19 Jul 2022 ctfcrypto

ImaginaryCTF 2022 Crypto Writeups

Changelog (last updated 2022-07-22)

Update 2022-07-22: Added Poker writeup, some touch-ups here and there.

Poker

By StealthyDev

In the early days of online poker rooms, the game would get cracked and players would play with perfect knowledge of the upcoming handsโฆ Can you deal out the next two games correctly?

Attachments: poker.py cards.22.07.16.txt

8 solves, 495 points

Challenge poker.py
python
from random import getrandbits
from math import prod

HEARTS = "๐ฑ๐ฒ๐ณ๐ด๐ต๐ถ๐ท๐ธ๐น๐บ๐ป๐ฝ๐พ"
DIAMONDS = "๐๐๐๐๐๐๐๐๐๐๐๐๐"
CLUBS = "๐๐๐๐๐๐๐๐๐๐๐๐๐"
DECK = SPADES+HEARTS+DIAMONDS+CLUBS  # Bridge Ordering of a Deck
ALPHABET52 = "ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnop_rstuvw{y}"

CARDS_PER_DEAL = 25
assert CARDS_PER_DEAL % 2 == 1
MAX_DEAL = prod(x for x in range(len(DECK) - CARDS_PER_DEAL + 1, len(DECK) + 1))
DEAL_BITS = MAX_DEAL.bit_length()

def text_from_cards(string):
return string.translate(string.maketrans(DECK, ALPHABET52))

def deal_game():
shuffle = getrandbits(DEAL_BITS) % MAX_DEAL
deck = list(DECK)
deal = ""
while shuffle > 0:
deal += deck.pop(shuffle % len(deck))
shuffle //= len(deck) + 1
while len(deal) < CARDS_PER_DEAL:
deal += deck.pop(0)
return deal

def print_puzzle():
with open("cards.txt", "w") as cards_file:
for i in range(750):
table = deal_game()
cards_file.write(f"Game {i+1}:\n")
for i in range((CARDS_PER_DEAL - 5) // 2):
cards_file.write(f"{i + 1}: {table[i * 2]}{table[i * 2 + 1]}  ")
cards_file.write("\nTable: {}{}{} {} {}\n\n".format(*table[CARDS_PER_DEAL - 5:]))


We get a poker simulator which deals cards based on the python random() PRNG and are told to predict two deals after the end of the file.

By re-running some of poker.py locally, we find:

plaintext
DEAL_BITS = 133
2 ** DEAL_BITS = 10889035741470030830827987437816582766592
MAX_DEAL       =  7407396657496428903767538970656768000000


With some regex and careful implementation, we can convert the dealt cards back to a number x for each deal such that x = getrandbits(DEAL_BITS) % MAX_DEAL.

For each deal, one of two cases have occurred:

1. getrandbits(DEAL_BITS) < MAX_DEAL and x = getrandbits(DEAL_BITS)
2. getrandbits(DEAL_BITS) >= MAX_DEAL and x = getrandbits(DEAL_BITS) - MAX_DEAL

We can split the first case depending on the value of x and rearrange to get three cases:

1. x < 2 ** DEAL_BITS - MAX_DEAL and getrandbits(DEAL_BITS) = x
2. x >= 2 ** DEAL_BITS - MAX_DEAL and getrandbits(DEAL_BITS) = x
3. x < 2 ** DEAL_BITS - MAX_DEAL and getrandbits(DEAL_BITS) = x + MAX_DEAL

Note that from our perspective with only knowledge of $x$, we are unable to differentiate between case 1 or 3. If our $x$ has a value that may have come from case 1 or 3, there are two possible values of getrandbits(DEAL_BITS). In case 2, this is only one possible value of getrandbits(DEAL_BITS). We can put this into a symbolic python PRNG cracker that supports unknown bits to recover the PRNG state. For case 1 and 3, we will only report bits which are the same in both x and x + MAX_DEAL, and any disagreeing bits will be labelled as unknown. In case 2, we know that getrandbits(DEAL_BITS) = x for sure and can report all bits of x to the PRNG solver.

To simplify implementation, only the low 128 bits were considered when recovering PRNG output and the upper 5 bits were discarded. Some quick estimations suggest that case 2 occurs around 36% of the time, giving an expected leak of 87 bits of PRNG state per deal. After testing, around 400 deal outputs are needed as input for the PRNG cracker to successfully recover the PRNG state, which can be confirmed by predicting the remaining 350 deals correctly. Finally, the flag is obtained by converting the 751st and 752nd deals to text.

