prime-factors

rust/prime-factors/miller-rabin.rs

@@ -0,0 +1,115 @@
+pub fn nth(n: u32) -> u32 {
+    let mut num = 1;
+    for _ in 0..=n {
+        loop {
+            num += 1;
+            if miller_rabin(num as u64) {
+                break;
+            }
+        }
+    }
+    num
+}
+
+fn miller_rabin(n: u64) -> bool {
+    const HINT: &[u64] = &[2];
+
+    // we have a strict upper bound, so we can just use the witness
+    // table of Pomerance, Selfridge & Wagstaff and Jeaschke to be as
+    // efficient as possible, without having to fall back to
+    // randomness. Additional limits from Feitsma and Galway complete
+    // the entire range of `u64`. See also:
+    // https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test#Testing_against_small_sets_of_bases
+    const WITNESSES: &[(u64, &[u64])] = &[
+        (2_046, HINT),
+        (1_373_652, &[2, 3]),
+        (9_080_190, &[31, 73]),
+        (25_326_000, &[2, 3, 5]),
+        (4_759_123_140, &[2, 7, 61]),
+        (1_112_004_669_632, &[2, 13, 23, 1662803]),
+        (2_152_302_898_746, &[2, 3, 5, 7, 11]),
+        (3_474_749_660_382, &[2, 3, 5, 7, 11, 13]),
+        (341_550_071_728_320, &[2, 3, 5, 7, 11, 13, 17]),
+        (3_825_123_056_546_413_050, &[2, 3, 5, 7, 11, 13, 17, 19, 23]),
+        (std::u64::MAX, &[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37]),
+    ];
+
+    if n % 2 == 0 {
+        return n == 2;
+    }
+    if n == 1 {
+        return false;
+    }
+
+    let mut d = n - 1;
+    let mut s = 0;
+    while d % 2 == 0 {
+        d /= 2;
+        s += 1
+    }
+
+    let witnesses = WITNESSES
+        .iter()
+        .find(|&&(hi, _)| hi >= n)
+        .map(|&(_, wtnss)| wtnss)
+        .unwrap();
+    'next_witness: for &a in witnesses.iter() {
+        let mut power = mod_exp(a, d, n);
+        assert!(power < n);
+        if power == 1 || power == n - 1 {
+            continue 'next_witness;
+        }
+
+        for _r in 0..s {
+            power = mod_sqr(power, n);
+            assert!(power < n);
+            if power == 1 {
+                return false;
+            }
+            if power == n - 1 {
+                continue 'next_witness;
+            }
+        }
+        return false;
+    }
+
+    true
+}
+
+fn mod_mul_(a: u64, b: u64, m: u64) -> u64 {
+    (u128::from(a) * u128::from(b) % u128::from(m)) as u64
+}
+
+fn mod_mul(a: u64, b: u64, m: u64) -> u64 {
+    match a.checked_mul(b) {
+        Some(r) => {
+            if r >= m {
+                r % m
+            } else {
+                r
+            }
+        }
+        None => mod_mul_(a, b, m),
+    }
+}
+
+fn mod_sqr(a: u64, m: u64) -> u64 {
+    if a < (1 << 32) {
+        let r = a * a;
+        if r >= m { r % m } else { r }
+    } else {
+        mod_mul_(a, a, m)
+    }
+}
+
+fn mod_exp(mut x: u64, mut d: u64, n: u64) -> u64 {
+    let mut ret: u64 = 1;
+    while d != 0 {
+        if d % 2 == 1 {
+            ret = mod_mul(ret, x, n)
+        }
+        d /= 2;
+        x = mod_sqr(x, n);
+    }
+    ret
+}