Data update

Task/Tonelli-Shanks-algorithm/Dart/tonelli-shanks-algorithm.dart

@@ -0,0 +1,122 @@
+class Pair {
+  final int n;
+  final int p;
+
+  Pair(this.n, this.p);
+}
+
+class Solution {
+  final int root1;
+  final int root2;
+  final bool isSquare;
+
+  Solution(this.root1, this.root2, this.isSquare);
+}
+
+int multiplyModulus(int a, int b, int modulus) {
+  a %= modulus;
+  b %= modulus;
+  if (b < a) {
+    var temp = a;
+    a = b;
+    b = temp;
+  }
+
+  int result = 0;
+  while (a > 0) {
+    if (a % 2 == 1) {
+      result = (result + b) % modulus;
+    }
+    b = (b << 1) % modulus;
+    a >>= 1;
+  }
+  return result;
+}
+
+int powerModulus(int base, int exponent, int modulus) {
+  if (modulus == 1) return 0;
+
+  base %= modulus;
+  int result = 1;
+  while (exponent > 0) {
+    if ((exponent & 1) == 1) {
+      result = multiplyModulus(result, base, modulus);
+    }
+    base = multiplyModulus(base, base, modulus);
+    exponent >>= 1;
+  }
+  return result;
+}
+
+int legendre(int a, int p) {
+  return powerModulus(a, (p - 1) ~/ 2, p);
+}
+
+Solution tonelliShanks(int n, int p) {
+  if (legendre(n, p) != 1) {
+    return Solution(0, 0, false);
+  }
+
+  // Factor out powers of 2 from p - 1
+  int q = p - 1;
+  int s = 0;
+  while (q % 2 == 0) {
+    q ~/= 2;
+    s += 1;
+  }
+
+  if (s == 1) {
+    int result = powerModulus(n, (p + 1) ~/ 4, p);
+    return Solution(result, p - result, true);
+  }
+
+  // Find a non-square z such that (z | p) = -1
+  int z = 2;
+  while (legendre(z, p) != p - 1) {
+    z += 1;
+  }
+
+  int c = powerModulus(z, q, p);
+  int t = powerModulus(n, q, p);
+  int m = s;
+  int result = powerModulus(n, (q + 1) ~/ 2, p);
+
+  while (t != 1) {
+    int i = 1;
+    int zTemp = multiplyModulus(t, t, p);
+    while (zTemp != 1 && i < m - 1) {
+      i += 1;
+      zTemp = multiplyModulus(zTemp, zTemp, p);
+    }
+
+    int b = powerModulus(c, 1 << (m - i - 1), p);
+    c = multiplyModulus(b, b, p);
+    t = multiplyModulus(t, c, p);
+    m = i;
+    result = multiplyModulus(result, b, p);
+  }
+
+  return Solution(result, p - result, true);
+}
+
+void main() {
+  final List<Pair> tests = [
+    Pair(10, 13),
+    Pair(56, 101),
+    Pair(1030, 1009),
+    Pair(1032, 1009),
+    Pair(44402, 100049),
+    Pair(665820697, 1000000009),
+    Pair(881398088036, 1000000000039),
+  ];
+
+  for (final test in tests) {
+    final solution = tonelliShanks(test.n, test.p);
+    print('n = ${test.n}, p = ${test.p}');
+    if (solution.isSquare) {
+      print('has solutions: ${solution.root1} and ${solution.root2}\n');
+    } else {
+      print('has no solutions because n is not a square modulo p\n');
+    }
+  }
+}