Exploit code
python
from math import prod
from random import seed, getrandbits
import re

# !wget \
#     https://raw.githubusercontent.com/icemonster/symbolic_mersenne_cracker/main/main.py \
#     -O untwister.py 2>/dev/null
from untwister import Untwister

HEARTS = "๐ฑ๐ฒ๐ณ๐ด๐ต๐ถ๐ท๐ธ๐น๐บ๐ป๐ฝ๐พ"
DIAMONDS = "๐๐๐๐๐๐๐๐๐๐๐๐๐"
CLUBS = "๐๐๐๐๐๐๐๐๐๐๐๐๐"
CARDS_PER_DEAL = 25
MAX_DEAL = prod(
x for x in range(len(DECK) - CARDS_PER_DEAL + 1, len(DECK) + 1)
)
DEAL_BITS = MAX_DEAL.bit_length()

print("deal bits", DEAL_BITS)

def deal_game_leak(frombits=None):
if frombits is None:
getrand = getrandbits(DEAL_BITS)
else:
getrand = frombits
shuffle = getrand % MAX_DEAL
shuffle_orig = shuffle
deck = list(DECK)
deal = ""
deal_idxs = []
while shuffle > 0:
deal_idx = shuffle % len(deck)
deal_idxs.append(deal_idx)
deal += deck.pop(deal_idx)
shuffle //= len(deck) + 1
while len(deal) < CARDS_PER_DEAL:
deal += deck.pop(0)
return deal, getrand, shuffle_orig, deal_idxs

def cards_to_deal_idxs(deal):
deck = list(DECK)
deal_idxs = []
for c in deal:
deal_idx = deck.index(c)
deck.pop(deal_idx)
deal_idxs.append(deal_idx)
return deal_idxs

def deal_idxs_to_getrand(deal_idxs):
cur_deck_len = len(DECK) - len(deal_idxs) + 1
acc = 0
for deal_idx in deal_idxs[::-1]:
acc = acc * cur_deck_len + deal_idx
cur_deck_len += 1
return acc

def getrand_to_randbits(getrand):
cap = DEAL_BITS // 32 * 32
bits = bin(getrand + 2 ** cap)[-cap:]
return bits

def getrand_to_randbits_mod(getrand):
if (getrand + MAX_DEAL) >= 2 ** DEAL_BITS:
return getrand_to_randbits(getrand)
a = getrand_to_randbits(getrand)
b = getrand_to_randbits(getrand + MAX_DEAL)
res = "".join([
i if i == j else "?"
for i, j in zip(a, b)
])
return res

# Sanity checks
test_seed = 0
seed(test_seed)
for _ in range(10):
deal = deal_game_leak()
deal_idxs = cards_to_deal_idxs(deal[0])
assert(deal_idxs == deal[3])
getrand = deal_idxs_to_getrand(deal_idxs)
randbits = getrand_to_randbits_mod(getrand)
assert(all([
i == j
for i, j in zip(randbits, bin(deal[1])[-128:])
if i != "?"
]))

def pipeline(cards):
known = getrand_to_randbits_mod(
deal_idxs_to_getrand(
cards_to_deal_idxs(
cards
)
)
)

return [
known[i:i+32]
for i in range(0, 128, 32)
][::-1] + ["?" * 32]

with open("poker/cards.22.07.16.txt", "r") as f:
deals = raw.strip().split("\n\n")
print("num deals", len(deals))

def parse_deal(deal, check_num):
deal_num = re.search(
r"Game (\d+):", deal,
re.UNICODE | re.MULTILINE
).group(1)
deal_num = int(deal_num)
assert check_num == deal_num

cards = re.sub(
r"[ -~\n]*", "", deal,
re.UNICODE | re.MULTILINE
)
assert(len(cards) == CARDS_PER_DEAL)
return cards

all_deals = [
parse_deal(deal, deal_idx + 1)
for deal_idx, deal in enumerate(deals)
]

print()
print("deals")
for i in all_deals[:5]:
print(i)

all_deals_pred = [
i
for deal in all_deals
for i in pipeline(deal)
]

print()
print("pipeline output")
for i in all_deals_pred[:5]:
print(i)

use = 400

print()
print("submitting")
ut = Untwister()
for i in all_deals_pred[:use * 5]:
ut.submit(i)

print("solving")
rng = ut.get_random()

print("verify")
for i in all_deals[use:]:
test = deal_game_leak(rng.getrandbits(DEAL_BITS))[0]
assert(test == i)

