"""Theoretical cost models used as reference curves (group operations) — see paper §5."""
import math

def rho_expected(n, class_size=1, r=20):
    """Random-mapping expectation √(π n /(2·class_size)); Teske's r-adding walk is within a few % of this for r ≥ 16."""
    return math.sqrt(math.pi * n / (2 * class_size))

def rho_dp_overhead(threads, dp_bits):
    """Extra operations after the collision in the vOW method: ≈ threads · 2^dp_bits."""
    return threads * (1 << dp_bits)

def bsgs_cost(n, m=None):
    m = m or math.isqrt(n) + 1
    return m + n / m, m  # (time, memory entries)

def kangaroo_expected(w):
    return 2 * math.sqrt(w)

def preprocessing_online(n, T):
    return math.sqrt(n / T)

def rayleigh_cv():
    """Coefficient of variation of the number of rho iterations (Rayleigh-like first-collision law)."""
    return math.sqrt((4 - math.pi) / math.pi)