Cryptanalysis of “Fully Homomorphic Encryption over the Binary Polynomials”

Turns out, I’m not he only one who was inspired to adapt the Fully Homomorphic Encryption over the Integers scheme by van Dijk, Gentry, Halevi and Vaikuntanathan to polynomials.  Gu Chunsheng posted a pre-print on the IACR eprint server this week which essentially instantiates the integer scheme over univariate polynomials over .  Below is my implementation (in Sage) of his somewhat homomorphic scheme:

class BinPolySHE:
    def __init__(self, n):
        n = n
        tau = n # choice here
        P = PolynomialRing(GF(2),'x')

        x = P.gen()

        s = P.random_element(degree=2*n+1)
        while not (s.is_irreducible() and s.degree()==2*n+1):
            s = P.random_element(degree=2*n+1)

        b = []

        a0 = P.random_element(2*n+1)
        if a0.degree() < 2*n+1:
            a0 += x**(2*n+1)
        e0 = P.random_element(degree=n-1)
        b0 = a0*s + x*e0 # deg: 4*n+2

        b.append(b0)

        for i in range(1,tau):
            ai = P.random_element(degree=n) # choice here
            ei = P.random_element(degree=n-1)
            bi = ai*s + x*ei # deg 3*n+1
            bi = bi % b0
            b.append(bi)

        self.n = n
        self.pk = b
        self.sk = s
        self.P = P

    def encrypt(self, m):
        T = []

        for i in range(1, len(self.pk)):
            if random() <= 0.5: # choice here
                T.append(i)

        c = self.P(m%2)

        x = self.P.gen()

        for i in T:
            e = self.P.random_element(degree=self.n-1)
            c += self.pk[i] + x*e

        return c % self.pk[0]

    def decrypt(self, c):
        x = self.P.gen()
        return (c % self.sk) % x

Regular readers of this blog might have noticed that the scheme looks like a bit like a univariate specialisation of this PollyCracker scheme. Indeed, just like this first PollyCracker scheme, Gu’s scheme is badly broken. Below, the source code of my attack:

    def attack(self, c):

        A = Matrix(GF(2),len(self.pk),4*self.n+2)

        x = self.P.gen()

        for i,b in enumerate(self.pk):
            for j in b.exponents():
                A[i,A.ncols()-j-1] = 1
        E = A.echelon_form(reduced=False)

        pk2 = []
        for i,r in enumerate(A.rows()):
            b = 0
            for j in range(A.ncols()):
                b += E[i,A.ncols()-j-1]*x**j
            pk2.append(b)

        for b in pk2:
            if c[b.degree()]:
                c -= b
        return c % x

The attack proceeds pretty much as discussed here: we can compute a triangular basis for the span of the public key and use that to perform all eliminations. Since the noise does not grow with each addition and does not affect the constant coefficient (which holds the message), we can essentially ignore it.

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