Here are our challenge writeups from the CryptoCTF 2020 competition. Members of the CryptoHack community played under the team “CryptoHackers” and came second overall, solving 18 of the 20 challenges during the 24 hour competition. This was the first time we all played a CTF together, and we will definitely be doing it again in the future. It was truly a pleasure to get so many cryptographic brains together in one chatroom and collaborate on cracking some mindbending puzzles. Thank you to everyone who played for the CryptoHackers, and to ASIS CTF for organising this enjoyable event. Congratulations to hellman for being an astonishingly quick one man army!

We will be publishing more writeups here as soon as they are finished. If you spot a mistake or an improvement that could be made, please ping jack or hyperreality on CryptoHack Discord.

## Challenges

Challenge Name Category Solved By Points
Trailing Bits Bitshifting willwam845 30
Amsterdam Encoding randomdude999, rkm0959 55
Gambler PRNG Cryptanalyse, willwam845 87
Three Ravens RSA TheBlueFlame121 90
Model RSA TheBlueFlame121, rkm0959, joachim 112
One Line Crypto Primes UnblvR 146
Abbot Maths rkm0959 194
Butterfly Effect PRNG rkm0959, Robin_Jadoul 209
Classic Classical Tux, hyperreality 226
Strip Primes pcback, rkm0959 285
Complex to Hell Hill Cipher pcback, joseph, UnblvR 300
Fatima Reversing pcback 316
Namura Knapsack Q7 375
Decent RSA RSA jack 423
Gengol Goldwasser, RSA pcback, hyperreality, UnblvR 450

## Trailing Bits

### Challenge

The text that includes the flag is transmitted while
unfortunately both of its head and tail bits are lost :(


We are given a file containing binary.

### Solution

From the challenge description, we can guess that the binary data provided has a couple bits removed from either end. Not to worry however, since the removed bits won’t affect the rest of the data at all.

We know that some bits have been removed, so we can just replace those with some decoy bits, and then try decoding from binary until we get readable text.

from Crypto.Util.number import *

i = 0
while True:
data = long_to_bytes(int(flag,2)*2**i)
if b'CCTF' in data:
print(data)
exit()
i += 1


Output:

basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represented as either 0or1, but other representations such as true/false, yes/no, +/−, or on/off are common. The flag is CCTF{it5_3n0u9h_jU5T_tO_sH1ft_M3} The correspondence between these values and the physical states of the underlying storage or device is a matter of convention, and different assignments may be used even within the same device or program. It may be physically implemented with a two-st

### Flag

CCTF{it5_3n0u9h_jU5T_tO_sH1ft_M3}

## Amsterdam

### Challenge

Is it normal to have such encoding?

from Crypto.Util.number import *
from functools import reduce
import operator
from secret import flag, n, k

def comb(n, k):
if k > n :
return 0
k = min(k, n - k)
u = reduce(operator.mul, range(n, n - k, -1), 1)
d = reduce(operator.mul, range(1, k + 1), 1)
return u // d

def encrypt(msg, n, k):
msg = bytes_to_long(msg.encode('utf-8'))
if msg >= comb(n, k):
return -1
m = ['1'] + ['0' for i in range(n - 1)]
for i in range(1, n + 1):
if msg >= comb(n - i, k):
m[i-1]= '1'
msg -= comb(n - i, k)
k -= 1
m = int(''.join(m), 2)
i, z = 0, [0 for i in range(n - 1)]
c = 0
while (m > 0):
if m % 4 == 1:
c += 3 ** i
m -= 1
elif m % 4 == 3:
c += 2 * 3 ** i
m += 1
m //= 2
i += 1
return c

enc = encrypt(flag, n, k)
print('enc =', enc)

enc = 5550332817876280162274999855997378479609235817133438293571677699650886802393479724923012712512679874728166741238894341948016359931375508700911359897203801700186950730629587624939700035031277025534500760060328480444149259318830785583493


### Solution

We see that there are two steps into solving this problem. First, we have to retrieve the value of $m$ from the encryption result. Then, we calculate the plaintext from $m$.

The first part can be done with recursion. By dividing cases on $c \pmod 3$, we can find which ‘if’ statement we entered on the first ‘while’ iteration. This also gives our value of $m \pmod{4}$. We continue this until we have our original value of $c$.

def recv(res):
if res == 1:
return 1
if res % 3 == 0:
return 2 * recv(res//3)
if res % 3 == 1:
return 1 + 2 * recv(res//3)
if res % 3 == 2:
return -1 + 2 * recv(res//3)

## computation result
m = 13037931070082386429043329808978789360911287214189289770230708339088698578551447560972351036453899271623903109387482345515668380476074788749548946464


Now we calculate the plaintext. Notice that $m$ was initially a bit string, which was then converted to an integer. Therefore, we start by changing $m$ into a bit string.

s = []
while m > 0:
s.append(m%2)
m //= 2
s = s[::-1]
n = len(s)


It can be proved with induction that after a successful (one that does not return $-1$) call of encrypt(), the value of ‘msg’ must be $0$. The key idea is Pascal’s Triangle. If we know the value of $k$ at the end of encrypt(), we can reverse engineer the plaintext since we know the result $m$. Brute force all possibilities for $k$, and we are done.

### Implementation

def comb(n, k):
if k > n :
return 0
k = min(k, n - k)
u = reduce(operator.mul, range(n, n - k, -1), 1)
d = reduce(operator.mul, range(1, k + 1), 1)
return u // d

def recv(res):
if res == 1:
return 1
if res % 3 == 0:
return 2 * recv(res//3)
if res % 3 == 1:
return 1 + 2 * recv(res//3)
if res % 3 == 2:
return -1 + 2 * recv(res//3)

## m = recv(enc)
m = 13037931070082386429043329808978789360911287214189289770230708339088698578551447560972351036453899271623903109387482345515668380476074788749548946464
s = []
while m > 0:
s.append(m%2)
m //= 2
s = s[::-1]
n = len(s)

ans = 0
for k in range(0, 400):
ans = 0
for i in range(0, n-1):
if s[n-1-i] == 1:
ans += comb(i, k)
k = k + 1
print(long_to_bytes(ans))


#### Flag

CCTF{With_Re3p3ct_for_Sch4lkwijk_dec3nt_Encoding!}

## Gambler

### Challenge

In this challenge, we have access to a server with the following options

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+ Hi, there is a strong relation between philosophy and the gambling!  +
+ Gamble as an ancient philosopher and find the flag :)                +
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
| Options:
|    [C]ipher flag!
|    [E]ncryption function!
|    [T]ry the encryption
|    [Q]uit


Where the encrypttion function is given by:

def encrypt(m, p, a, b):
assert m < p and isPrime(p)
return (m ** 3 + a * m + b) % p


### Solution

The goal is to decrypt the flag by recovering the hidden parameters $a,b,p$ and then solving the polynomial used in encrypt.

We can recover all parameters quite easily with the function to encrypt our own message.

We can obtain from the server the value of

$y(x) = x^3 + ax + b \mod p$

for any input $x$.

We can recover $b$ by encrypting 0 as

$y(0) = 0^3 + a*0 + b = b \mod p$

Where we are assuming that $a,b < p$.

With the value of $b$, we can calculate

$y(1) = 1 + a + b \mod p, \quad \Rightarrow \quad a = y(1) - 1 - b$

Finally, with both $a,b$ recovered, we need to find the modulus $p$. If we encrypt a fairly small message, such that $y(x) > p$ we can use that

$x^3 + ax + b = y(x) + kp, \quad \Rightarrow \quad kp = x^3 + ax + b - y(x)$

Since we know a and b, we can compute all the terms on the right hand side of this equation and recover $k p$. All that remains is solving for $k$, which is pretty fast as $k$ is so small.

With all parameters known, we can request the encrypted flag from the server and solve the cubic equation with Sage: $x^3 + ax + b = c \mod p$

### Implementation

import os
os.environ["PWNLIB_NOTERM"] = "True"
from pwn import *
from Crypto.Util.number import long_to_bytes

debug = False

r.recvuntil(b'|\t[Q]uit\n')

r.sendline('C')
data = r.recvuntil(b'|\t[Q]uit\n')
enc = int(data.split().decode().strip())

def encrypt_int(n):
r.sendline('T')
r.sendline(str(n))
data = r.recvuntil(b'|\t[Q]uit\n')
b = int(data.split().decode().strip())
return b

b = encrypt_int(0)
c = encrypt_int(1)
a = c - b - 1
enc_kp = encrypt_int(100)
kp = (100**3 + a*100 + b) - enc_kp

if debug:
print(a)
print(b)
print(kp)

p = max(f for f in factor(kp))
PR.<x> = PolynomialRing(GF(p))
f = x^3 + a * x + b - enc
rts = f.roots()
print(rts)

for root in rts:
flag = root
print(long_to_bytes(flag))

r.interactive()


### Flag

CCTF{__Gerolamo__Cardano_4N_itaLi4N_p0lYma7H}

## Three Ravens

### Challenge

There were three ravens sat on a tree, Downe a downe, hay downe, a downe, They were as black as they might be.

