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from __future__ import division
import hashlib
from hkdf import Hkdf
from .six import integer_types
from .util import (size_bits, size_bytes, unbiased_randrange,
bytes_to_number, number_to_bytes)
"""Interface specification for a Group.
A cyclic abelian group, in the mathematical sense, is a collection of
'elements' and a (binary) operation that takes two elements and produces a
third. It has the following additional properties:
* there is an 'identity' element named 0, and X+0=X
* there is a distinguished 'generator' element G
* adding G to 0 'n' times is called scalar multiplication: Y=n*G
* this addition loops around after 'q' times, called the 'order'
* so (n+k*q)*X = n*X
* scalar multiplication is associative, n*(X+Y) = n*X+n*Y
* 'scalar division' is really multiplying by (q-n)
A 'scalar' is an integer in [0,q-1] inclusive. You can add scalars to each
other, invert them, and multiply them by elements. There is a one-to-one
mapping between scalars and elements. It is trivial to go from a scalar to an
element, but hard (in the cryptographic sense) to go from element to scalar.
You can ask for a random scalar, and you can convert scalars to bytes and
back again.
The form of an 'element' depends upon the group (there are integer-element
groups, and ECC groups). You can add elements together, invert them
(scalarmult by -1), and subtract them (invert then add). You can ask for a
random element (found by choosing a random scalar, then multiplying). You can
convert elements to bytes and back.
There is a distinguished element called 'Base', which is the generator
(or 'base point' in ECC groups).
Two final operations are provided. The first produces an 'arbitrary element'
from a seed. This is somewhat like a random element, but with the additional
important property that nobody knows what the corresponding scalar is. The
second takes a password (an arbitrary bytestring) and produces a scalar.
The functions that produce random scalars/elements require an entropy
function, which is expected to behave like os.urandom. The only reason to not
use os.urandom is for deterministic unit tests.
g = I2048Group # or Ed25519Group
s = g.random_scalar(entropy_f)
s = g.bytes_to_scalar(bytes)
bytes = g.scalar_to_bytes()
s = g.password_to_scalar(password)
e = g.bytes_to_element(bytes)
e = g.arbitrary_element(seed)
e = g.Base # this is an Element too, with all the methods below
e3 = e1.add(e2)
e3 = e1.scalarmult(s) # takes int, positive or negative
bytes = e.to_bytes()
# equality tests work: e1 == e2, e1 != e2
"""
def expand_password(data, num_bytes):
h = Hkdf(salt=b"", input_key_material=data, hash=hashlib.sha256)
info = b"SPAKE2 pw"
return h.expand(info, num_bytes)
def password_to_scalar(pw, scalar_size_bytes, q):
assert isinstance(pw, bytes)
# the oversized hash reduces bias in the result, so
# uniformly-random passwords give nearly-uniform scalars
oversized = expand_password(pw, scalar_size_bytes+16)
assert len(oversized) >= scalar_size_bytes
i = bytes_to_number(oversized)
return i % q
def expand_arbitrary_element_seed(data, num_bytes):
h = Hkdf(salt=b"", input_key_material=data, hash=hashlib.sha256)
info = b"SPAKE2 arbitrary element"
return h.expand(info, num_bytes)
class _Element:
def __init__(self, group, e):
self._group = group
self._e = e
def add(self, other):
return self._group._add(self, other)
def scalarmult(self, s):
return self._group._scalarmult(self, s)
def to_bytes(self):
return self._group._element_to_bytes(self)
class IntegerGroup:
def __init__(self, p, q, g):
self.q = q # the subgroup order, used for scalars
self.scalar_size_bytes = size_bytes(self.q)
_s = self.scalar_to_bytes(self.password_to_scalar(b""))
assert isinstance(_s, bytes)
assert len(_s) >= self.scalar_size_bytes
self.Zero = _Element(self, 1)
self.Base = _Element(self, g) # generator of the subgroup
# these are the public system parameters
self.p = p # the field size
self.element_size_bits = size_bits(self.p)
self.element_size_bytes = size_bytes(self.p)
# double-check that the generator has the right order
assert pow(g, self.q, self.p) == 1
def order(self):
return self.q
def random_scalar(self, entropy_f):
return unbiased_randrange(0, self.q, entropy_f)
def scalar_to_bytes(self, i):
# both for hashing into transcript, and save/restore of
# intermediate state
assert isinstance(i, integer_types)
assert 0 <= 0 < self.q
return number_to_bytes(i, self.q)
def bytes_to_scalar(self, b):
# for restore of intermediate state
assert isinstance(b, bytes)
assert len(b) == self.scalar_size_bytes
i = bytes_to_number(b)
assert 0 <= i < self.q, (0, i, self.q)
return i
def password_to_scalar(self, pw):
return password_to_scalar(pw, self.scalar_size_bytes, self.q)
def arbitrary_element(self, seed):
