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[python]basics of elliptic curve cryptography
# Basics of Elliptic Curve Cryptography implementation on Python
import collections
def inv(n, q):
"""div on PN modulo a/b mod q as a * inv(b, q) mod q
>>> assert n * inv(n, q) % q == 1
"""
for i in range(q):
if (n * i) % q == 1:
return i
pass
assert False, "unreached"
pass
def sqrt(n, q):
"""sqrt on PN modulo: returns two numbers or exception if not exist
>>> assert (sqrt(n, q)[0] ** 2) % q == n
>>> assert (sqrt(n, q)[1] ** 2) % q == n
"""
assert n < q
for i in range(1, q):
if i * i % q == n:
return (i, q - i)
pass
raise Exception("not found")
Coord = collections.namedtuple("Coord", ["x", "y"])
class EC(object):
"""System of Elliptic Curve"""
def __init__(self, a, b, q):
"""elliptic curve as: (y**2 = x**3 + a * x + b) mod q
- a, b: params of curve formula
- q: prime number
"""
assert 0 < a and a < q and 0 < b and b < q and q > 2
assert (4 * (a ** 3) + 27 * (b ** 2)) % q != 0
self.a = a
self.b = b
self.q = q
# just as unique ZERO value representation for "add": (not on curve)
self.zero = Coord(0, 0)
pass
def is_valid(self, p):
if p == self.zero: return True
l = (p.y ** 2) % self.q
r = ((p.x ** 3) + self.a * p.x + self.b) % self.q
return l == r
def at(self, x):
"""find points on curve at x
- x: int < q
- returns: ((x, y), (x,-y)) or not found exception
>>> a, ma = ec.at(x)
>>> assert a.x == ma.x and a.x == x
>>> assert a.x == ma.x and a.x == x
>>> assert ec.neg(a) == ma
>>> assert ec.is_valid(a) and ec.is_valid(ma)
"""
assert x < self.q
ysq = (x ** 3 + self.a * x + self.b) % self.q
y, my = sqrt(ysq, self.q)
return Coord(x, y), Coord(x, my)
def neg(self, p):
"""negate p
>>> assert ec.is_valid(ec.neg(p))
"""
return Coord(p.x, -p.y % self.q)
def add(self, p1, p2):
"""<add> of elliptic curve: negate of 3rd cross point of (p1,p2) line
>>> d = ec.add(a, b)
>>> assert ec.is_valid(d)
>>> assert ec.add(d, ec.neg(b)) == a
>>> assert ec.add(a, ec.neg(a)) == ec.zero
>>> assert ec.add(a, b) == ec.add(b, a)
>>> assert ec.add(a, ec.add(b, c)) == ec.add(ec.add(a, b), c)
"""
if p1 == self.zero: return p2
if p2 == self.zero: return p1
if p1.x == p2.x and (p1.y != p2.y or p1.y == 0):
# p1 + -p1 == 0
return self.zero
if p1.x == p2.x:
# p1 + p1: use tangent line of p1 as (p1,p1) line
l = (3 * p1.x * p1.x + self.a) * inv(2 * p1.y, self.q) % self.q
pass
else:
l = (p2.y - p1.y) * inv(p2.x - p1.x, self.q) % self.q
pass
x = (l * l - p1.x - p2.x) % self.q
y = (l * (p1.x - x) - p1.y) % self.q
return Coord(x, y)
def mul(self, p, n):
"""n times <mul> of elliptic curve
>>> m = ec.mul(p, n)
>>> assert ec.is_valid(m)
>>> assert ec.mul(p, 0) == ec.zero
"""
r = self.zero
m2 = p
# O(log2(n)) add
while 0 < n:
if n & 1 == 1:
r = self.add(r, m2)
pass
n, m2 = n >> 1, self.add(m2, m2)
pass
# [ref] O(n) add
#for i in range(n):
# r = self.add(r, p)
# pass
return r
def order(self, g):
"""order of point g
>>> o = ec.order(g)
>>> assert ec.is_valid(a) and ec.mul(a, o) == ec.zero
>>> assert o <= ec.q
"""
assert self.is_valid(g) and g != self.zero
for i in range(1, self.q + 1):
if self.mul(g, i) == self.zero:
return i
pass
raise Exception("Invalid order")
pass
class ElGamal(object):
"""ElGamal Encryption
pub key encryption as replacing (mulmod, powmod) to (ec.add, ec.mul)
- ec: elliptic curve
- g: (random) a point on ec
"""
def __init__(self, ec, g):
assert ec.is_valid(g)
self.ec = ec
self.g = g
self.n = ec.order(g)
pass
def gen(self, priv):
"""generate pub key
- priv: priv key as (random) int < ec.q
- returns: pub key as points on ec
"""