#!/usr/bin/python

from Crypto.Util.number import *
from flag import flag

def keygen(nbit):
while True:
p, q, r = [getPrime(nbit) for _ in range(3)]
if isPrime(p + q + r):
pubkey = (p * q * r, p + q + r)
privkey = (p, q, r)
return pubkey, privkey

def encrypt(msg, pubkey):
enc = pow(bytes_to_long(msg.encode('utf-8')), 0x10001, pubkey * pubkey)
return enc

nbit = 512
pubkey, _ = keygen(nbit)
print('pubkey =', pubkey)

enc = encrypt(flag, pubkey)
print('enc =', enc)


### Solution

The challenge encrypts the flag with a modulus

$N = (p*q*r)*(p+q+r)$

and gives the output $n = pqr$, $k = p+q+r$. To totally break the cryptosystem, we would want to find the totient of the modulus

$\phi(N) = (p-1)(q-1)(r-1)(p+q+r - 1)$

but we can simplify this when the encrypted message $m$ is small enough. If we have $m < k$, we can instead find $\phi(k) = k-1$, and find $e^{-1} \mod \phi(k)$, and solve!

Observe that for any $n, l$, as long as $l | n$, any equivalence $a \equiv b \pmod n$ also holds mod $l$: $a \equiv b \pmod l$ (but note that the other way around does not necessarily hold). To fully see this, we can write

\begin{align*} a &\equiv b \pmod n \\ \Leftrightarrow \\ a &= b + kn \\&=b + ktl \quad(\text{since } l|n\text{, so } n = tl)\\ \Rightarrow\\ a &\equiv b \pmod l \end{align*}

So, since $k|n$, we can solve $m \equiv m^{1+\phi(k)} \pmod k$ as if it was a single-prime RSA problem. And because $m < k$, the residue $[m]_k$ (the rest when dividing $m$ by $k$) is exactly equal to $m$.

### Implementation

from Crypto.Util.number import *

k = 31678428119854378475039974072165136708037257624045332601158556362844808093636775192373992510841508137996049429030654845564354209680913299308777477807442821
c = 8218052282226011897229703907763521214054254785275511886476861328067117492183790700782505297513098158712472588720489709882417825444704582655690684754154241671286925464578318013917918101067812646322286246947457171618728341255012035871158497984838460855373774074443992317662217415756100649174050915168424995132578902663081333332801110559150194633626102240977726402690504746072115659275869737559251377608054255462124427296423897051386235407536790844019875359350402011464166599355173568372087784974017638074052120442860329810932290582796092736141970287892079554841717950791910180281001178448060567492540466675577782909214
e = 0x10001

phi = k-1
d = pow(e,-1,phi)

flag = pow(c,d,k)
print(long_to_bytes(flag))


### Flag

CCTF{tH3_thr3E_r4V3n5_ThRe3_cR0w5}

## Model

### Challenge

def genkey(nbit):
while True:
p, q = getPrime(nbit), getPrime(nbit)
if gcd((p-1) // 2, (q-1) // 2) == 1:
P, Q = (q-1) // 2, (p-1) // 2
r = inverse(Q, P)
e = 2 * r * Q  - 1
return(p, q, e)

def encrypt(msg, pubkey):
e, n = pubkey
return pow(bytes_to_long(msg), e, n)


### Solution

The key to solving this challenge is that

$e \equiv 2 Q^{-1}Q - 1 \equiv 2 - 1 \equiv 1 \pmod P$

So encrypting a message m we have, for some integer $k$,

$m^e \equiv m^{1 + \frac{k(q-1)}{2}} \equiv m \cdot m^\frac{k(q-1)}{2} \pmod q$

Looking at the second term, it can be reduced using Fermat’s little theorem as follows:

$m^{\frac{k*(q-1)}{2}}\equiv (m^{(q-1)})^{\frac{k}{2}}\equiv 1^{\frac{k}{2}} \equiv 1^{\frac{1}{2}} \pmod q$

$1$ can have only two roots here, namely $\pm 1$. Therefore the encryption becomes

$m^e \equiv m^{1 + \frac{k(q-1)}{2}} \equiv \pm m \pmod q$

and we can compute $q = \text{gcd}(m^e \mp m, n)$ to factorize $n$ because $m^e \mp m \equiv 0 \pmod q$ is a multiple of $q$.

One last step that needs attention is that we recover $q$, not $p$, which matters for the recovery of $e$, as the primes are not interchangeable there.

### Implementation

from Crypto.Util.number import *
import math

def derive_e(p,q):
P, Q = (q-1) // 2, (p-1) // 2
r = pow(Q, -1, P)
e = 2 * r * Q  - 1
return e

N = 17790613564907955318126717576181316624843451677921227941389832111093895513875496295594784102148835715126789396535470416868485674231839509486983792844881941097589192520877472968227711640216343330193184235164710328845507199362646489303138765492026284976828397217700058854699501312701069031398507487060508966602815218264215778115331187180105972920333780067280854048113094622799996118383376340217782122945586262887450863620856214375258659362300743471229410735400189992359220551961441580630740022857304514895745174813529758766758733506538696933950282130984955594881517339093338779101106466633380921338845195921235252323721
flag_enc = 8216344743331409189205831776342200252705923796193752552649425282859227400617284746437075756157249953578189229459392338128783031841882560801175367779263048253787547952450480816724222285583987363793884961526545550108790689158473753461378651141379053427506957702375732452598640804768960184186521954448243004125900395894265450073650101942224629389391631821735998886688813393717718376391743836798122485350719355567861201466641767009303179260141365766023680788250688524528992952859061172438083227729190577738108854783536925967748199734513738782142055609101950770816942854252284355975365351013803601963990179403849614198536

m = bytes_to_long(b'0')
ct = 8131881080215090371487466406674376044247120209816118806949066423689730735519795472927218783473464525260814227606067990363574576048132004742403517775620572793232598693334765641758830271460405790617624271060522834683042735967050146871067065889288923913486919193720360254923458500009885281654478144592942337767754315130844294762755237864506689552987776560881357285727629827190391683150994461127468196118126587159811046890420456603820675085450111755868116701855834309297184745623927049652098555126342100576188575279791066071616897443075423425299542959405192350563251777193668273523389978129359003036691597884885020756981

q = math.gcd(ct - m, N)
assert isPrime(q)
p = N // q
e = derive_e(p,q)
d = pow(e, -1, (p-1)*(q-1))
m = pow(flag_enc,d,N)
print(long_to_bytes(m))


### Flag

CCTF{7He_mA1n_iD34_0f_pUb1iC_key_cryPto9raphy_iZ_tHa7_It_l3ts_y0u_puBli5h_4N_pUbL!c_k3y_wi7hOuT_c0mprOmi5InG_y0Ur_5ecr3T_keY}

## One Line Crypto

### Challenge

A profile, a look, a voice, can capture a heart ♥ in no time at all.”

We’re given this very short snippet:

#!/usr/bin/python

from Crypto.Util.number import *
from secret import m, n, x, y, flag

p, q = x**(m+1) - (x+1)**m, y**(n+1) - (y+1)**n
assert isPrime(p) and isPrime(q) and p < q < p << 3 and len(bin(p*q)[2:]) == 2048
enc = bytes_to_long(flag)
print(pow(enc, 0x10001, p*q))


and the output of the print function, which consists of just a single, giant number. This is typical RSA encryption, with a slight twist: n is not given!

### Solution

The prime selection for p and q is identical. Two secret numbers x and m are used, such that x^(m+1) - (x+1)^m is a prime. Calculating q uses differently named constants, but there’s no cross-dependencies between the parameters.

Knowing that p*q contains 2048 bits, counting from the MSB, puts some rather strict bounds on the parameters. Just playing around with random numbers, it quickly becomes clear that m/n can’t be very big.

The plan simply becomes:

1. Start off with a low bound like 500 or 1000
2. Brute-force all m and x less than the bound, such that x^(m+1) - (x+1)^m is max 2048 bits.
3. When we got ourselves a small pool of candidate values, pair up two and two random values from the pool
4. Check if their product is 2048 bits, and try to decrypt the ciphertext.

