### Baby Step Giant Step Algorithm Implementation in Python The Baby Step Giant Step (BSGS) algorithm provides an efficient method for solving the discrete logarithm problem within finite cyclic groups. The core idea involves splitting the search space into smaller segments and using precomputation to reduce overall complexity. Below is a straightforward implementation of BSGS: ```python def baby_step_giant_step(g, h, p): m = int(p ** 0.5) + 1 # Precompute baby steps g^j mod p where j ranges from 0 to m-1. baby_steps = {} current_value = 1 for j in range(m): baby_steps[current_value] = j current_value = (current_value * g) % p # Compute giant step multiplier inverse inv_g_m = pow(pow(g, m, p), -1, p) # Search through possible values with giant steps current_value = h for i in range(m): if current_value in baby_steps: return i * m + baby_steps[current_value] current_value = (current_value * inv_g_m) % p return None # No solution found ``` This code defines `baby_step_giant_step` which takes three parameters: base \(g\), target value \(h\) and prime modulus \(p\). It calculates square root of \(p\) as segment size \(m\). For each iteration up to \(m\): - Store computed powers of \(g\) along with their exponents in dictionary named `baby_steps`. Then calculate multiplicative inverse of \(g^{m}\mod{p}\). Iterate again over another set of iterations not exceeding \(m\), checking against stored baby steps until match occurs or all possibilities exhausted.