print("flag")
ALPHABET52 = "ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnop_rstuvw{y}"
def text_from_cards(string):
return string.translate(string.maketrans(DECK, ALPHABET52))

for i in range(3):
deal = deal_game_leak(rng.getrandbits(DEAL_BITS))[0]
print(text_from_cards(deal), end="")
print()

output
deal bits 133
num deals 750

deals
๐ฆ๐๐ญ๐๐๐ฝ๐ก๐๐๐พ๐๐๐๐ฒ๐ง๐๐๐๐๐๐ณ๐ฑ๐ธ๐๐ช
๐๐๐ก๐๐๐ฃ๐๐๐๐ฅ๐๐ง๐๐๐๐ฑ๐จ๐ฎ๐๐ข๐ณ๐ซ๐๐ฉ๐ด
๐บ๐ฑ๐๐๐๐ณ๐จ๐๐๐ช๐ข๐๐ฒ๐ด๐๐๐๐ฅ๐ฎ๐น๐ป๐ต๐๐ง๐ฉ
๐๐๐ข๐๐๐ต๐๐ฝ๐๐๐ฒ๐๐ฎ๐ท๐๐บ๐ช๐๐ณ๐ฆ๐ธ๐๐๐ค๐ด
๐ข๐๐๐๐ป๐๐ธ๐๐๐น๐๐๐ฑ๐๐ฅ๐ซ๐ง๐ต๐๐ญ๐๐ฉ๐ท๐๐

pipeline output
0??11?00010101101101110001001101
1?111??1?????0000??01?0?100?01??
0010????1100???0????00???011??0?
0?010???1????1????0???010111????
????????????????????????????????

submitting

STT> Solving...

solving

STT> Solved! (in 8.13s)

verify
flag
ictf{CheaTINg_PsuEdoRAnDOm_BITgenEratioN}bcdfhjklpGJTYgCOrVswXhAj{aQnSPckfW


Secure Encoding: Base64

By puzzler7

Base64 is my favorite encryption scheme.

Attachments: encode.py out.txt

10 solves, 492 points

Challenge encode.py
python
from base64 import b64encode
from random import shuffle

charset = b'abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ0123456789+/='
shuffled = [i for i in charset]
shuffle(shuffled)

d = {charset[i]:v for(i,v)in enumerate(shuffled)}

while "\n\n\n" in pt:
pt = pt.replace("\n\n\n", '\n\n')
while '  ' in pt:
pt = pt.replace('  ', ' ')

assert all(ord(i)<128 for i in pt)

ct = bytes(d[i] for i in b64encode(pt.encode()))
f = open('out.txt', 'wb')
f.write(ct)


Most fun challenge of the CTF IMO.

We get a file encode.py and a very large base64 encoded text out.txt. By reading encode.py we understand that it does the following:

• Reads text from a file
• Trims every instance of 3 or more newlines to 2 newlines
• Trims every instance of 2 or more spaces to 1 space
• Asserts that all text has ord(c) < 128
• Encodes text as base64
• Substitutes each base64 character to another unique base64 character

Our task is to recover the original text despite the base64 character set being substituted.

The first observation is to ensure valid base64 padding. We see the = padding character in the middle of the text which is invalid base64, and also that the 2 character only appears at the end of out.txt. We can make a naive swap of = with 2 to make the padding valid.

We need to solve for some permutation / mapping of 64 encrypted characters to 64 original characters. The next observation to make is that the more correct assignments of shuffled charset to original charset we have, the more the base64 decoded text will โlookโ like a valid input text. This suggests a simulated annealing approach to recover the shuffle permutation.

First we make a mutation function which swaps 2 (or rarely 3 or 4) characters with each other inside the ciphertext, allowing us to iterate through permutation search space.

We then make a fitness function which grades decryptions on how โvalidโ it is:

• We assign a heavy penalty to decryptions containing patterns that definitely cannot appear in the plaintext, such as
• Illegal characters ord(c) >= 128
• Illegal substrings such as three newlines "\n\n\n" and two spaces " " (but I skipped these in the final exploit)
• Assuming the text is printable ASCII, we assign a small penalty to non-printable characters that have ord(c) < 128
• Assuming the text is english, we add a bonus to decryptions containing text patterns which frequently appear in english such as
• Whitespace (" ", "\n")
• Alphabets
• Common english bigrams ("th", "he") and trigrams ("the", "and")
• Valid english punctuation (", ", ". ")
• And several other patterns. See here for the stats used.

