feat(Maths): add baby-step giant-step discrete log

Classic BSGS discrete logarithm for prime moduli with Vitest coverage.

Signed-off-by: Felipe Fernandes <felipe.of.dev@gmail.com>

Author: felipeofdev-ai

Maths/BabyStepGiantStep.js

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+/**
+ * Baby-step giant-step discrete logarithm modulo a prime.
+ * https://en.wikipedia.org/wiki/Baby-step_giant-step
+ *
+ * Solves base^x ≡ target (mod modulus) for the smallest non-negative x.
+ */
+
+/**
+ * @param {number} base
+ * @param {number} target
+ * @param {number} modulus prime (or modulus where base is invertible)
+ * @returns {number} smallest non-negative discrete log
+ */
+export function babyStepGiantStep(base, target, modulus) {
+  if (
+    typeof base !== 'number' ||
+    typeof target !== 'number' ||
+    typeof modulus !== 'number' ||
+    !Number.isInteger(base) ||
+    !Number.isInteger(target) ||
+    !Number.isInteger(modulus)
+  ) {
+    throw new TypeError('Arguments must be integers')
+  }
+  if (modulus <= 1) throw new RangeError('modulus must be > 1')
+
+  base = ((base % modulus) + modulus) % modulus
+  target = ((target % modulus) + modulus) % modulus
+
+  if (target === 1) return 0
+  if (base === 0) {
+    if (target === 0) return 1
+    throw new RangeError('no discrete log')
+  }
+
+  const modPow = (b, e, mod) => {
+    let r = 1
+    b = ((b % mod) + mod) % mod
+    while (e > 0) {
+      if (e % 2 === 1) r = (r * b) % mod
+      b = (b * b) % mod
+      e = Math.floor(e / 2)
+    }
+    return r
+  }
+
+  const m = Math.ceil(Math.sqrt(modulus - 1))
+  const baby = new Map()
+  let value = 1
+  for (let j = 0; j < m; j++) {
+    if (!baby.has(value)) baby.set(value, j)
+    value = (value * base) % modulus
+  }
+
+  // factor = base^{-m} mod modulus (Fermat inverse assumes prime modulus)
+  const invBase = modPow(base, modulus - 2, modulus)
+  const factor = modPow(invBase, m, modulus)
+
+  let gamma = target
+  for (let i = 0; i < m; i++) {
+    if (baby.has(gamma)) {
+      return i * m + baby.get(gamma)
+    }
+    gamma = (gamma * factor) % modulus
+  }
+
+  throw new RangeError('no discrete log')
+}

Maths/test/BabyStepGiantStep.test.js

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+import { babyStepGiantStep } from '../BabyStepGiantStep'
+
+describe('babyStepGiantStep', () => {
+  it.each([
+    [2, 1, 5, 0],
+    [2, 3, 5, 3], // 2^3 = 8 ≡ 3 (mod 5)
+    [5, 8, 13, 3], // 5^3 = 125 ≡ 8 (mod 13)
+    [3, 13, 17, 4] // 3^4 = 81 ≡ 13 (mod 17)
+  ])('solves %i^x ≡ %i (mod %i)', (base, target, modulus, expected) => {
+    expect(babyStepGiantStep(base, target, modulus)).toBe(expected)
+  })
+
+  it('throws when no solution exists', () => {
+    expect(() => babyStepGiantStep(2, 0, 5)).toThrow(RangeError)
+  })
+
+  it('throws for non-integer inputs', () => {
+    expect(() => babyStepGiantStep('2', 3, 5)).toThrow(TypeError)
+  })
+})