### Implementation

#!/usr/bin/python3
from Crypto.Util.number import long_to_bytes
from gmpy2 import invert, is_prime
from tqdm import tqdm

primes = []

for xy in tqdm(range(500)):
for mn in range(500):
prime = xy**(mn+1) - (xy+1)**mn
if prime.bit_length() > 2048: break
if is_prime(prime):
primes.append(prime)

c = 14608474132952352328897080717325464308438322623319847428447933943202421270837793998477083014291941466731019653023483491235062655934244065705032549531016125948268383108879698723118735440224501070612559381488973867339949208410120554358243554988690125725017934324313420395669218392736333195595568629468510362825066512708008360268113724800748727389663826686526781051838485024304995256341660882888351454147057956887890382690983135114799585596506505555357140161761871724188274546128208872045878153092716215744912986603891814964771125466939491888724521626291403272010814738087901173244711311698792435222513388474103420001421

for i in range(len(primes)):
for j in range(i, len(primes)):
pq = primes[i]*primes[j]
if len(bin(pq)[2:]) == 2048:
try:
d = invert(0x10001, (primes[i]-1)*(primes[j]-1))
dec = long_to_bytes(pow(c, d, pq))
if b"CCTF" in dec:
print(dec)
except ValueError:
pass


### Flag

CCTF{0N3_1!nE_CrYp7O_iN_202O}

## Butterfly Effect

### Challenge

Have you heard of the butterfly effect in chaos theory? We have a very clear sample!

def hq_prng(x, p, g):
rng = 0
for _ in range(getRandomNBitInteger(14)):
x = pow(g, x, p)
for i in range(nbit):
x = pow(g, x, p)
if x < (p-1) // 2:
rng += 2**i - 1
elif x > (p-1) // 2:
rng -= 2**i + 1
else:
rng ^= 2**(i + 1)
if rng <= 0:
return -rng
return rng

def keygen(p, g):
r, s = hq_prng(getRandomNBitInteger(nbit), p, g), hq_prng(getRandomNBitInteger(nbit), p, g)
u, v = gmpy2.next_prime(r**2 + s**2), gmpy2.next_prime(2*r*s)
e, n = 0x10001, u * v
return e, n

def encrypt(msg, e, n):
return pow(bytes_to_long(msg.encode('utf-8')), e, n)

| encrypt(flag, e, n) = 117667582947026307482709850318214820165964980495414423711608614681075036546959172088083682928734150238100258554942348560628319669155021151214088627854571799267413754120136204715904365400299634909310436631535142237485014585244686834739846288478469145431250711508705770591654103989678439960043788163169606969942

| (p, g, n) = (68396932999729141946282927360590169590631231980913314894620521363257317833167L, 11148408907716636563689390048104567047599159688378384611325239859308138541650L, 174421003123810033381790236221837856515717326914686209725144085385038416771831243218121939343204739172142573392914539954561702192037384452671829208544052005809111211272767217997373349539621608610104198849289553199153550766664805075406312493730029215085806581713874721280621739864343621647737777392917211017939L)


### Solution

We start by realizing that the given value of $g$ is not actually a generator of $\mathbb{F}_p$. In reality, $g$ generates a very small subgroup, having order of $31337$. This can be easily computed with Sage or brute force in Python.

This implies that there are at most $31337$ possible values for the outputs for the PRNG. We calculate them all, then sort. Denote the result as $x_1 \le x_2 \le \cdots \le x_{31337}$.

We now want to find a pair $(i, j)$ such that $N = \text{nxt}(x_i^2 + x_j^2) \cdot \text{nxt}(2 x_ix_j)$, where $\text{nxt}(n)$ is the smallest prime larger than $n$. This gives a solution with approximately $31337^2/2$ calls to the $\text{nxt}$ function. This is too slow to solve the problem.

To optimize, we use two tricks. First, due to small prime gap, one may assume that

$(x_i^2+x_j^2) \cdot 2x_ix_j \le N \le (x_i^2+x_j^2+1000) \cdot (2x_ix_j+1000)$ holds true. This cuts the number of pairs to compute. Also, if we fix $x_i$, the values of $x_j$ that satisfy this inequality forms an interval. Therefore, one can use binary search to efficiently compute this interval. This is efficient enough for the task.

### Implementation

from Crypto.Util.number import *

p = 68396932999729141946282927360590169590631231980913314894620521363257317833167
g = 11148408907716636563689390048104567047599159688378384611325239859308138541650
N = 174421003123810033381790236221837856515717326914686209725144085385038416771831243218121939343204739172142573392914539954561702192037384452671829208544052005809111211272767217997373349539621608610104198849289553199153550766664805075406312493730029215085806581713874721280621739864343621647737777392917211017939
c = 117667582947026307482709850318214820165964980495414423711608614681075036546959172088083682928734150238100258554942348560628319669155021151214088627854571799267413754120136204715904365400299634909310436631535142237485014585244686834739846288478469145431250711508705770591654103989678439960043788163169606969942
e = 0x10001

def hq_prng(x, p, g):
global rsf;
rng = 0
for i in range(256):
x = rsf[x]
if x < (p-1) // 2:
rng += 2**i - 1
elif x > (p-1) // 2:
rng -= 2**i + 1
else:
rng ^= 2**(i + 1)
x = x % 31337
if rng <= 0:
return -rng
return rng

rsf = 1
for i in range(1, 31337):
rsf[i] = (g * rsf[i-1]) % p
WOW =  * 31337
for i in range(0, 31337):
WOW[i] = hq_prng(i, p, g)
WOW.sort()
for i in range(0, 31337):
lef = i+1
rig = 31336
mid, best = 0, 0
while lef <= rig:
mid = (lef + rig) // 2
if (WOW[i]*WOW[i] + WOW[mid] * WOW[mid]) * 2 * WOW[i] * WOW[mid] >= n:
best = mid
rig = mid -1
else:
lef = mid + 1
if best == 0:
continue
for j in range(best-1, min(best+30, 31337)):
u = WOW[i] * WOW[i] + WOW[j] * WOW[j]
v = 2 * WOW[i] * WOW[j]
if u * v <= n and n <= (u+1000) * (v+1000):
u = nextprime(u)
v = nextprime(v)
if u * v == n:
break

u = 13207168490744652956999406596846767472614127517045010655090178723910296606220559473477009696618646553552917605315941229263316963221556883007951846286507319
v = 13206540315287197799978983146788490475082830408392129019383447128092673850363700139125558344894148410716241976023782262109119063597770109472702331423302981

phi = (u-1)*(v-1)
d = inverse_mod(e,phi)
flag = pow(c,d,N)
print(long_to_bytes(flag))


### Flag

CCTF{r341Ly_v3ryYyyyYY_s3cUrE___PRNG___}

## Abbot

### Challenge

It isn’t the big troubles in life that require character. Anybody can rise to a crisis and face a crushing tragedy with courage, but to meet the petty hazards of the day with a laugh.

import string
import random
from fractions import Fraction as frac
from secret import flag

def me(msg):
if len(msg) == 1 :
return ord(msg)
msg = msg[::-1]
reducer = len(msg) - 1
resultNum, resultDen = frac(ord(msg), reducer).denominator, frac(ord(msg), reducer).numerator
reducer -= 1
for i in range(1, len(msg)-1):
result =  ord(msg[i]) +  frac(resultNum, resultDen)
resultDen, resultNum  = result.denominator, result.numerator
resultDen, resultNum =  resultNum, reducer * resultDen
reducer -= 1
result = ord(msg[-1]) + frac(resultNum, resultDen)
resultDen, resultNum  = result.denominator, result.numerator
return (resultNum, resultDen)

def you(msg):
if len(msg) == 1 :
return ord(msg)
msg = msg[::-1]
reducer = (-1) ** len(msg)
result = frac(ord(msg), reducer)
resultNum, resultDen = result.denominator, result.numerator
reducer *= -1
for i in range(1, len(msg)-1):
result =  ord(msg[i]) +  frac(resultNum, resultDen)
resultDen, resultNum  = result.denominator, result.numerator
resultDen, resultNum =  resultNum, reducer * resultDen
reducer *= -1

result = ord(msg[-1]) + frac(resultNum, resultDen)
resultDen, resultNum  = result.denominator, result.numerator
return (resultNum, resultDen)

def us(msg):
if len(msg) == 1 :
return ord(msg)
msg = msg[::-1]
reducer = (-1) ** int(frac(len(msg), len(msg)**2))
result = frac(ord(msg), reducer)
resultNum, resultDen = result.denominator, result.numerator
reducer **= -1
reducer = int(reducer)
for i in range(1, len(msg)-1):
result =  ord(msg[i]) +  frac(resultNum, resultDen)
resultDen, resultNum  = result.denominator, result.numerator
resultDen, resultNum =  resultNum, reducer * resultDen
reducer **= -1
reducer = int(reducer)
result = ord(msg[-1]) + frac(resultNum, resultDen)
resultDen, resultNum  = result.denominator, result.numerator
return (resultNum, resultDen)

dict_encrypt = {
1: me,
2: you,
3: us,
4: you,
5: me
}
cipher = [[] for _ in range(5)]
S = list(range(1,6))
random.shuffle(S)
print("enc = [")
for i in range(4):
cipher[i] = dict_encrypt[S[i]](flag[int(i * len(flag) // 5) : int(i * len(flag) // 5 + len(flag) // 5)])
print(cipher[i])
print(", ")
i += 1
cipher[i] = dict_encrypt[S[i]](flag[int(i * len(flag) // 5) : int(i * len(flag) // 5 + len(flag) // 5)])
print(cipher[i])
print( " ]")

enc = [(4874974328610108385835995981839358584964018454799387862L, 72744608672130404216404640268150609115102538654479393L),(39640220997840521464725453281273913920171987264976009809L, 366968282179507143583456804992018400453304099650742276L),(145338791483840102508854650881795321139259790204977L, 1529712573230983998328149700664285268430918011078L),(84704403065477663839636886654846156888046890191627L, 717773708720775877427974283328022404459326394028L),(287605888305597385307725138275886061497915866633976011L, 8712550395581704680675139804565590824398265004367939L)]