return self.ec.mul(g, priv)
def enc(self, plain, pub, r):
"""encrypt
- plain: data as a point on ec
- pub: pub key as points on ec
- r: randam int < ec.q
- returns: (cipher1, ciper2) as points on ec
"""
assert self.ec.is_valid(plain)
assert self.ec.is_valid(pub)
return (self.ec.mul(g, r), self.ec.add(plain, self.ec.mul(pub, r)))
def dec(self, cipher, priv):
"""decrypt
- chiper: (chiper1, chiper2) as points on ec
- priv: private key as int < ec.q
- returns: plain as a point on ec
"""
c1, c2 = cipher
assert self.ec.is_valid(c1) and ec.is_valid(c2)
return self.ec.add(c2, self.ec.neg(self.ec.mul(c1, priv)))
pass
class DiffieHellman(object):
"""Elliptic Curve Diffie Hellman (Key Agreement)
- ec: elliptic curve
- g: a point on ec
"""
def __init__(self, ec, g):
self.ec = ec
self.g = g
self.n = ec.order(g)
pass
def gen(self, priv):
"""generate pub key"""
assert 0 < priv and priv < self.n
return self.ec.mul(self.g, priv)
def secret(self, priv, pub):
"""calc shared secret key for the pair
- priv: my private key as int
- pub: partner pub key as a point on ec
- returns: shared secret as a point on ec
"""
assert self.ec.is_valid(pub)
assert self.ec.mul(pub, self.n) == self.ec.zero
return self.ec.mul(pub, priv)
pass
class DSA(object):
"""ECDSA
- ec: elliptic curve
- g: a point on ec
"""
def __init__(self, ec, g):
self.ec = ec
self.g = g
self.n = ec.order(g)
pass
def gen(self, priv):
"""generate pub key"""
assert 0 < priv and priv < self.n
return self.ec.mul(self.g, priv)
def sign(self, hashval, priv, r):
"""generate signature
- hashval: hash value of message as int
- priv: priv key as int
- r: random int
- returns: signature as (int, int)
"""
assert 0 < r and r < self.n
m = self.ec.mul(self.g, r)
return (m.x, inv(r, self.n) * (hashval + m.x * priv) % self.n)
def validate(self, hashval, sig, pub):
"""validate signature
- hashval: hash value of message as int
- sig: signature as (int, int)
- pub: pub key as a point on ec
"""
assert self.ec.is_valid(pub)
assert self.ec.mul(pub, self.n) == self.ec.zero
w = inv(sig[1], self.n)
u1, u2 = hashval * w % self.n, sig[0] * w % self.n
p = self.ec.add(self.ec.mul(self.g, u1), self.ec.mul(pub, u2))
return p.x % self.n == sig[0]
pass
if __name__ == "__main__":
# shared elliptic curve system of examples
ec = EC(1, 18, 19)
g, _ = ec.at(7)
assert ec.order(g) <= ec.q
# ElGamal enc/dec usage
eg = ElGamal(ec, g)
# mapping value to ec point
# "masking": value k to point ec.mul(g, k)
# ("imbedding" on proper n:use a point of x as 0 <= n*v <= x < n*(v+1) < q)
mapping = [ec.mul(g, i) for i in range(eg.n)]
plain = mapping[7]
priv = 5
pub = eg.gen(priv)
cipher = eg.enc(plain, pub, 15)
decoded = eg.dec(cipher, priv)
assert decoded == plain
assert cipher != pub
# ECDH usage
dh = DiffieHellman(ec, g)
apriv = 11
apub = dh.gen(apriv)
bpriv = 3
bpub = dh.gen(bpriv)
cpriv = 7
cpub = dh.gen(cpriv)
# same secret on each pair
assert dh.secret(apriv, bpub) == dh.secret(bpriv, apub)
assert dh.secret(apriv, cpub) == dh.secret(cpriv, apub)
assert dh.secret(bpriv, cpub) == dh.secret(cpriv, bpub)
# not same secret on other pair
assert dh.secret(apriv, cpub) != dh.secret(apriv, bpub)
assert dh.secret(bpriv, apub) != dh.secret(bpriv, cpub)
assert dh.secret(cpriv, bpub) != dh.secret(cpriv, apub)
# ECDSA usage
dsa = DSA(ec, g)
priv = 11
pub = eg.gen(priv)
hashval = 128
r = 7
sig = dsa.sign(hashval, priv, r)
assert dsa.validate(hashval, sig, pub)
pass
# Appendix: improved modulo calc
def inv(n, q):
"""div on PN modulo a/b mod q as a * inv(b, q) mod q
>>> assert n * inv(n, q) % q == 1
"""