Next we create a temperature evaluation function to probabilistically accept or reject new permutations based on a โtemperatureโ parameter and the improvement of fitness from last iteration. We should accept any permutations that improve the fitness, but at the same time we want to give permutations with poorer fitness a small chance to be accepted to avoid being stuck in local minima.

Finally we create a scheduler function which keeps track of overall best permutations and decryption so far while adjusting temperature over time, also known as a cooling schedule.

To keep runtime fast, we trim the ciphertext to the first 40 thousand base64 characters (i.e. first 30 thousand bytes of plaintext).

After around 5 minutes of compute over 4 threads, most of the text is readable and we can find most of the flag as well!

Exploit code
python
from collections import Counter
from base64 import b64decode
from random import choice, sample, random, seed
from string import printable, ascii_letters
from numpy import linspace
from tqdm.auto import tqdm
from multiprocessing import Pool

with open("b64/out.txt", "r") as f:

# 2 is the original padding character
print("Least common", Counter(raw).most_common()[-3:])
print("Tail", raw[-5:])

# swap 2 and =
raw = raw.replace("2", "@").replace("=", "2").replace("@", "=")
print("Total length", len(raw))

# https://www3.nd.edu/~busiforc/handouts/cryptography/cryptography%20hints.html

non_printable_charset = set(range(128)) - set(printable.encode())
whitespace_charset = set(" \n".encode())
alphabet_charset = set(ascii_letters.encode())

one_score = {
i: 0
for i in range(256)
}
for i in range(128, 256):
one_score[i] = -1000
for i in non_printable_charset:
one_score[i] = -10
for i in whitespace_charset:
one_score[i] = 4
for i in alphabet_charset:
one_score[i] = 2

two_patterns_more = [
". ", ", ", ".\n", "\n"
# whoops the last "\n" should have been "\n\n"
]
two_patterns = [
"th", "er", "on", "an", "re", "he", "in", "ed", "nd", "ha", "at",
"en", "es", "of", "or", "nt", "ea", "ti", "to", "it", "st", "io",
"le", "is", "ou", "ar", "as", "de", "rt", "ve",
]
two_patterns_less = [
" T", " O", " A", " W", " B", " C", " D", " S",
"E ", "S ", "T ", "D ", "N ", "R ", "Y ", "F ",
]

two_score = {}
for i in two_patterns_more:
two_score[i.encode()] = 10
for i in two_patterns:
two_score[i.encode()] = 5
for i in two_patterns_less:
two_score[i.encode()] = 1

three_patterns = [
"the", "and", "tha", "ent", "ion", "tio", "for", "nde", "has",
"nce", "edt", "tis", "oft", "sth", "men", " a ", " i ",
]
three_score = {
i.encode(): 10
for i in three_patterns
}

four_patterns = [
" of ", " to ", " in ", " it ", " is ", " be ", " as ", " at ",
" so ", " we ", " he ", " by ", " or ", " on ", " do ", " if ",
" me ", " my ", " up ", " an ", " go ", " no ", " us ", " am ",
]
four_score = {
i.encode(): 10
for i in four_patterns
}

def swap(x, *swaps):
assert(len(swaps) >= 2)
sub = {i: j for i, j in zip(swaps, swaps[1:] + swaps[:1])}
return "".join([sub.get(i, i) for i in x])

b64_charset = 'abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ0123456789+/'
def mutate(x):
while True:
swap_len = choice([2, 2, 2, 2, 3, 4])
swaps = sample(b64_charset, swap_len)
x = swap(x, *swaps)
if choice([True, True, True, False]):
return x

def fitness(x):
x = b64decode(x)
n = len(x)

acc = 0

for i in range(n):
c = x[i]
cc = x[i:i+2]
ccc = x[i:i+3]
cccc = x[i:i+4]

acc += (
one_score[c] +
two_score.get(cc, 0) +
three_score.get(ccc, 0) +
four_score.get(cccc, 0)
)

return acc / n

def mutate_and_fitness(x):
x = mutate(x)
return x, fitness(x)

def check_temp(fit_delta, temp):
if fit_delta > 0:
return True
return - temp * random() < fit_delta

take = 40000
print("Using first b64 characters", take)
cur = raw[:take]
cur_fit = fitness(cur)

best = cur
best_fit = cur_fit

# Nothing up my sleeve, https://remywiki.com/1116
seed(1116)

schedule = sum([
list(linspace(1, 0.0001, 200)) * 30,
list(linspace(0.5, 0.0001, 200)) * 10,
list(linspace(0.1, 0.0001, 200)) * 10,
], [])