### Solution

The code breaks the flag to five pieces, applies three types of encryption function to them, then combines them into an array. Therefore, it suffices to write codes that reverse these encryptions. Then, we can try to reverse each ciphertext with all types of encryption function, and see what strings it gives us.

We’ll start with the me encryption function and work from there. The key intuition is that the value of resultNum / resultDen is kept between 0 and 1 at the end of each iteration. If this holds, this implies that we can calculate each character of the plaintext by simply taking the integer part of the fraction and using chr() function. Why would this be true? Well, we can guess that the length of the each broken message is less than, say, 30. Since each character of the string is readable, the ASCII value of them will be at least 33. With this idea in hand, one can prove the claim by induction. Reversing is also straightforward. Take the integer part, change it into character, take the remaining fractional part and reverse it accordingly.

you encryption is a bit trickier since negative values appear. By inspection, we can see that resultNum / resultDen changes its sign every iteration. The absolute value is kept between 0 and 1. The key idea remains the same, but we need to be careful.

us encryption is easy, since the value of reducer equals 1 the entire time. Noting this, the code remains very similar to the one of me encryption.

### Implementation

def me(resultNum, resultDen, dg):
st = ""
if resultNum == 0 or resultDen == 0:
return ""
if dg == 0:
TT = resultNum // resultDen
if TT < 0 or TT > 256:
return ""
st = chr(TT)
resultNum = resultNum - TT * resultDen
st += me(resultNum, resultDen, dg+1)
return st
acnum = resultDen * dg
acden = resultNum
if acden == 0:
return ""
TT = acnum // acden
if TT < 0 or TT > 256:
return ""
st = chr(TT)
acnum = acnum - TT * acden
st += me(acnum, acden, dg+1)
return st

def you(resultNum, resultDen, dg, cv):
st = ""
if resultNum == 0 or resultDen == 0:
return ""
if cv == 0:
TT = resultNum // resultDen
if TT < 0 or TT > 256:
return ""
st = chr(TT)
resultNum %= resultDen
st += you(resultNum, resultDen, dg, cv+1)
return st
acnum = resultDen
acden = resultNum * dg
if acden < 0:
acden = -acden
acnum = acnum
if acden == 0:
return ""
if cv % 2 == 0:
TT = acnum // acden
if TT < 0 or TT > 256:
return ""
st = chr(TT)
acnum %= acden
st += you(acnum, acden, dg*-1, cv+1)
return st
else:
TT = (acnum + acden - 1) // acden
if TT < 0 or TT > 256:
return ""
st = chr(TT)
acnum = acnum - TT * acden
st += you(acnum, acden, dg*-1, cv+1)
return st

def us(resultNum, resultDen, dg):
st = ""
if resultNum == 0 or resultDen == 0:
return ""
if dg == 0:
TT = resultNum // resultDen
if TT < 0 or TT > 256:
return ""
st = chr(TT)
resultNum %= resultDen
st += us(resultNum, resultDen, dg+1)
return st
acnum = resultDen
acden = resultNum // dg
if acden == 0:
return ""
TT = acnum // acden
if TT < 0 or TT > 256:
return ""
st = chr(TT)
acnum %= acden
st += us(acnum, acden, dg)
return st

for i in range(0, 5):
print(me(enc[i], enc[i], 0))
print(you(enc[i], enc[i], -1, 0))
print(us(enc[i], enc[i], 0))



### Flag

 CCTF{This_13_n0t_Arthur_Who_l0ves_Short_st0ries_This_IS___ASIS___Crypto_CTF____with_very_m0d3rn_arthur_Enc0d1ng!!_D0_you_Enj0y_IT_as_w311??}

### Challenge

A dream is not reality, but who’s to say which is which?

import random
from secret import p, flag

def transpose(x):
result = [[x[j][i] for j in range(len(x))] for i in range(len(x))]
return result

def multiply(A, B):
if len(A) != len(B):
return None
result = []
for i in range(len(A)):
r = []
for j in range(len(B)):
r.append(0)
result.append(r)
for i in range(len(A)):
for j in range(len(B)):
for k in range(len(B)):
result[i][j] += A[i][k] * B[k][j]
return result

def sum_matrix(A, B):
result = []
for i in range(len(A)):
r = []
for j in range(len(A)):
r.append(A[i][j]+B[i][j])
result.append(r)
return result

def keygen(p):
d = random.randint(1, 2**64)
if p % 4 == 1:
Q = []
for i in range(p):
q = []
for j in range(p):
if i == j:
q.append(0)
elif pow((i-j), int ((p-1) // 2), p) == 1:
q.append(1)
else:
q.append(-1)
Q.append(q)
Q_t = transpose(Q)
H = []
r = []
r.append(0)
r.extend([1 for i in range(p)])
H.append(r)
for i in range(1, p + 1):
r = []
for j in range(p + 1):
if j == 0:
r.append(1)
else:
r.append(Q[i-1][j-1])
H.append(r)

H2 = [[0 for j in range(2*(p+1))] for i in range(2*(p+1))]
for i in range(0, p+1):
for j in range(0, p+1):
if H[i][j] == 0:
H2[i*2][j*2] = 1
H2[i*2][j*2+1] = -1
H2[i*2+1][j*2] = -1
H2[i*2+1][j*2+1] = -1
elif H[i][j] == 1:
H2[i*2][j*2] = 1
H2[i*2][j*2+1] = 1
H2[i*2+1][j*2] = 1
H2[i*2+1][j*2+1] = -1
else:
H2[i*2][j*2] = -1
H2[i*2][j*2+1] = -1
H2[i*2+1][j*2] = -1
H2[i*2+1][j*2+1] = +1
ID = [[(-1)**d if i == j else 0 for i in range(len(H2))] for j in range(len(H2))]
H2 = multiply(ID, H2)
return(H2, d)
else:
Q = []
for i in range(p):
q = []
for j in range(p):
if i == j:
q.append(0)
elif pow( (i-j), int ((p-1) // 2), p) == 1:
q.append(1)
else:
q.append(-1)
Q.append(q)
Q_t = transpose(Q)
Q_Q_t = multiply(Q, Q_t)
H1 = []
H1.append([1 for i in range(p+1)])
for i in range(1, p +1):
r = []
for j in range(p +1):
if j == 0:
r.append(-1)
elif i == j:
r.append(1 + Q[i-1][j-1])
else:
r.append(Q[i-1][j-1])
H1.append(r)
ID = [[(-1)**d if i == j else 0 for i in range(len(H1))] for j in range(len(H1))]
H1 = multiply(ID, H1)
return(H1, d)

def encrypt(msg, key):
matrix = key
d = key
m = [[ord(char) for char in msg ]]
de = [[-d for i in range(len(msg))]]
C = multiply(m, matrix)
cipher = sum_matrix(C, de)
return cipher

key = keygen(p)
flag = flag + (len(key) - len(flag)) * flag[-1]
cipher = encrypt(flag, key)
print('cipher =', cipher)


### Solution

By analyzing the dimensions of the ciphertext, it’s straightforward to find $p=37$. Since the matrix part of the secret key is only determined by $p$ and the parity of $d$, we have two possible matrices to consider. Also, if we fix $d$, we can compute the plaintext by solving a system of linear equations. We proceed this way.

If we iterate over $d$ simply as integers, the solution of the equation may contain rational, non-integer numbers. This is slow and prone to floating-point errors, (unless we take proper care) so we will use another trick.

