , false, false, false, false, false, false, false, true , false, false, false, false, false, false, false, true , false, false, true , false, false, false, false, true , false, false, false, false, false, false, false, true , false, false, false, false, false, false, false, true , false, false, false, false, false, false]; /// Return `true` if `n` is a perfect square. /// /// This function works by taking the first byte of `n`, and /// checking to see if it is a candidate for being a perfect square. /// If it is not, `false` is returned. If it is, the square root is /// taken. If the root is an integral, `n` is a perfect square and `true` /// is returned, otherwise `false` is returned. /// /// # Examples /// /// ``` /// use reikna::factor::perfect_square; /// assert_eq!(perfect_square(435), false); /// assert_eq!(perfect_square(81), true); /// ``` pub fn perfect_square(n: u64) -> bool { if !GOOD_BYTES[(n & 0xff) as usize] { return false; } let root = (n as f64).sqrt() as u64; root * root == n } /// Return `true` if `n` is a perfect cube. /// /// This function works by checking if the digital root of `n` /// is equal to zero, one, eight, or nine. If it is not, `n` cannot /// be a perfect cube and the function returns `false`. If the /// digital root is a valid number, then the cube root of `n` is taken. /// If the root is an integer, then `n` is a perfect cube and `true` is /// returned, otherwise `false` is returned. /// /// # Examples /// /// ``` /// use reikna::factor::perfect_cube; /// assert_eq!(perfect_cube(216), true); /// assert_eq!(perfect_cube(9), false); /// ``` pub fn perfect_cube(n: u64) -> bool { if n == 0 { return true; } let dr = n - 9 * ((n - 1) as f64 / 9.0) as u64; if dr == 0 && dr != 1 && dr != 8 && dr != 9 { return false; } let root = (n as f64).cbrt(); if (root - root.round()).abs() > 0.000000001 { return false; } let root_i = root.round() as u64; root_i * root_i * root_i == n } /// Extract a factor of `val` using `entropy` as a seed /// value. /// /// This function will extract a non-trivial factor of `val` /// using Brent's modification of Pollard's Rho Algorithm. /// /// This is one of the functions used by `quick_factorize()`, /// it is applied if value being factor is considered to have /// "large" magnitude. /// /// This function is not very useful on its own, and should be /// integrated into a more general factorization function rather than /// used directly. pub fn rho(val: u64, entropy: u64) -> u64 { if val == 0 { return 1; } let entropy = entropy.wrapping_mul(val); let c = entropy & 0xff; let u = entropy & 0x7f; let mut r: u64 = 1; let mut q: u64 = 1; let mut y: u64 = entropy & 0xf; let mut fac = 1; let mut y_old = 0; let mut x = 0; let f = |x: u64| (x.wrapping_mul(x) + c) % val; while fac == 1 { x = y; for _ in 0..r { y = f(y); } let mut k = 0; while k < r && fac == 1 { y_old = y; for _ in 0..min(u, r - k) { y = f(y); if x > y { q = q.wrapping_mul(x - y) % val; } else { q = q.wrapping_mul(y - x) % val; } } fac = gcd(q, val); k += u; } r *= 2; } while fac == val || fac <= 1 { y_old = f(y_old); if x > y_old { fac = gcd(x - y_old, val); } else if x < y_old { fac = gcd(y_old - x, val); } else { // the algorithm has failed for this entropy, // return the factor as-is return fac; } } fac } /// The largest number considered "small" by `quick_factorize_wsp()`. /// /// Values less than this will be factored with `prime::factorize_wp()`, /// this is also the value used as the maximum argument to `prime_sieve()` /// in the `quick_factorize()` helper function. pub const MAX_SMALL_NUM: u64 = 65_536; /// Return a `Vec<u64>` of `value`'s prime factorization, /// using `sprimes` as a list of small primes; /// /// `sprimes` should be a sorted list of the prime numbers in /// `[1, MAX_SMALL_NUM]`, or else this function will not /// behave properly. A suitable list can be generated using /// `prime::prime_sieve(MAX_SMALL_NUM)`. /// /// Alternatively, if only a few values are being factored, /// `quick_factorize()` can be used in lieu of this function and /// an explicit list of primes. /// /// This function will factor "small" values, i.e., those less /// than `MAX_SMALL_NUM`, using `prime::factorize_wp()`, which /// in turn factors by trial division over a list of primes. This /// is the fastest way of factoring relatively small values. /// /// Large values are factored using Brent's modification of /// Pollard's Rho Algorithm, implemented in the function `rho()`. /// /// Note this function can take a long time if the value being /// factored is a large prime or a value with one very large factor. /// The correct factorization will be returned for these values, /// but it is best to filter them out of any data set being factored /// before hand. /// /// The factor list this function returns is sorted. /// /// # Examples /// /// ``` /// use reikna::factor::*; /// use reikna::prime; /// let sprimes = prime::prime_sieve(MAX_SMALL_NUM); /// assert_eq!(quick_factorize_wsp(65_536, &sprimes), vec![2; 16]); /// assert_eq!