since_new_best = 0
stall_limit = 5

pbar = tqdm(schedule)
with Pool(processes=4) as pool:
for temp in pbar:
muts = pool.map(mutate_and_fitness, [cur] * 4)
mut, mut_fit = max(muts, key=lambda x: x[1])

pbar.set_description(
f"Best: {best_fit:.03f} "
f"Temp: {temp:.02f} "
f"Cur: {cur_fit:.03f} "
f"Stall: {since_new_best} "
f"Mut: {mut_fit:.03f} "
)

if mut_fit > best_fit:
best, best_fit = mut, mut_fit
since_new_best = 0
continue

since_new_best += 1

if since_new_best == stall_limit:
since_new_best = 0
cur, cur_fit = best, best_fit
continue

if check_temp(mut_fit - cur_fit, temp):
cur, cur_fit = mut, mut_fit

pln = b64decode(best)
print("First 1000 characters")
print("=" * 80)
print(pln[:1000].decode())
print("=" * 80)
print("Found ictf", b"ictf" in pln)

if b"ictf" in pln:
flag_pos = pln.index(b"ictf")
print(pln[flag_pos-10:flag_pos+50])

output
Least common [('F', 57), ('T', 46), ('2', 2)]
Tail hoP22
Total length 597924
Using first b64 characters 40000
0%|          | 0/10000 [00:00<?, ?it/s]
First 1000 characters
================================================================================
The Project Gutenberg eBooj of The Picture of Dorian Gray, by Nscar Wilde

This eBook is for the use of anyone anywhere in the United States and
most other parts of the world at no cost and with almost no restrictions
whatsoever. You may copy it, give it away or re-use it under the terms
of the Prozect Gutenberg License included with this eBook or online at
www.gutenberg.org. If you are not located in the United States, you
will have to check the laws of the country where you are located before
using this eBooj.

Title+ The Picture of Dorian Gray

Author+ Oscar Wilde

Release Date+ October, 19=4 [eBooj #174]
[Most recently updated+ February 3, 2022]

Language: English

Produced by+ Judith Boss. HTML version by Al Haines.

**: START OF THE PROJECT GUTEOBERG EBONK THE PICTURE NF DORIAN GRAY :**

The Picture of Dorian Gray

by Nscar Wilde

Contents

THE PREFACE
CHAPTER I.
CHAPTER II.
CHAPTER III.
CHAPTER IV.
CHAPTER V.
CHAPTER VI.
CHAPTER VII.
CHAPTER VIII.
CHAPTER IX.
CHAPTER
================================================================================
Found ictf True
b'useless.\n\nictf~all_encodings_are_nothing_but_art}\n\nOSCAR WIL'


Living Without Expectations

Sometimes you just gotta have some fun implementing bare hardness assumptions.

Attachments: lwe.py output.txt

10 solves, 492 points

lwe.py has rand() that provides a deterministic stream of integers from 0 to 7. The script generates a 69-dimension $s$ vector with elements 0 or 1. For each bit of the flag, we are given:

• 69x69 matrix $A$ with random elements $< q$
• 69-dimension vector $b$ equal to $A \times s + \text{rand()} \mod q$ if the flag bit is 0; or
• 69-dimension vector $b$ with random elements $< q$ if the flag bit is 1

This looks like learning with errors, where $s$ is the secret key.

Assuming that the flag begins with ictf{, the first bit of i should be 0 and we should get the $b = A \times s + \text{rand()} \mod q$ case. We can solve for $s$ using Babaiโs closest vector algorithm. Notably, both the noise as well as the secret key $s$ is small in comparison to the magnitude of $q$ and the elements in $A$ and $b$. The closest lattice point is therefore likely to reveal $s$. After this, we can check for each flag bit using $A \times s - b \mod q$ and inspecting if recovered vector is a possible error from $\text{rand()}$ by looking at its magnitude.