Since all the ordinal values in the plaintext are between $0$ and $256$, we will take this entire problem into $\mathbb{F}_{257}$. This way, we can try $257$ different values of $d$, solve the system without worrying about floating error, and retrieve our answer.

### Implementation

## keygen(d) : simply keygen with p=37, d's parity

MAT0 = keygen(0)
MAT1 = keygen(1)
MM0 = Matrix(GF(257), MAT0)
MM1 = Matrix(GF(257), MAT1)
cipher = [-3459749918754130611, -3459749918754138177, -3459749918754137803, -3459749918754138385, -3459749918754138025, -3459749918754138097, -3459749918754138073, -3459749918754138245, -3459749918754138183, -3459749918754138445, -3459749918754137991, -3459749918754138597, -3459749918754138309, -3459749918754138309, -3459749918754138279, -3459749918754138771, -3459749918754138327, -3459749918754138485, -3459749918754138233, -3459749918754138389, -3459749918754138207, -3459749918754138555, -3459749918754138141, -3459749918754138501, -3459749918754138677, -3459749918754138297, -3459749918754138563, -3459749918754138439, -3459749918754138429, -3459749918754138041, -3459749918754138611, -3459749918754138469, -3459749918754138217, -3459749918754138585, -3459749918754138403, -3459749918754138177, -3459749918754137777, -3459749918754138587, -3459749918754138231, -3459749918754138677, -3459749918754138127, -3459749918754138679, -3459749918754137789, -3459749918754138305, -3459749918754138025, -3459749918754138301, -3459749918754137941, -3459749918754138489, -3459749918754137583, -3459749918754138297, -3459749918754137949, -3459749918754138475, -3459749918754137879, -3459749918754138813, -3459749918754137981, -3459749918754138395, -3459749918754138201, -3459749918754138459, -3459749918754138195, -3459749918754138617, -3459749918754138003, -3459749918754138557, -3459749918754138429, -3459749918754138499, -3459749918754137951, -3459749918754138673, -3459749918754137975, -3459749918754138341, -3459749918754138121, -3459749918754138375, -3459749918754137869, -3459749918754138459, -3459749918754137739, -3459749918754138405, -3459749918754137921, -3459749918754138775]
res = vector(GF(257), cipher)
XX = MM0.solve_right(res)
YY = MM1.solve_right(res)
for i in range(0, 257):
stX = ""
stY = ""
for j in range(0, 76):
for j in range(0, len(XX)):
if (int)(XX[j]) <= 255:
stX = stX + chr((int)(XX[j]))
for j in range(0, len(YY)):
if (int)(YY[j]) <= 255:
stY = stY + chr((int)(YY[j]))
if "CCTF" in stX:
print(stX)
if "CCTF" in stY:
print(stY)


### Flag

CCTF{TH13_i3_Hadamard_rip_y0ung_&_bri11iant_Paley!}

## Classic

### Challenge

Classic is Easy but Essential!

b7UkM iK2L0 PUVnZ Ho79I tDAf0 PUvfQ G5jHo 7GwLG wL9It vfQHo 7G5j0 PUvfQ 9Ithd JkMiK 2LU2b 0PUkM B8Nih dJK2L GwL0P UHo7U 2bK2L
...


### Solution

Taking trigrams from the ciphertext (i.e. splitting it up into groups of three characters), this becomes a classic monoalphabetic substitution cipher. See the script for inline comments about the solving methodology we used on this more open-ended challenge.

### Implementation

import string
from collections import Counter
from cipher_solver.simple import SimpleSolver

with open('enc.txt') as f:

def chunks(l, n):
n = max(1, n)
return [l[i:i+n] for i in range(0, len(l), n)]

# 1. We suspect that groupings of 5 are there to confuse us.
# Let's break chars into groups of different sizes and look at
# the size of the set
for i in range(1,10):
unique = len(set(chunks(ctext, i)))
print(f"{unique} unique groups when split into groups of length {i}")

# 2. Breaking into groups of 3 (trigrams) gives much less unique chars
chunked = chunks(ctext, 3)
freq = Counter(chunked).most_common()
print(freq)

# 3. Build a substitution table for each trigram to a different letter
# only important thing is that the most frequent trigram corresponds to a space
subs = {}
alphabet = " " + string.ascii_lowercase + string.ascii_uppercase
cur = 0
for trigram in freq:
subs[trigram] = alphabet[cur]
cur += 1
print(subs)

# 4. Make the substitutions
substituted = "".join([subs[c] for c in chunked])
print(substituted)

# 5. Use any algorithm for solving substitution ciphers (quipqiup also works)
s = SimpleSolver(substituted)
s.solve()

# 6. It's readable and the flag is visible after a couple of manual
# substitutions
ptext = s.plaintext()
ptext = ptext.replace('z', 'T')
ptext = ptext.replace('q', '_')
print(ptext)


### Flag

CCTF{The_main_classical_cipher_types_are_substitution_ciphers}

## Heaven

### Challenge

from bitstring import BitArray
from heaven import seventh_seal, oh, no, new_testament

def matthew_effect(shire, rohan):
gandalf = ''
for every, hobbit in enumerate(shire):
gandalf += oh if ord(hobbit) ^ ord(rohan[every]) == 0 else no
return gandalf

def born_to_die(isengard):
luke = 0
for book in new_testament:
luke ^= ord(isengard[book])
lizzy_grant = oh + isengard[:-1] if luke == 0 else no + isengard[:-1]
return lizzy_grant

david = len(seventh_seal)
elf = seventh_seal
bilbo = len(lord)
matthew = 0
princess_leia = ''
destiny = bilbo // david
apocalypse = bilbo % david
for i in range(32):
elf = born_to_die(elf)
while matthew < destiny:
princess_leia += matthew_effect(elf, lord[matthew * david : (matthew + 1) * david])
elf = born_to_die(elf)
matthew += 1
princess_leia += matthew_effect(elf[:apocalypse], lord[matthew * david :])
res = open('flag.enc', 'wb')
res.write(bytes(int(princess_leia[i : i + 8], 2) for i in range(0, bilbo, 8)))


### Solution

After some renaming and minor reverse engineering of the challenge logic, we see that a JPEG image has been xor’ed with a keystream generated from an LFSR. Each time a key-sized block is xored, and the key is forwarded one step in the LFSR.

Xor the encrypted file with a JFIF jpg file header to try to recover the current state of the LFSR.

from bitstring import BitArray

pt = BitArray(bytes=bytes.fromhex('FFD8FFE000104A4649460001')).bin
with open('flag.enc', 'rb') as f:
ct = BitArray(bytes=content).bin

key = ''

for i, j in zip(pt, ct):
key += str(int(i) ^ int(j))
print(key)


Then we can get

1100011100101011000
0110001110010101100
0011000111001010110
0001100011100101011
1000110001110010101
0


Since we know from the source code that encryptions under consecutive keys share almost the entire key (x...xa and bx...x), we can recover the length of the key from this. We can observe this rotation already in the above listing of the bits, thanks to our insertion of newlines in the right positions.

Finally we can brute force the polynomial to recover the original image file.

### Implementation

from bitstring import BitArray
key = '1100011100101011000'
seventh_seal = key
oh = '0'
no = '1'

def matthew_effect(shire, rohan):
gandalf = ''
for every, hobbit in enumerate(shire):
gandalf += oh if ord(hobbit) ^ ord(rohan[every]) == 0 else no
return gandalf

new_testament = []

def born_to_die(isengard):
luke = 0
for book in new_testament:
luke ^= ord(isengard[book])
lizzy_grant = oh + isengard[:-1] if luke == 0 else no + isengard[:-1]
return lizzy_grant

a = b'1100011100101011000'
b = b'0110001110010101100'
c = b'0011000111001010110'
d = b'0001100011100101011'
e = b'1000110001110010101'

for i in range(19):
for j in range(i+1, 19):
for k in range(j+1, 19):
for l in range(k+1, 19):
if a[i] ^ a[j] ^ a[k] ^ a[l] == \
b[i] ^ b[j] ^ b[k] ^ b[l] == \
c[i] ^ c[j] ^ c[k] ^ c[l] == \
e[i] ^ e[j] ^ c[k] ^ c[l] == 0:
if d[i] ^ d[j] ^ d[k] ^ d[l] == 1:
print(i, j, k, l)
seventh_seal = '1100011100101011000'
new_testament = [i, j, k, l]

david = len(seventh_seal)
elf = seventh_seal
bilbo = len(lord)
matthew = 0
princess_leia = ''
destiny = bilbo // david
apocalypse = bilbo % david
# for i in range(32):
#     elf = born_to_die(elf)
while matthew < destiny:
princess_leia += matthew_effect(elf, lord[matthew * david: (matthew + 1) * david])
elf = born_to_die(elf)
matthew += 1
princess_leia += matthew_effect(elf[:apocalypse], lord[matthew * david:])
res = open(f'flag_{i}-{j}-{k}-{l}.jpg', 'wb')
res.write(bytes(int(princess_leia[i: i + 8], 2) for i in range(0, bilbo, 8)))
res.close()



### Flag

CCTF{0Ne_k3y_t0_rU1e_7hem_A11_4Nd_7o_d3crYp7_th3_fl4g!}

## Strip

### Challenge

Sometimes it is the people no one imagines anything of who do the things that no one can imagine.