# we do *not* know the discrete log of this one. Nobody should.
assert isinstance(seed, bytes)
processed_seed = expand_arbitrary_element_seed(seed,
self.element_size_bytes)
assert isinstance(processed_seed, bytes)
assert len(processed_seed) == self.element_size_bytes
# The larger (non-prime-order) group (Zp*) we're using has order
# p-1. The smaller (prime-order) subgroup has order q. Subgroup
# orders always divide the larger group order, so r*q=p-1 for
# some integer r. If h is an arbitrary element of the larger
# group Zp*, then e=h^r will be an element of the subgroup. If h
# is selected uniformly at random, so will e, and nobody will
# know its discrete log. We can enforce this for pre-selected
# parameters by choosing h as the output of a hash function.
r = (self.p - 1) // self.q
assert r * self.q == self.p - 1
h = bytes_to_number(processed_seed) % self.p
element = _Element(self, pow(h, r, self.p))
assert self._is_member(element)
return element
def _is_member(self, e):
if not e._group is self:
return False
if pow(e._e, self.q, self.p) == 1:
return True
return False
def _element_to_bytes(self, e):
# for sending to other side, and hashing into transcript
assert isinstance(e, _Element)
assert e._group is self
return number_to_bytes(e._e, self.p)
def bytes_to_element(self, b):
# for receiving from other side: test group membership here
assert isinstance(b, bytes)
assert len(b) == self.element_size_bytes
i = bytes_to_number(b)
if i <= 0 or i >= self.p: # Zp* excludes 0
raise ValueError("alleged element not in the field")
e = _Element(self, i)
if not self._is_member(e):
raise ValueError("element is not in the right group")
return e
def _scalarmult(self, e1, i):
if not isinstance(e1, _Element):
raise TypeError("E*N requires E be an element")
assert e1._group is self
if not isinstance(i, integer_types):
raise TypeError("E*N requires N be a scalar")
return _Element(self, pow(e1._e, i % self.q, self.p))
def _add(self, e1, e2):
if not isinstance(e1, _Element):
raise TypeError("E*N requires E be an element")
assert e1._group is self
if not isinstance(e2, _Element):
raise TypeError("E*N requires E be an element")
assert e2._group is self
return _Element(self, (e1._e * e2._e) % self.p)
# This 1024-bit group originally came from the J-PAKE demo code,
# http://haofeng66.googlepages.com/JPAKEDemo.java . That java code
# recommended these 2048 and 3072 bit groups from this NIST document:
# http://csrc.nist.gov/groups/ST/toolkit/documents/Examples/DSA2_All.pdf
# L=1024, N=160
I1024 = IntegerGroup(
p=0xE0A67598CD1B763BC98C8ABB333E5DDA0CD3AA0E5E1FB5BA8A7B4EABC10BA338FAE06DD4B90FDA70D7CF0CB0C638BE3341BEC0AF8A7330A3307DED2299A0EE606DF035177A239C34A912C202AA5F83B9C4A7CF0235B5316BFC6EFB9A248411258B30B839AF172440F32563056CB67A861158DDD90E6A894C72A5BBEF9E286C6B,
q=0xE950511EAB424B9A19A2AEB4E159B7844C589C4F,
g=0xD29D5121B0423C2769AB21843E5A3240FF19CACC792264E3BB6BE4F78EDD1B15C4DFF7F1D905431F0AB16790E1F773B5CE01C804E509066A9919F5195F4ABC58189FD9FF987389CB5BEDF21B4DAB4F8B76A055FFE2770988FE2EC2DE11AD92219F0B351869AC24DA3D7BA87011A701CE8EE7BFE49486ED4527B7186CA4610A75,
)
# L=2048, N=224
I2048 = IntegerGroup(
p=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
q=0x90EAF4D1AF0708B1B612FF35E0A2997EB9E9D263C9CE659528945C0D,
g=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
)
# L=3072, N=256
I3072 = IntegerGroup(
p=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
q=0xCFA0478A54717B08CE64805B76E5B14249A77A4838469DF7F7DC987EFCCFB11D,
g=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
)