Exploit code
sage
import numpy as np

# https://jgeralnik.github.io/writeups/2021/08/12/Lattices/
def babai_closest_vector(B, target):
# Babai's Nearest Plane algorithm
M = B.LLL()
G = M.gram_schmidt()[0]
small = target
for _ in range(1):
for i in reversed(range(M.nrows())):
c = ((small * G[i]) / (G[i] * G[i])).round()
small -= M[i] * c
return target - small

q = 2**142 + 217
n = 69
nseeds = 142

with open("lwe/output.txt", "r") as f:

lines = raw.strip().split("\n")
print("data size", len(raw), len(lines))

def parse_arr(arr):
arr = arr.strip(" ][").split(", ")
return np.array([
int(i.strip("'")[2:], 16)
for i in arr
])

def parse_line(line):
a, _, b = line.strip(" []").partition("] [")
return parse_arr(a), parse_arr(b)

# take first bit, should be 0
which = 0

a, b = parse_line(lines[which])
a = a.reshape((n, n))

# use only first 10 elements of b
samples = 10
lock = 1
lockval = 2 ** 200

mat = [
[0 for x in range(n + samples + lock)]
for y in range(n + samples + lock)
]

for i in range(n):
mat[i][i] = 1
for j in range(samples):
mat[i][n+j] = a[j][i]

for i in range(samples):
mat[n+i][n+i] = q
for i in range(samples):
mat[n+samples][n+i] = -b[i]
mat[n+samples][n+samples] = lockval

vec = [0] * n + [0] * samples + [lockval]

mat = Matrix(mat)
vec = vector(vec)

cv = babai_closest_vector(mat, vec)
s = list(cv)[0:n]
s = np.array(s)

print("s", s)

res = ""
for which in range(len(lines)):
a, b = parse_line(lines[which])
a = a.reshape((n, n))

if -7 * n < ((a @ s % q) - b).sum() <= 0:
res += "0"
else:
res += "1"

print("flag bin", res[:50], "...")

pln = bytes.fromhex(hex(int(res, 2))[2:])
print(pln)

output
data size 67729514 336
s [1 1 1 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 0 0 1 1
1 1 1 0 1 0 0 1 0 0 0 0 1 0 0 1 0 1 0 1 1 0 0 0 0 1 1 1 1 1 0 1]
flag bin 01101001011000110111010001100110011110110110110000 ...
b'ictf{l4tt1c3_crypt0_t4k1ng_0v3r_th3_w0rld}'


Lorge

I guess smoll needed a revenge after all ๐ญ

Attachments: lorge.py output.txt

29 solves, 433 points

From lorge.py, we see that $p-1$ and $q-1$ are $2^{25}$ smooth. We first try to factor $n$ using Pollard p-1 attack and generator $g=2$, but at some point $g$ goes from a coprime number to $1$ straight away. This is because some largest prime factor $x$ of $p-1$ and $q-1$ are the same. Since the initial run stopped at the largest prime factor $x$, we can instead start with generator $g = 2^x \mod n$ and re-try the attack which succeeds.

Exploit code
python
from tqdm.auto import tqdm
from gmpy2 import mpz, next_prime, log, floor, powmod, gcd

with open("lorge/output.txt", "r") as f:
n, e, ct = [int(i.partition(" = ")[2]) for i in raw.strip().split("\n")]

n = mpz(n)

cap = mpz(2 ** 25)

def pollard_rho_step(g, q):
r = mpz(floor(log(cap) / log(q)))
z = q ** r
return powmod(g, z, n)

leak = []
while True:
g = mpz(2)
cur = mpz(2)
for i in leak:
g = pollard_rho_step(g, i)

pbar = tqdm(total=int(cap))
pbar.update(int(cur))
while cur < cap:
g = pollard_rho_step(g, cur)

check = gcd(g - 1, n)
if check != 1:
break

nx = next_prime(cur)
delta = nx - cur
pbar.update(int(delta))
cur = nx

if g == 1:
print("leak common prime factor", cur)
leak.append(cur)
else:
break

p = gcd(g-1, n)
q = n // p
assert(p * q == n)

phi = (p-1) * (q-1)
d = powmod(e, -1, phi)

pln = powmod(ct, d, n)
bytes.fromhex(hex(int(pln))[2:])

output
  0%|          | 0/33554432 [00:00<?, ?it/s]
leak common prime factor 19071329
0%|          | 0/33554432 [00:00<?, ?it/s]


stream

By Eth007

HELP! I encrypted my files with this program I downloaded from the internet. Can you recover my 42-byte flag?

Attachments: stream out.txt

37 solves, 390 points

Open stream binary with ghidra and grok.