#!/usr/bin/python

from Crypto.Util.number import *
from secret import flag, r

def a(n): # WARNING: very slow implementation...
if n <= 2:
return n
elif n == 3:
return 24
else:
return (6*a(n-1)**2*a(n-3) - 8*a(n-1)*a(n-2)**2) // (a(n-2)*a(n-3))

def strip(n):
return int(bin(n)[2:].rstrip('0'), 2)

def encrypt(msg, r):
n = strip(a(r))
return pow(bytes_to_long(msg.encode('utf-8')), 0x10001 + 0x02, n)

print(encrypt(flag, r))


### Solution

The challenge’s a(n) is the A028365 sequence on OEIS, which has a simpler form (written in PARI) is a(n) = prod(k=1, n, 2^k-1)*2^((n^2+n)/2), and after stripped a(n) is only prod(k=1, n, 2^k-1).

We estimated r and got r >= 605, factored a(r) by FactorDB's API, but the huge modulus took forever for calculating. So we took 2 prime factors from the factors of a(r) and let their product be the new modulus, and we finally got the flag.

Alternate solution uses the estimate of r >= 60. We can take all prime factors of a(60) that are less than 500000, and solve x ** e = enc (mod p) for each of them by brute force. If we find a unique solution, we recover the value of flag (mod p). We combine these with Chinese Remainder Theorem to finish.

### Implementation

#### Solution 1

from factordb.factordb import FactorDB
from itertools import combinations

fac = []
for i in range(1, 606):
print(i, end=' ')
f = FactorDB(2^i - 1)
f.connect()
fac += f.get_factor_list()

fac2 = sorted(list(set(fac)), reverse=True)

n_ = 1
phi_ = 1
for i,j in combinations(fac2, 2):
n_ = i*j
phi_ = (i - 1)*(j - 1)
d_ = inverse_mod(e, phi_)
m = long_to_bytes(pow(c, d_, n_))
if b'CCTF' in m:
print(m)
break


#### Solution 2

from factordb.factordb import FactorDB
from tqdm import tqdm
from Crypto.Util.number import long_to_bytes

## modular inverse of a mod b, can be replaced with Crypto.Util.number's inverse
def minv(a, b):
if a == 1:
return 1
return b - minv(b%a, a) * b // a

## Chinese Remainder Theorem
def CRT(a, b, c, d):
na = d * minv(d, b) * a + b * minv(b, d) * c
nb = b * d
na %= nb
assert na % b == a
assert na % d == c
return na, nb

n = 1
e = 0x10001 + 0x02
totph = 1
wow = []
res = 1
for i in tqdm(range(2, 68)):
res = res * (2**i - 1)
f = FactorDB(2**i - 1)
f.connect()
A = f.get_factor_list()
for num in A:
if num in wow:
totph = totph * num
else:
totph = totph * (num - 1)
wow.append(num)

AA = 0
BB = 1

wow.sort()
for i in tqdm(range(0, len(wow))):
print(wow[i])
if wow[i] > 50000:
continue
cnt = 0
whi = 0
for j in range(0, wow[i]):
if pow(j, e, wow[i]) == enc % wow[i]:
cnt = cnt + 1
whi = j
if cnt == 1:
AA, BB = CRT(AA, BB, whi, wow[i])
print(long_to_bytes(AA))


### Flag

CCTF{R3arR4n9inG_7He_9Iv3n_eQu4t10n_T0_7h3_mUcH_MOrE_traCt4bLe_0n3}

## Complex to Hell

### Challenge

I Already Know I’m Going to Hell

At This Point, It’s Really Go Big Or Go Home!

import math
import string
import random
from secret import flag, key

mapstr = "0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ!{}_"

def multiply(A ,B):
ac,ar,bc,br = len(A), len(A), len(B), len(B)
if ac != br:
return None
result = []
for i in range(ar):
r = []
for j in range(bc):
r.append(0)
result.append(r)
for i in range(ar):
for j in range(bc):
for k in range(br):
result[i][j] += A[i][k] * B[k][j]
return result

def comple_congruent (z):
a = z.real % len(mapstr)
b = z.imag % len(mapstr)
return a + b * 1j

def plain_to_matrix(msg ,n):
p = int(math.ceil(len(msg) // (2 * n))) + 1

matrix_row_size = n
matrix_col_size = p
index = 0
matrix_plain = []
for i in range(matrix_row_size):
col = []
for j in range(matrix_col_size):
if index >= len(msg):
col.append(0 + 0.j)
elif index == len(msg)-1:
col.append(mapstr.index(msg[index]) + 0.j)
index += 1
else:
col.append(mapstr.index(msg[index]) + mapstr.index(msg[index+1]) * 1.j)
index += 2
matrix_plain.append(col)
return matrix_plain

def encrypt(flag ,key):
n = len(key)
p = int(math.ceil(len(flag) // (2 * n))) + 1
matrix_plain = plain_to_matrix(flag, n)
key_congruent = []
for i in range(n):
r = []
for j in range(n):
r.append(comple_congruent(key[i][j]))
key_congruent.append(r)
cipher = multiply (key_congruent, matrix_plain)
result = []
for i in range(n):
r = []
for j in range(p):
r.append(comple_congruent(cipher[i][j]))
result.append(r)
return result

cipher = encrypt(flag, key)
print("cipher = ", cipher)


### Solution

tldr;

• Use flag format to set up system of equations.
• Bruteforce the first two entries on the 2nd row.
• Many keys allow decryption of the first row.
• Use last entries in first row, with knowledge of padding to bruteforce the full, correct key.

plain_to_matrix(msg, n) takes a message string as input and a row count n, and returns a matrix with n rows that has the characters of msg as complex numbers as entries (one complex number represents two characters). If the message doesn’t fill the matrix completely, it is padded with 0s.

encrypt(msg, key) encrypts the given message by left multiplying the message (as a matrix) by the $2 \times 2$ key. The size of the key space is $66^8$ which is infeasible to bruteforce.

We can use the flag format to reduce the amount of bruteforce required. Let the key be

$K = \begin{bmatrix} a+bi & c+di \\ e+fi & g+hi \end{bmatrix}$

where $a,b,c,d,e,f,g \in \mathbb{Z}/66\mathbb{Z}$

Write the plaintext flag matrix as

$M = \begin{bmatrix} m_0 + m_1i & m_2 + m_3 i & \cdots & m_{32} + m_{33} i \\ m_{34} + m_{35} i & m_{36} + m_{37} i & \cdots & m_{66} + m_{67} i \end{bmatrix}$

and the ciphertext matrix as

$C = \begin{bmatrix} c_0 + c_1i & c_2 + c_3 i & \cdots & c_{32} + c_{33} i \\ c_{34} + c_{35} i & c_{36} + c_{37} i & \cdots & c_{66} + c_{67} i \end{bmatrix}$

(all coefficients in $\mathbb{Z}/66\mathbb{Z}$).

So $C = KM$.

From this we get the equations

\begin{aligned} c_0 + c_1 i &= (a + bi)(m_0 + m_1 i) + (c + d_i)(m_{34} + m_{35} i) \\ c_2 + c_3 i &= (a + bi)(m_2 + m_3 i) + (c + d_i)(m_{36} + m_{37} i) \\ c_{34} + c_{35} i &= (e + fi)(m_0 + m_1 i) + (g + hi)(m_{34} + m_{35} i) \\ c_{36} + c_{37} i &= (e + fi)(m_2 + m_3 i) + (g + hi)(m_{36} + m_{37} i) \end{aligned}

so

\begin{aligned} c_0 &= am_0 - bm_1 + cm_{34} - dm_{35} \\ c_1 &= am_1 + bm_0 + cm_{35} + dm_{34} \\ c_2 &= am_2 - bm_3 + cm_{36} - dm_{37} \\ c_3 &= am_3 + bm_2 + cm_{37} + dm_{36} \\ c_{34} &= em_0 - fm_1 + gm_{34} - hm_{35} \\ c_{35} &= em_1 + fm_0 + gm_{35} + hm_{34} \\ c_{36} &= em_2 - fm_3 + gm_{36} - hm_{37} \\ c_{37} &= em_3 + fm_2 + gm_{37} + hm_{36} \end{aligned}

We’ll bruteforce $66^4$ values for $m_{34}, m_{35}, m_{36}$ and $m_{37}$ and solve for the 8 key values with the 8 equations. We already know $m_0, m_1, m_2$ and $m_3$ from the flag format.

