Cellular Automata

148

It's hard to reverse a step in a cellular automata, but solvable if done right.

https://cellularautomata.web.ctfcompetition.com/

As the rules state we are dealing with a Rule 126 automata. Patterns 000 and 111 produce a 0 bit while all others produce 1.

The problem with reversing cellular automata is that a lot of different steps correspond to the step you are trying to reverse. Straight bruteforcing will take forever - 64-bit steps are just too bit. However, the bits in the reverse step are not random, and are limited by 2 rules:

1. 0 has to reverse to patterns 000 or 111, and 1 to all others
2. If a pattern is selected the patterns that follow it are limited to 2 possibilities (because they have to overlap by 2 bits). For example, pattern 000 can only be followed by patterns 000 and 001.

If we apply these 2 rules we can generate all possibilities very quickly with the help of the following script:

import sys

if len(sys.argv) != 2:
    print "Please supply a hex number on the command line"
    quit()
    
hexval = sys.argv[1]
bit_size = len(hexval)*4

# convert hex value to bit string
bitstr = (bin(int(hexval,16))[2:]).zfill(bit_size)

# map from bits to patterns that generate it
patterns_generating_bit = {"0":[0,7], "1":[1,2,3,4,5,6]}

# valid patterns that can follow each pattern; for example, pattern 010 can be followed 
#  only by 100 and 101 because they must overlap with its last 2 digits (10)
valid_next_patterns = {0:[0,1], 1:[2,3], 2:[4,5], 3:[6,7], 4:[0,1], 5:[2,3], 6:[4,5], 7:[6,7]}

# mid bits in each pattern
pattern_mid_bits = {0:"0", 1:"0", 2:"1", 3:"1", 4:"0", 5:"0", 6:"1", 7:"1"}

def reverse_rule126(bitstr, depth, valid_patterns, patterns_in_step):
    
    # walk through all patterns that generate the current bit
    for pattern in patterns_generating_bit[bitstr[depth]]:
        # make sure the pattern is valid based on previously seen patterns
        if pattern in valid_patterns: 
            
            # if we are not at the last bit - keep going recursively
            if depth < (bit_size-1):
                reverse_rule126(bitstr, depth+1, valid_next_patterns[pattern], patterns_in_step+[pattern])

            # if we are at the last bit...
            if depth == (bit_size-1):
                # ...and the last pattern wraps around properly to the beginning of the step string 
                if patterns_in_step[0] in valid_next_patterns[pattern]:
                    
                    # generate the full bitstring for the step and print it out
                    found_step = ""
                    for x in patterns_in_step:
                        found_step += pattern_mid_bits[x]
                    found_step += pattern_mid_bits[pattern]
                            
                    print hex(int(found_step,2))[2:]

reverse_rule126(bitstr, 0, [0,1,2,3,4,5,6,7], [])

The script generates about 10K possibilities, which we can try to determine if there is a flag in the output:

#!/bin/sh

for i in $(python solve.py 66de3c1bf87fdfcf); do
    echo "$i" > /tmp/plain.key; xxd -r -p /tmp/plain.key > /tmp/enc.key
    echo "U2FsdGVkX1/andRK+WVfKqJILMVdx/69xjAzW4KUqsjr98GqzFR793lfNHrw1Blc8UZHWOBrRhtLx3SM38R1MpRegLTHgHzf0EAa3oU
eWcQ=" | openssl enc -d -aes-256-cbc -pbkdf2 -md sha1 -base64 --pass file:/tmp/enc.key 2>/dev/null | grep CTF
done

When we run the script we quickly get the flag decoded:

$ ./solve.sh
CTF{reversing_cellular_automatas_can_be_done_bit_by_bit}

The flag is CTF{reversing_cellular_automatas_can_be_done_bit_by_bit}.

Original writeup (https://0xd13a.github.io/ctfs/gctf2019/cellular-automata/).