Decompile of stream binary

The binary encrypts text with this procedure:

• Read an input file specified in argv[1]
• Process 8 bytes of input at a time as a 64 bit unsigned integer
• Xor the input integer with a 64 bit unsigned integer key specified in argv[2]
• Square the key
• Repeat until all of the input file is processed
• Write the output to a file specified in argv[3]

What happens to input files that need padding to the nearest 8 bytes? We can run the script locally with a dummy key to test.

bash
$python -c "print('\x00' * 42, end='')" > zeros.txt$ ./stream zeros.txt 7 zeros.enc.txt
$hexdump -C -v zeros.txt 00000000 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 |................| 00000010 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 |................| 00000020 00 00 00 00 00 00 00 00 00 00 |..........| 0000002a$ hexdump -C -v zeros.enc.txt
00000000  07 00 00 00 00 00 00 00  31 00 00 00 00 00 00 00  |........1.......|
00000010  61 09 00 00 00 00 00 00  c1 f6 57 00 00 00 00 00  |a.........W.....|
00000020  81 7d 05 a5 39 1e 00 00  01 3b 9b 6e 58 33 3c 30  |.}..9....;.nX3<0|
00000030
\$ python -c "print(hex(pow(7, 2 ** 5, 2 ** 64)))"
0x303c33586e913b01


It seems like the trailing bits are padded with zeros before xor. Since the flag is 42 bytes there will be 6 zero bytes of padding. These tail 6 bytes of enc then leaks the high 6 bytes of key during the final iteration.

Assuming the flag starts with ictf{, we can use this to leak the low 5 bytes of key during the first iteration. The key * key operation is done on a unsigned long long, which acts like multiplication modulo 2**64. As a result of this nice modulo, the $k$ lower bits of key in later iterations only depends on the same $k$ lower bits of key in previous iterations. We can manually square key from first iteration to get the low 5 bytes of key in the final iteration. This way, we have leaked key.

We can recover key of earlier iteration using iterative square root modulo prime power of key from later iterations. The exploit uses z3 to calculate this square root. We can speed this up by early pruning if some key produces a non-printable flag.

Exploit code
python
import struct
from z3 import *
from tqdm.auto import tqdm
from string import printable
from itertools import product

# for some reason sage is slow, so z3 to the rescue
def sqrt_mod_2_64(x):
x = int(x)
res = []
a = BitVec("a", 64)
while True:
s = Solver()
for soln in  res:
if str(s.check()) == "unsat":
break
soln = s.model()[a].as_long()
res.append(soln)
return res

with open("stream/out.txt", "rb") as f:

enc_raw, len(enc_raw)
enc = struct.unpack("<" + "Q" * 6, enc_raw)

key_low_first = (struct.unpack("<Q", b"ictf{\x00\x00\x00")[0] ^ enc[0]) & 0xffffffffff
key_low_final = pow(key_low_first, 2 ** 5, 2 ** 64) & 0xffffffffff
print("final key low 5 bits".ljust(22), hex(key_low_final).rjust(18))

key_high_final = enc[5] & 0xffffffffffff0000
print("final key high 6 bits".ljust(22), hex(key_high_final))

key_final = key_low_final | key_high_final
print("final key".ljust(22), hex(key_final))

charset = set(printable.encode())

def search(idx, key, data=""):
pln_chunk = bytes.fromhex(hex((enc[idx] ^ key) + 2 ** 64)[3:])
if idx == 5:
pln_chunk = pln_chunk[-2:]
if not all([i in charset for i in pln_chunk]):
return None

data = pln_chunk[::-1].decode() + data
print(idx, data.strip())
if idx == 0:
return

prev_keys = sqrt_mod_2_64(key)
for prev_key in prev_keys:
search(idx-1, prev_key, data)

search(5, key_final)

output
final key low 5 bits          0x241d06d81
final key high 6 bits  0x2297b40241d00000
final key              0x2297b40241d06d81
5 }
4 901bf2e4}
3 ystream_901bf2e4}
2 ed_my_keystream_901bf2e4}
1 _rec0vered_my_keystream_901bf2e4}
0 ictf{y0u_rec0vered_my_keystream_901bf2e4}


hash

By Eth007

My passwords are safe and secure with the use of sha42!

nc hash.chal.imaginaryctf.org 1337

Attachments: hash.py jbox.txt

39 solves, 378 points

sha42 has poor confusion. Each byte of the hash output is effectively the xor of some subset of the initial input bytes, depending on the length of the input byte array. We can do a (dumb) symbolic run of the hash to identify which input bytes affect each output byte, then use these correlations with z3 to solve for a possible input which produces some hash.

Since the challenge service does not compare input lengths, we can fix a length of 30 and will always get (some) solution since we have 30 unknowns and 21 equations.

