Data update

Task/Tonelli-Shanks-algorithm/Rust/tonelli-shanks-algorithm.rs

@@ -0,0 +1,120 @@
+struct Pair {
+    n: u64,
+    p: u64,
+}
+
+struct Solution {
+    root1: u64,
+    root2: u64,
+    is_square: bool,
+}
+
+fn multiply_modulus(a: u64, b: u64, modulus: u64) -> u64 {
+    let mut a = a % modulus;
+    let mut b = b % modulus;
+
+    if b < a {
+        std::mem::swap(&mut a, &mut b);
+    }
+
+    let mut result = 0;
+    while a > 0 {
+        if a % 2 == 1 {
+            result = (result + b) % modulus;
+        }
+        b = (b << 1) % modulus;
+        a >>= 1;
+    }
+    result
+}
+
+fn power_modulus(base: u64, exponent: u64, modulus: u64) -> u64 {
+    if modulus == 1 {
+        return 0;
+    }
+
+    let mut base = base % modulus;
+    let mut result = 1;
+    let mut exponent = exponent;
+
+    while exponent > 0 {
+        if (exponent & 1) == 1 {
+            result = multiply_modulus(result, base, modulus);
+        }
+        base = multiply_modulus(base, base, modulus);
+        exponent >>= 1;
+    }
+    result
+}
+
+fn legendre(a: u64, p: u64) -> u64 {
+    power_modulus(a, (p - 1) / 2, p)
+}
+
+fn tonelli_shanks(n: u64, p: u64) -> Solution {
+    if legendre(n, p) != 1 {
+        return Solution { root1: 0, root2: 0, is_square: false };
+    }
+
+    // Factor out powers of 2 from p - 1
+    let mut q = p - 1;
+    let mut s = 0;
+    while q % 2 == 0 {
+        q /= 2;
+        s += 1;
+    }
+
+    if s == 1 {
+        let result = power_modulus(n, (p + 1) / 4, p);
+        return Solution { root1: result, root2: p - result, is_square: true };
+    }
+
+    // Find a non-square z such as ( z | p ) = -1
+    let mut z = 2;
+    while legendre(z, p) != p - 1 {
+        z += 1;
+    }
+
+    let mut c = power_modulus(z, q, p);
+    let mut t = power_modulus(n, q, p);
+    let mut m = s;
+    let mut result = power_modulus(n, (q + 1) >> 1, p);
+
+    while t != 1 {
+        let mut i = 1;
+        let mut z = multiply_modulus(t, t, p);
+        while z != 1 && i < m - 1 {
+            i += 1;
+            z = multiply_modulus(z, z, p);
+        }
+        let b = power_modulus(c, 1 << (m - i - 1), p);
+        c = multiply_modulus(b, b, p);
+        t = multiply_modulus(t, c, p);
+        m = i;
+        result = multiply_modulus(result, b, p);
+    }
+
+    Solution { root1: result, root2: p - result, is_square: true }
+}
+
+fn main() {
+    let tests = vec![
+        Pair { n: 10, p: 13 },
+        Pair { n: 56, p: 101 },
+        Pair { n: 1030, p: 1009 },
+        Pair { n: 1032, p: 1009 },
+        Pair { n: 44402, p: 100049 },
+        Pair { n: 665820697, p: 1000000009 },
+        Pair { n: 881398088036, p: 1000000000039 },
+    ];
+
+    for test in tests {
+        let solution = tonelli_shanks(test.n, test.p);
+        print!("n = {}, p = {}", test.n, test.p);
+        if solution.is_square {
+            println!(" has solutions: {} and {}\n", solution.root1, solution.root2);
+        } else {
+            println!(" has no solutions because n is not a square modulo p\n");
+        }
+    }
+}