Doing this quickly reveals the first row of the plaintext: CCTF{This_0n3_Is_State_0f_th3_4rt_.

With this, we can reduce the bruteforce amount to at most $66^3$. Fortunately for us, it turns out that the last 4 characters of the plaintext are }000, so we have enough information to enumerate possible keys with minimal bruteforce. We can use the exact same setup as above, except instead of bruteforcing $m_{34}, m_{35}, m_{36}$ and $m_{37}$, we take them to be }000. Solving the system will give us a vector $\mathbf{k} = (a,b,c,d,e,f,g,h)$, but this might not be the correct key.

Any vector of the form $\mathbf{k} + \mathbf{t}$ where $\mathbf{t}$ is in the kernel of the coefficients matrix, $A$, will satisfy the system. We can find all vectors in the kernel of $A$ by finding a basis for the kernel modulo each of the prime factors of $66$, and then combining them with the Chinese Remainder Theorem. In this case, the nullity of $A$ in $\mathbb{F}_3$ and $\mathbb{F}_{11}$ is $0$, and the nullity of $A$ in $\mathbb{F}_2$ is $4$. This means we’ll need to enumerate at most $2^4$ possible keys.

Note on inv(key): I couldn’t find a way to use Sage’s built-ins to find the inverse of a matrix with complex entries so I just used the following theory:

Suppose $K^{-1}$ exists. Write

$K^{-1} = \begin{bmatrix} a' + b'i & c' + d'i \\ e' + f'i & g' + h'i \end{bmatrix}$

Then, by definition

$\begin{bmatrix} a + bi & c + di \\ e + fi & g + hi \end{bmatrix} \begin{bmatrix} a' + b'i & c' + d'i \\ e' + f'i & g' + h'i \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

so

\begin{aligned} aa' - bb' + ce' - df' &= 1 \\ ab' + ba' + cf' + de' &= 0 \\ ac' - bd' + cg' - dh' &= 0 \\ ad' + bc' + ch' + dg' &= 0 \\ ea' - fb' + ge' - hf' &= 0 \\ eb' + fa' + gf' + he' &= 0 \\ ec' - fd' + gg' - hh' &= 1 \\ ed' + fc' + gh' + hg' &= 0 \end{aligned}

which is a system of 8 equations in 8 unknowns that can be easily solved.

### Implementation

from itertools import product

mapstr = "0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ!{}_"
cipher =  [ [(24+36j), (41+47j), (3+27j), (36+41j), (57+58j), (11+24j), (33+7j), (52+64j), (26+23j), (30+35j), (64+39j), (52+19j), (39+45j), (33+31j), (3+17j), (21+32j), (15+55j)], [(33+44j), (15+39j), (64+50j), (44+41j), (39+20j), 42j, (16+12j), (63+27j), (9+52j), (39+64j), (5+18j), (53+25j), (47+31j), (5+49j), (24+8j), (57+9j), (38+16j)] ]

F = IntegerModRing(66)

def multiply(A ,B):
ac,ar,bc,br = len(A), len(A), len(B), len(B)
if ac != br:
return None
result = []
for i in range(ar):
r = []
for j in range(bc):
r.append(0)
result.append(r)
for i in range(ar):
for j in range(bc):
for k in range(br):
result[i][j] += A[i][k] * B[k][j]
return result

def inv(key):
a,b,c,d,e,f,g,h = key
M = [[a,-b,0,0,c,-d,0,0],
[b,a,0,0,d,c,0,0],
[0,0,a,-b,0,0,c,-d],
[0,0,b,a,0,0,d,c],
[e,-f,0,0,g,-h,0,0],
[f,e,0,0,h,g,0,0,],
[0,0,e,-f,0,0,g,-h],
[0,0,f,e,0,0,h,g]]
M = Matrix(F,M)
t = vector(F,[1,0,0,0,0,0,1,0])
i = M.solve_right(t)
a,b,c,d,e,f,g,h = map(ZZ, i)
I = [[a+b*1j, c+d*1j],[e+f*1j, g+h*1j]]
return I

def cong(z):
a = z.real() % 66
b = z.imag() % 66
return a + b*1j

def decrypt(key):
Kinv = inv(key)
key = [[cong(Kinv[i][j]) for j in range(2)] for i in range(2)]
M = multiply(key, cipher)
res = []
flag = ''
for i in range(2):
for j in range(17):
a = cong(M[i][j]).real()
b = cong(M[i][j]).imag()
flag += mapstr[int(a)] + mapstr[int(b)]
return flag

# first round, bruteforce m34,m35,m36,m37
# c0, c1, c2, c3 = 24, 36, 41, 17
# c34, c35, c36, c37 = 33, 44, 15, 19
# m0, m1, m2, m3 = 38, 38, 55, 41
# RECOVERS first row: CCTF{This_0n3_Is_State_0f_th3_4rt_

c0, c1, c2, c3 = 21, 32, 15, 55
c34, c35, c36, c37 = 57, 9, 38, 16
m0, m1, m2, m3 = 4, 27, 29, 65
m34, m35, m36, m37 = 64, 0, 0, 0

A = [[m0,-m1,m34,-m35,0,0,0,0],
[m1, m0, m35, m34,0,0,0,0],
[m2, -m3,m36,-m37,0,0,0,0],
[m3,m2,m37,m36,0,0,0,0],
[0,0,0,0,m0,-m1,m34,-m35],
[0,0,0,0,m1,m0,m35,m34],
[0,0,0,0,m2,-m3,m36,-m37],
[0,0,0,0,m3,m2,m37,m36]]
v = [c0,c1,c2,c3,c34,c35,c36,c37]
A = Matrix(F, A)
v = vector(F, v)
x = A.solve_right(v)
A2 = Matrix(GF(2), A)
A2K = Matrix(F, A2.right_kernel_matrix())
# A3 and A11 have 0 nullity

for lc in product(range(2), repeat=A2.right_nullity()):
try:
t2 = A2K.linear_combination_of_rows(lc)
t = vector([crt([int(a2), 0, 0], [2, 3, 11]) for a2 in t2])
key = x + t
flag = decrypt(key)
print(flag)
except ValueError:
pass
except KeyboardInterrupt:
exit()


### Flag

CCTF{This_0n3_Is_State_0f_th3_4rt_and_C0mplex_is_Truly_compl3x!!}

## Fatima

### Challenge

I think we should all learn elliptic curves and fatima is a good start, enjoy!

#!/usr/bin/env python3
# -*- coding: utf-8 -*-

from fastecdsa.curve import Curve
from fastecdsa.point import Point
import math, random
from flag import flag
import time

def multiply(A, B):
ac, ar, bc, br = len(A), len(A), len(B), len(B)
if ac != br:
return None
result = []
for i in range(ar):
r = []
for j in range(bc):
r.append(0)
result.append(r)
for i in range(ar):
for j in range(bc):
for k in range(br):
result[i][j] += A[i][k] * B[k][j]
return result

def pow_matrix(A, n):
R = circulant( + [0 for i in range(len(A)-1)])
for _ in range(n):
R = multiply(R, A)
return R

def circulant(v):
C, n = [], len(v)
for i in range(n):
C.append(v)
tmp = []
tmp.append (v[-1])
tmp.extend(v[:-1])
v = tmp
return C

def spiral(A):
row = len(A)
col = len(A)
top = 0
left = 0
tmp = []

while (top < row and left < col) :
for i in range(left,col) :
tmp.append(A[top][i])
top += 1
for i in range(top,row) :
tmp.append(A[i][col - 1])
col -= 1
if ( top < row) :
for i in range(col - 1,(left - 1),-1) :
tmp.append(A[row - 1][i])
row -= 1

if (left < col) :
for i in range(row - 1,top - 1,-1) :
tmp.append(A[i][left])
left += 1
result = []
for i in range(len(A)):
r = []
for j in range(len(A)):
r.append(tmp[i*len(A) + j])
result.append(r)
return result

def revspiral(A):
tmp = sum(spiral(A),[])
tmp = tmp[::-1]
result = []
for i in range(len(A)):
r = []
for j in range(len(A)):
r.append(tmp[i*len(A) + j])
result.append(r)
return result

def sinwaveform(A):
row = len(A)
col = len(A)
tmp = []
for j in range(col):
if j%2 == 0:
for i in range(row):
tmp.append(A[i][j])
else:
for i in range(row-1,-1,-1 ):
tmp.append(A[i][j])
result = []
for i in range(len(A)):
r = []
for j in range(len(A)):
r.append(tmp[i*len(A) + j])
result.append(r)
return result

