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I have been studying a cryptosystem using Mersenne primes. More specifically, this paper.

I have implemented this cryptosystem in Python, but I am missing the key encapsulation system.

On page 12, they refer to something known as an "expandable hash function". It should take as input a $\lambda$-bit string and output a uniformly random $n$-bit string ($\lambda<<n$) of Hamming weight $h$. This weight $h$ is already determined (actually $h=\lambda$).

I am kind of new to this stuff. Is there a way to implement this hash function in Python?

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    $\begingroup$ @kelalaka but what about the Hamming weight? $\endgroup$ May 24 at 20:29
  • $\begingroup$ Hmm, would this just be $H'(m)\leftarrow_\$ S(\text{1}^h\parallel\text{0}^\left(n-h\right))$? All you gotta do is find a way to use $H$ to uniformly randomly select a permutation of '1' * h + '0' * (n-h). Have you got any candidates for expandable hash functions currently? (This question may prove informative or helpful) $\endgroup$ May 25 at 17:31
  • $\begingroup$ @JamesTheAwesomeDude no, I have not. How would you implement it in python? $\endgroup$ May 25 at 18:32
  • $\begingroup$ I was too dumb to properly understand or implement it, but it appears that this paper provides a general method for doing so (or, at least, constructing a function that constructs functions that do so) $\endgroup$ May 26 at 15:13
  • $\begingroup$ While I haven't yet figured out the permutation generator, It looks like pycryptodome includes a reputable expandable-output hash function: “Are there any variable length hash functions available for Python? $\endgroup$ May 27 at 4:27
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Remember: a random permutation (or, when taken bitwise, "a hamming-weight-preserving one-way function") is known in layman's terms as a shuffle — there are trivially correct algorithms to do this — Python itself, though, makes it quite convenient to just leverage its shuffle implementation by subclassing Random with your choice of DRBG:

from Crypto.Hash import SHAKE256
from random import Random
from functools import reduce
from itertools import repeat, islice


def deterministic_shuffle(seq, key, alg=SHAKE256):
    """Applies a pseudorandom permutation from key to seq"""
    SpongeBasedRandom(key, alg.new).shuffle(seq)


class SpongeBasedRandom(Random):
    def __init__(self, seed, spongefactory, blocksize=1):
        sponge = spongefactory(seed)
        self._randbits = _ibytestobits(map(sponge.read, repeat(blocksize)))
    def getrandbits(self, k):
        return _concatbits(islice(self._randbits, k))
    # Fix the following functions to prevent implementation-dependency
    def randbytes(self, n):
        return self.getrandbits(n * 8).to_bytes(n, 'big')
    def _randbelow(self, n):
        """Version of Python 3000's Random._randbelow that doesn't waste bits"""
        if n <= 1:
            return 0
        getrandbits = self.getrandbits
        k = (n-1).bit_length()
        r = getrandbits(k)
        while r >= n:
            r = getrandbits(k)
        return r
    def shuffle(self, x):
        """Modern Fisher-Yates shuffle"""
        randbelow = self._randbelow
        for i in reversed(range(1, len(x))):
            j = randbelow(i + 1)
            x[i], x[j] = x[j], x[i]


def _ibytestobits(ibytes):
    yield from (((i & (0b1 << k)) >> k) for byte in ibytes for i in byte for k in reversed(range(8)))


def _concatbits(x):
    return reduce((lambda acc, cur: ((acc << 1) | cur)), x)

SHAKE256 was used in the example code; it should be easily repurposeable to any bit generator. See this answer for some other ideas. To use this in your code would be something like:

k = b'Hyper Secret Input Key'
h = len(k) * 8
n = 4096
assert n > (8 * h)

# An n-element bit sequence of hamming weight h
bitstream = ([1] * h) + ([0] * (n - h))
print(_concatbits(bitstream).to_bytes(n // 8, 'big').hex())

deterministic_shuffle(bitstream, k)
print(_concatbits(bitstream).to_bytes(n // 8, 'big').hex())
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