(quick_factorize_wsp(9_223_372_036_854_775_807, &sprimes), /// vec![7, 7, 73, 127, 337, 92737, 649657]); /// ``` pub fn quick_factorize_wsp(mut val: u64, sprimes: &[u64]) -> Vec<u64> { if val < MAX_SMALL_NUM { return prime::factorize_wp(val, sprimes); } let mut factors: Vec<u64> = Vec::with_capacity(64); while val & 0x01 == 0 { val >>= 1; factors.push(2); } let mut e = 2; while val > 1 { if prime::is_prime(val) { factors.push(val); break; } let factor = rho(val, e); if factor == val || factor == 1 { e += 1; continue; } else if prime::is_prime(factor) { factors.push(factor); } else { factors.extend_from_slice( &quick_factorize_wsp(factor, sprimes)); } val /= factor; } factors.sort(); factors } /// Return a `Vec<u64>` of `value`'s prime factorization. /// /// This is a helper function that calls `quick_factorize_wsp()`, /// using `value` and a generated list of primes for the arguments. /// See `quick_factorize_wsp()` for more information. /// /// Because this function generates a list of primes each time it /// is called, it is preferable to use `quick_factorize_wsp()` /// directly with an explicit list of primes if numerous factorizations /// are being computed. /// /// # Panics /// /// Panics if `prime_sieve()`, see the documentation for /// this function for more information. /// /// # Examples /// /// ``` /// use reikna::factor::quick_factorize; /// assert_eq!(quick_factorize(65_536), vec![2; 16]); /// assert_eq!(quick_factorize(9_223_372_036_854_775_807), /// vec![7, 7, 73, 127, 337, 92737, 649657]); /// ``` pub fn quick_factorize(value: u64) -> Vec<u64> { quick_factorize_wsp(value, &prime::prime_sieve(MAX_SMALL_NUM)) } #[cfg(test)] mod tests { use super::*; use super::super::prime::is_prime; #[test] fn t_gcd() { assert_eq!(gcd(0, 0), 0); assert_eq!(gcd(0, 10), 10); assert_eq!(gcd(10, 0), 10); assert_eq!(gcd(24, 12), 12); assert_eq!(gcd(8, 12), 4); assert_eq!(gcd(5125215, 890898), 3); assert_eq!(gcd(5125215, 890898), 3); } #[test] fn t_gcd_all() { assert_eq!(gcd_all(&vec![]), 0); assert_eq!(gcd_all(&vec![0, 0, 0]), 0); assert_eq!(gcd_all(&vec![0, 1, 0, 1]), 1); assert_eq!(gcd_all(&vec![0, 2, 6, 8]), 2); assert_eq!(gcd_all(&vec![1, 2, 3, 4]), 1); assert_eq!(gcd_all(&vec![9, 27, 81]), 9); assert_eq!(gcd_all(&vec![2, 4, 6, 8]), 2); } #[test] fn t_coprime() { assert_eq!(coprime(0, 0), false); assert_eq!(coprime(1, 0), true); assert_eq!(coprime(1, 10), true); assert_eq!(coprime(4, 9), true); assert_eq!(coprime(12, 9), false); } #[test] fn t_lcm() { assert_eq!(lcm(0, 0), 0); assert_eq!(lcm(0, 15), 0); assert_eq!(lcm(5, 2), 10); assert_eq!(lcm(13, 5), 65); assert_eq!(lcm(1, 35), 35); } #[test] fn t_lcm_all() { assert_eq!(lcm_all(&vec![]), 1); assert_eq!(lcm_all(&vec![0, 0, 0]), 0); assert_eq!(lcm_all(&vec![0, 1, 2, 3]), 0); assert_eq!(lcm_all(&vec![1, 2, 3, 4]), 12); assert_eq!(lcm_all(&vec![2, 2, 2]), 2); } #[test] fn t_perfect_square() { assert_eq!(perfect_square(0), true); assert_eq!(perfect_square(1), true); assert_eq!(perfect_square(2), false); assert_eq!(perfect_square(7), false); assert_eq!(perfect_square(15), false); assert_eq!(perfect_square(144), true); assert_eq!(perfect_square(145), false); assert_eq!(perfect_square(1_073_741_824), true); assert_eq!(perfect_square(1_073_741_823), false); assert_eq!(perfect_square(4_611_686_018_427_387_904), true); assert_eq!(perfect_square(4_611_686_018_427_387_905), false); } #[test] fn t_perfect_cube() { assert_eq!(perfect_cube(0), true); assert_eq!(perfect_cube(1), true); assert_eq!(perfect_cube(3), false); assert_eq!(perfect_cube(8), true); assert_eq!(perfect_cube(27), true); assert_eq!(perfect_cube(28), false); assert_eq!(perfect_cube(125), true); assert_eq!(perfect_cube(126), false); assert_eq!(perfect_cube(262_144), true); assert_eq!(perfect_cube(262_143), false); assert_eq!(perfect_cube(8_589_934_592), true); assert_eq!(perfect_cube(8_589_934_593), false); assert_eq!(perfect_cube(11_529_2150_460_6846_976), true); assert_eq!(perfect_cube(11_529_2150_460_6846_975), false); } #[test] fn t_quick_factorize() { assert_eq!(quick_factorize(0), Vec::new()); assert_eq!(quick_factorize(1), Vec::new()); let test_vals = vec![125, 97, 168, 256, 1789, 34567, 97020, 103685, 653123, 4593140, 13461780, 982357223, 72314573234, 517825353462, 8735263124568, 128735128735049, 1302131490435579, 90977992317385808, (2f64.powf(63.0)) as u64 - 1]; for val in test_vals.iter() { let factors = quick_factorize(*val); let prod: u64 = factors.iter().fold(1, |acc, x| acc * *x); assert_eq!(*val, prod); for fac in factors.iter() { assert_eq!(is_prime(*fac), true); } } } #[test] #[ignore] fn t_quick_factorize_long() { let test_vals = vec![(2f64.powf(61.0)) as u64 - 1, (2f64.powf(31.0)) as u64 - 1, (2f64.powf(19.0)) as u64 - 1, (2f64.powf(17.0)) as u64 - 1,]; for val in test_vals.iter() { let factors = quick_factorize(*val); let prod = factors.iter().fold(1, |acc, x| acc * *x); assert_eq!(*val, prod); for fac in factors.iter() { assert_eq!(super::super::prime::is_prime(*fac), true); } } } }