# n*inv % q = 1 => n*inv = q*m + 1 => n*inv + q*-m = 1
# => egcd(n, q) = (inv, -m, 1) => inv = egcd(n, q)[0] (mod q)
return egcd(n, q)[0] % q
#[ref] naive implementation
#for i in range(q):
# if (n * i) % q == 1:
# return i
# pass
#assert False, "unreached"
#pass
def egcd(a, b):
"""extended GCD
returns: (s, t, gcd) as a*s + b*t == gcd
>>> s, t, gcd = egcd(a, b)
>>> assert a % gcd == 0 and b % gcd == 0
>>> assert a * s + b * t == gcd
"""
s0, s1, t0, t1 = 1, 0, 0, 1
while b > 0:
q, r = divmod(a, b)
a, b = b, r
s0, s1, t0, t1 = s1, s0 - q * s1, t1, t0 - q * t1
pass
return s0, t0, a
def inv2(n, q):
"""another PN invmod: from euler totient function
- n ** (q - 1) % q = 1 => n ** (q - 2) % q = n ** -1 % q
"""
assert q > 2
s, p2, p = 1, n, q - 2
while p > 0:
if p & 1 == 1: s = s * p2 % q
p, p2 = p >> 1, pow(p2, 2, q)
pass
return s
def sqrt(n, q):
"""sqrt on PN modulo: returns two numbers or exception if not exist
>>> assert (sqrt(n, q)[0] ** 2) % q == n
>>> assert (sqrt(n, q)[1] ** 2) % q == n
"""
assert n < q
for i in range(1, q):
if pow(i, 2, q) == n:
return (i, q - i)
pass
raise Exception("not found")
def sqrt2(n, q):
"""sqrtmod for bigint
- Algorithm 3.34 of http://www.cacr.math.uwaterloo.ca/hac/about/chap3.pdf
"""
import random
# b: some non-quadratic-residue
b = 0
while b == 0 or jacobi(b, q) != -1:
b = random.randint(1, q - 1)
pass
# q = t * 2^s + 1, t is odd
t, s = q - 1, 0
while t & 1 == 0:
t, s = t >> 1, s + 1
pass
assert q == t * pow(2, s) + 1 and t % 2 == 1
ni = inv(n, q)
c = pow(b, t, q)
r = pow(n, (t + 1) // 2, q)
for i in range(1, s):
d = pow(pow(r, 2, q) * ni % q, pow(2, s - i - 1, q), q)
if d == q - 1: r = r * c % q
c = pow(c, 2, q)
pass
return (r, q - r)
def jacobi(a, q):
"""jacobi symbol: judge existing sqrtmod (1: exist, -1: not exist)
- j(a*b,q) = j(a,q)*j(b,q)
- j(a*q+b, q) = j(b, q)
- j(a, 1) = 1
- j(0, q) = 0
- j(2, q) = -1 ** (q^2 - 1)/8
- j(p, q) = -1 ^ {(p - 1)/2 * (q - 1)/2} * j(q, p)
"""
if q == 1: return 1
if a == 0: return 0
if a % 2 == 0: return (-1) ** ((q * q - 1) // 8) * jacobi(a // 2, q)
return (-1) ** ((a - 1) // 2 * (q - 1) // 2) * jacobi(q % a, a)
def jacobi2(a, q):
"""quick jacobi symbol
- algorithm 2.149 of http://www.cacr.math.uwaterloo.ca/hac/about/chap2.pdf
"""
if a == 0: return 0
if a == 1: return 1
a1, e = a, 0
while a1 & 1 == 0:
a1, e = a1 >> 1, e + 1
pass
m8 = q % 8
s = -1 if m8 == 3 or m8 == 5 else 1 # m8 = 0,2,4,6 and 1,7
if q % 4 == 3 and a1 % 4 == 3: s = -s
return s if a1 == 1 else s * jacobi2(q % a1, a1)
@djerryz
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djerryz commented Sep 12, 2019

already know :
a = 16546484, b = 4548674875, q= 15424654874903,g = (6478678675,5636379357093), n = 546768
---n_a(private_key)= random.randrange(1, n)---
let n_a = n, want to know public key:

solve.py: (failed)
<><><>
ec = EC(16546484,4548674875,15424654874903)
g = Coord(6478678675,5636379357093)
eg = ElGamal(ec, g)
priv = 546768
pub = eg.gen(priv)
<><><>

but reference to: https://asecuritysite.com/encryption/ecdh2
<><><>
def make_keypair():
rewrite-> private_key = curve.n

def scalar_mult(k, point):
delete-> if k % self.curve.n == 0 or point is None:
delete-> return None
<><><>
i can get solve about public key is (13957031351290, 5520194834100), where is the problem?

@BagusMiftahRizqullah
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BagusMiftahRizqullah commented Oct 19, 2019

heyy all i make enkripsi plain text with eliptic curve

https://github.com/BagusMiftahRizqullah/Enkripsi-Eliptic-Curve

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