Exploit code
python
from collections import defaultdict
from tqdm.auto import tqdm
from z3 import *
from pwn import *

# original implementation
config = [
[int(a) for a in n.strip()]
]

def sha42(s: bytes, rounds=42):
out = [0]*21
for round in range(rounds):
for c in range(len(s)):
if config[((c//21)+round)%len(config)][c%21] == 1:
out[(c+round)%21] ^= s[c]
return bytes(out).hex()

# generate confusion relations
confusion = defaultdict(list)
# confusion[hash_index] = [input_char_indexes]

length = 30
for target in range(length):
probe = [0 for _ in range(length)]
probe[target] = 1
probe = bytes(probe)

res = sha42(probe)
res = bytes.fromhex(res)

for idx, f in enumerate(res):
if f == 1:
confusion[idx].append(target)

# unhash solver
def unhash(chal):
solver = Solver()

cs = [
BitVec(f"c{i}", 8)
for i in range(length)
]

for sink in range(21):
rhs = 0
for s in confusion[sink]:
rhs = rhs ^ cs[s]
solver.check()
model = solver.model()
res = bytes([
model[s].as_long()
for s in cs
])
return res

# sanity check
chal = sha42(b"some_16char_pass")
chal = bytes.fromhex(chal)
assert sha42(unhash(chal)) == chal.hex()

# run
conn = remote("hash.chal.imaginaryctf.org", 1337)
for _ in tqdm(range(50)):
chal = conn.recvline()
chal = bytes.fromhex(chal.strip().decode())
soln = unhash(chal)
conn.sendline(soln.hex().encode())
print(conn.recvall())

output
[x] Opening connection to hash.chal.imaginaryctf.org on port 1337
[x] Opening connection to hash.chal.imaginaryctf.org on port 1337: Trying 34.90.15.213
[+] Opening connection to hash.chal.imaginaryctf.org on port 1337: Done
0%|          | 0/50 [00:00<?, ?it/s]
[x] Receiving all data
[x] Receiving all data: 16B
[x] Receiving all data: 95B
[+] Receiving all data: Done (95B)
[*] Closed connection to hash.chal.imaginaryctf.org port 1337


otp

By Eth007

Encrypt your messages with our new OTP service. Your messages will never again be readable to anyone.

nc otp.chal.imaginaryctf.org 1337

Attachments: otp.py

48 solves, 316 points

secureRand returns more 1 bits than 0 bits, specifically giving 1 bits around 70% of the time. Connect multiple times to get many flag ^ secureRand() and the true flag bits will be the less common bit at each position.

Exploit code
python
from pwn import *
from tqdm.auto import tqdm
from collections import Counter

def char_to_bin(x):
return bin(0x100 + x)[3:]

# simulate secureRand bias
jumbler = []
jumbler.extend([2**n for n in range(300)])
jumbler.extend([3**n for n in range(300)])
jumbler.extend([4**n for n in range(300)])
jumbler.extend([5**n for n in range(300)])
jumbler.extend([6**n for n in range(300)])
jumbler.extend([7**n for n in range(300)])
jumbler.extend([8**n for n in range(300)])
jumbler.extend([9**n for n in range(300)])
print(
"Probability of 1",
sum([1 if int(str(i)[0]) < 5 else 0 for i in jumbler]) / len(jumbler)
)

conn = remote("otp.chal.imaginaryctf.org", 1337)

# leak many xor-ed flag
n_leaks = 100
data_raw = []
for _ in tqdm(range(n_leaks)):
conn.sendline(b"FLAG")
conn.recvuntil(b"Encrypted flag: ")
res = conn.recvline()
data_raw.append(
bytes.fromhex(res.decode().strip())
)

# recover flag
data = [
"".join([char_to_bin(j)for j in i])
for i in data_raw
]
pln_bin = "".join([
"1" if Counter(i).most_common(1)[0][0] == "0" else "0"
for i in zip(*data)
])
pln = bytes.fromhex(hex(int(pln_bin, 2))[2:])
print(pln)

output
Probability of 1 0.69875
[x] Opening connection to otp.chal.imaginaryctf.org on port 1337
[x] Opening connection to otp.chal.imaginaryctf.org on port 1337: Trying 35.204.97.207
[+] Opening connection to otp.chal.imaginaryctf.org on port 1337: Done
0%|          | 0/100 [00:00<?, ?it/s]
b'ictf{benfords_law_catching_tax_fraud_since_1938}\n'