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CVP

CVP is a particularly important issue in Lattice-based cryptography.

The basic definition of the problem is as follows: Given a set of bases and vectors \mathbf{v} for L, find the nearest vector to \mathbf{v} on L.

Algorithms

Babai's nearest plane algorithm

The algorithm inputs a set of lattice L (rank is n) base B and a target vector \mathbf{t} to output an approximate solution to the CVP problem.

  • The approximation factor is \gamma = 2^{\frac{n}{2}}

Specific algorithm:

  • where c_j is the rounding of the coefficients in the Gram-schmidt orthogonalization, which is the rounding of proj_{b_{j}}(b).

For the personal understanding of the second step of the algorithm: find a linear combination closest to \mathbf{t} in the base B after the lattice basis and the orthogonalization.

Babai’s Rounding Technique

This algorithm is a variant of Babai's nearest plane algorithm.

The steps can be expressed as:

N = rank(B), w = target

- B' = LLL(B)

- Find a linear combination [l_0, ... l_N] such that w = sum(l_i * b'_i).

* (b'_i is the i-th vector in the LLL-reduced basis B')

- Round each l_i to it's closest integer l'_i.

- Result v = sum(l'_i * b'_i)

Hidden number problem

The definition of HNP is as follows:

Given the prime p, many t \in \mathbb{F}_p and each corresponding MSB_{l,p}(\alpha t), find the corresponding \alpha.

  • MSB_{l,p}(x) means any integer $u that satisfies \lvert (x \mod p) - u \rvert \le \frac{p}{2^{l+1}} $, which is approximately l most significant digits of x \mod p.

According to the description in Reference 3, when l \approx \log^{\frac{1}{2}}{p}, the following algorithm can solve HNP:

We can turn this problem into a CVP problem on the lattice generated by the matrix:

\left[ \begin{matrix} p & 0 & \dots & 0 & 0 \\ 0 & p & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & \vdots \\ 0 & 0 & \dots & p & 0 \\ t_1 & t_2 & \dots & t_{n} & \frac{1}{2^{l+1}} \end{matrix} \right]

We need to find the nearest vector from \mathbf{u}=(u_1, u_2, \dots, u_{n}, 0) on the lattice, so here we can use Babai's nearest plane algorithm. Finally we can get a set of vectors \mathbf{v}=(\alpha \cdot t_1 \mod p, \alpha \cdot t_2 \mod p, \dots, \frac{\alpha}{2^{l+1} }), which calculates \alpha.

BCTF 2018 - guess_number

The topic provides server-side code:

import random, sys

from flag import FLAG

import gmpy2



def msb(k, x, p):

delta = p >> (k + 1)
ui = random.randint (x-delta, x + delta)
    return ui



def main():

    p = gmpy2.next_prime(2**160)

    for _ in range(5):

        alpha = random.randint(1, p - 1)

        # print(alpha)

        t = []

u = []
k = 10
        for i in range(22):

            t.append(random.randint(1, p - 1))

            u.append(msb(k, alpha * t[i] % p, p))

        print(str(t))

print (p (u))
        guess = raw_input('Input your guess number: ')

        guess = int(guess)

        if guess != alpha:

            exit(0)



if __name__ == "__main__":

    main()

    print(FLAG)

As you can see, the program performs a total of 5 rounds. In each round, the program generates a random \alpha and 22 random t_i. For each t_i, the program will take u_i = MSB_{10,p}(\alpha\cdot{t_i\mod{p}}) and send it to the client. We need to calculate the corresponding \alpha based on the provided t_i and u_i. As you can see, the problem is a typical Hidden number problem, so you can use the above algorithm to solve:

import socket

import ast
import telnetlib



#HOST, PORT = 'localhost', 9999

HOST, PORT = '60.205.223.220', 9999



s = socket.socket()

s.connect((HOST, PORT))

f = s.makefile('rw', 0)



def recv_until(f, delim='\n'):

buf = ''
    while not buf.endswith(delim):

        buf += f.read(1)

    return buf



p = 1461501637330902918203684832716283019655932542983

k = 10


def solve_hnp(t, u):

    # http://www.isg.rhul.ac.uk/~sdg/igor-slides.pdf

    M = Matrix(RationalField(), 23, 23)

    for i in xrange(22):
        M[i, i] = p

        M[22, i] = t[i]



    M[22, 22] = 1 / (2 ** (k + 1))



    def babai(A, w):

        A = A.LLL(delta=0.75)

        G = A.gram_schmidt()[0]

        t = w

        for i in reversed(range(A.nrows())):

            c = ((t * G[i]) / (G[i] * G[i])).round()

            t -= A[i] * c

        return w - t



    closest = babai(M, vector(u + [0]))

    return (closest[-1] * (2 ** (k + 1))) % p



for i in xrange(5):

    t = ast.literal_eval(f.readline().strip())

    u = ast.literal_eval(f.readline().strip())

    alpha = solve_hnp(t, u)

    recv_until(f, 'number: ')

    s.send(str(alpha) + '\n')



t = telnetlib.Telnet()

t.sock = s

t.interact()

Reference


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