# Performance comparison #

## The base case ##

Let's assume we have a secret of length _m_. The splitting takes _n_ evaluations of a polynomial of order _k_ (over Galois field 256) for each byte, leading to _O(n\*k\*m)_ finite field multiplications. Reconstruction of the constant parameters during joining first precomputes parts of the Lagrange polynomial and then reuses them for each byte, taking _O(k\*k + k\*m)_ multiplications.

<table>

    <tr>

        <th>Revision</th>

        <th>Features</th>

        <th>k / n parameters</th>

        <th>Split</th>

        <th>Join</th>

    </tr>

    <tr>

        <td rowspan="2">a47ae3e113cc</td>

        <td rowspan="2">-</td>

        <td>2 / 3</td>

        <td>5.02e-06</td>

        <td>4.12e-05</td>

    </tr>

    <tr>

        <td>254 / 254</td>

        <td>0.0125</td>

        <td>0.00175</td>

    </tr>

    <tr>

        <td rowspan="2">c6815615a077</td>

        <td rowspan="2">gfmul() caching</td>

        <td>2 / 3</td>

        <td>5.38e-06</td>

        <td>4.28e-05</td>

    </tr>

    <tr>

        <td>254 / 254</td>

        <td>0.00741</td>

        <td>0.00156</td>

    </tr>

import math

import cmath

import itertools

from functools import cache

from .gf256 import gfmul, gfpow

# divisors of 255 and their factors in natural numbers

DIVISORS = [3, 5, 15, 17, 51, 85, 255]

FACTORS = {3: [3], 5: [5], 15: [3, 5], 17: [17], 51: [3, 17], 85: [5, 17], 255: [3, 5, 17]}

# values of n-th square roots in GF256

SQUARE_ROOTS = {3: 189, 5: 12, 15: 225, 17: 53, 51: 51, 85: 15, 255: 3}

def ceil_size(n):

	assert n <= DIVISORS[-1]

	for (i, ni) in enumerate(DIVISORS):

		if ni >= n:

			break

	return ni

@cache

def precompute_x(n):

	"""Return a geometric sequence [1, w, w**2, ..., w**(n-1)], where w**n==1.

	This can be done only for certain values of n."""

	assert n in SQUARE_ROOTS, n

	w = SQUARE_ROOTS[n]  # primitive N-th square root of 1

	return list(itertools.accumulate([1]+[w]*(n-1), gfmul))

def complex_dft(p):

	"""Quadratic formula from the definition. The basic case in complex numbers."""

	N = len(p)

	w = cmath.exp(-2*math.pi*1j/N)  # primitive N-th square root of 1

	y = [0]*N

	for k in range(N):

		xk = w**k

		for n in range(N):

			y[k] += p[n] * xk**n

	return y

def dft(p):

	"""Quadratic formula from the definition. In GF256."""

	N = len(p)

	x = precompute_x(N)

	y = [0]*N

	for k in range(N):

		for n in range(N):

			y[k] ^= gfmul(p[n], gfpow(x[k], n))

	return y

def compute_inverse(N1, N2):

	for i in range(N2):

		if N1*i % N2 == 1:

			return i

	raise ValueError("Failed to find an inverse to {0} mod {1}.".format(N1, N2))

def prime_fft(p, divisors, basic_dft=dft):

	"""https://en.wikipedia.org/wiki/Prime-factor_FFT_algorithm"""

	if len(divisors) == 1:

		return basic_dft(p)

	N = len(p)

	N1 = divisors[0]

	N2 = N//N1

	N1_inv = compute_inverse(N1, N2)

	N2_inv = compute_inverse(N2, N1)

	ys = []

	for n1 in range(N1):  # compute rows

		p_ = [p[(n2*N1+n1*N2) % N] for n2 in range(N2)]

		ys.append(prime_fft(p_, divisors[1:], basic_dft))

	for k2 in range(N2):  # compute cols

		p_ = [row[k2] for row in ys]

		y_ = basic_dft(p_)

		for (yi, row) in zip(y_, ys):  # update col

			row[k2] = yi

	# remap and output

	res = [0]*N

	for k1 in range(N1):

		for k2 in range(N2):

			res[(k1*N2*N2_inv+k2*N1*N1_inv) % N] = ys[k1][k2]

	return res

def evaluate(coefs, n):

	ni = ceil_size(n)

	extended_coefs = coefs + [0]*(ni-len(coefs))

	ys = prime_fft(extended_coefs, FACTORS[ni])