def helical(A):
row = len(A)
col = len(A)
tmp = []
dir = 0
for k in range(0,row):
if dir == 0:
i = k
for j in range(0,k+1):
tmp.append(A[i][j])
i -= 1
dir = 1
else:
j = k
for i in range(0,k+1):
tmp.append(A[i][j])
j -= 1
dir = 0
for k in range(1, row):
if dir == 0:
i = row - 1
for j in range(k, row):
tmp.append(A[i][j])
i -= 1
dir = 1
else:
j = row - 1
for i in range(k, row):
tmp.append(A[i][j])
j -= 1
dir = 0
result = []
for i in range(len(A)):
r = []
for j in range(len(A)):
r.append(tmp[i*len(A) + j])
result.append(r)
return result

def revhelical(A):
tmp = sum(helical(A),[])
tmp = tmp[::-1]
result = []
for i in range(len(A)):
r = []
for j in range(len(A)):
r.append(tmp[i*len(A) + j])
result.append(r)
return result

dict_traversal = {
1: spiral,
2: revspiral,
3: sinwaveform,
4: helical,
5: revhelical
}

def c2p(c, G):
C = ord(c) * G
return bin(C.x)[2:].zfill(8) + bin(C.y)[2:].zfill(8)

def aux(msg, G):
enc = ''
for c in msg:
enc += c2p(c, G)
return enc

def enmat(c, l):
s = int(math.sqrt(len(c) // l))
return [[int(c[i*l:i*l+l], 2) for i in range(s * j, s * (j + 1))] for j in range(s) ]

def encrypt(msg):
name = 'curve'.encode('utf-8')
p, a, b, q, gx, gy, aux = 241, 173, 41, 256, 53, 192, ''
curve = Curve(name, p, a, b, q, gx, gy)
G = Point(gx, gy, curve = curve)

for c in msg:
aux += c2p(c, G)
B = enmat(aux, 3)
S = list(range(1,6))
random.shuffle(S)
for i in range(5):
B = dict_traversal[S[i]](B)
C = circulant([0 for i in range(len(B)-1)] + )
a, l = [random.randint(2, len(B)) for _ in '01']
CL = pow_matrix(C, l)
CAL = pow_matrix(CL, a)
enc = (CL, multiply(B, CAL))
return enc
print("enc = ", encrypt(flag))


### Solution

Note that pow_matrix(C, i) == circulant(*(100 - i) +  + *(i - 1)) with C = circulant([0 for i in range(len(B)-1)] + ), with knowing that we can recover the plaintext step-by-step.

enc = B*C^(a*l)
enc*C^k = B*C^(a*l + k)


Therefore if we can find k such that a*l + k == 0 (mod 100), which mean C^(a*l + k) is the identity matrix, we can recover B. Note that C^k has the form circulant(*(100 - i) +  + *(i - 1)) so we can just bruteforce i and check for each case which produces the right plaintext. The traversal operators and EC part can be reversed using look-up tables.

### Implementation

from Crypto.Util.number import *
import itertools
import tqdm
from fastecdsa.curve import Curve
from fastecdsa.point import Point
import math, random

############ Reuse code of the challenge ############
def multiply(A, B):
ac, ar, bc, br = len(A), len(A), len(B), len(B)
if ac != br:
return None
result = []
for i in range(ar):
r = []
for j in range(bc):
r.append(0)
result.append(r)
for i in range(ar):
for j in range(bc):
for k in range(br):
result[i][j] += A[i][k] * B[k][j]
return result

def pow_matrix(A, n):
R = circulant( + [0 for i in range(len(A)-1)])
for _ in range(n):
R = multiply(R, A)
return R

def circulant(v):
C, n = [], len(v)
for i in range(n):
C.append(v)
tmp = []
tmp.append (v[-1])
tmp.extend(v[:-1])
v = tmp
return C

def spiral(A):
row = len(A)
col = len(A)
top = 0
left = 0
tmp = []

while (top < row and left < col) :
for i in range(left,col) :
tmp.append(A[top][i])
top += 1
for i in range(top,row) :
tmp.append(A[i][col - 1])
col -= 1
if ( top < row) :
for i in range(col - 1,(left - 1),-1) :
tmp.append(A[row - 1][i])
row -= 1

if (left < col) :
for i in range(row - 1,top - 1,-1) :
tmp.append(A[i][left])
left += 1
result = []
for i in range(len(A)):
r = []
for j in range(len(A)):
r.append(tmp[i*len(A) + j])
result.append(r)
return result

def revspiral(A):
tmp = sum(spiral(A),[])
tmp = tmp[::-1]
result = []
for i in range(len(A)):
r = []
for j in range(len(A)):
r.append(tmp[i*len(A) + j])
result.append(r)
return result

def sinwaveform(A):
row = len(A)
col = len(A)
tmp = []
for j in range(col):
if j%2 == 0:
for i in range(row):
tmp.append(A[i][j])
else:
for i in range(row-1,-1,-1 ):
tmp.append(A[i][j])
result = []
for i in range(len(A)):
r = []
for j in range(len(A)):
r.append(tmp[i*len(A) + j])
result.append(r)
return result

def helical(A):
row = len(A)
col = len(A)
tmp = []
dir = 0
for k in range(0,row):
if dir == 0:
i = k
for j in range(0,k+1):
tmp.append(A[i][j])
i -= 1
dir = 1
else:
j = k
for i in range(0,k+1):
tmp.append(A[i][j])
j -= 1
dir = 0
for k in range(1, row):
if dir == 0:
i = row - 1
for j in range(k, row):
tmp.append(A[i][j])
i -= 1
dir = 1
else:
j = row - 1
for i in range(k, row):
tmp.append(A[i][j])
j -= 1
dir = 0
result = []
for i in range(len(A)):
r = []
for j in range(len(A)):
r.append(tmp[i*len(A) + j])
result.append(r)
return result

def revhelical(A):
tmp = sum(helical(A),[])
tmp = tmp[::-1]
result = []
for i in range(len(A)):
r = []
for j in range(len(A)):
r.append(tmp[i*len(A) + j])
result.append(r)
return result

dict_traversal = {
1: spiral,
2: revspiral,
3: sinwaveform,
4: helical,
5: revhelical
}
#####################################################

enc =  # REDACTED

# Create lookup tables for all permutations
def qn(n=100):
lookup = list(range(n**2))
res = []
for it in tqdm.tqdm(itertools.permutations(range(1, 6))):
B = [lookup[i:i+n] for i in range(0, n**2, n)]
for i in it:
B = dict_traversal[i](B)
B = sum(B, [])
B = [B.index(i) for i in range(n**2)]
res.append([B[i:i+n] for i in range(0, n**2, n)])
return res

def apply_lookup(A, lk):
n = len(A)
A = sum(A, [])
lk = sum(lk, [])
return [[A[j] for j in lk[i:i+n]] for i in range(0, n**2, n)]

kk = qn()  #  Around 6 mins, can be cached

# Look-up table for ec part
rev = [[None for i in range(256)] for j in range(256)]
for i in range(256):
t = i*G
rev[t.x][t.y] = chr(i)

for i in tqdm.trange(99, -1, -1):
CC = circulant(*i +  + *(100 - i - 1))
B = multiply(enc, CC)
for lk in kk:
BB = [x[::] for x in B]
BB = apply_lookup(BB, lk)
BB = sum(BB, [])
s = ''
for x in BB:
s += bin(x)[2:].zfill(3)
m = ''
for j in range(0, len(s), 16):
px, py = int(s[j:j+8], 2), int(s[j+8:j+16], 2)
if rev[px][py] is not None:
m += rev[px][py]
continue
if 'CCTF' in m:
print(m)
break


And this is the output, a paragraph from Our Lady of Fátima.

Beginning in the spring of 1917, the children reported apparitions of an Angel, and starting in May 1917, apparitions of the Virgin Mary, whom the children described as the Lady more brilliant than the Sun. The children reported a prophecy that prayer would lead to an end to the Great War, and that on 13 October that year the Lady would reveal her identity and perform a CCTF{Elliptic_Curv31s_fun&_simpLE_Circulaitng_it_make_it_funnier!!} so that all may believe. Newspapers reported the prophecies, and many pilgrims began visiting the area. The children’s accounts were deeply controversial, drawing intense criticism from both local secular and religious authorities. A provincial administrator briefly took the children into custody, believing the prophecies were politically motivated in opposition to the officially secular First Portuguese Republic established in 1910. The events of 13 October became known as the Miracle of the Sun…

### Flag

CCTF{Elliptic_Curv3_1s_fun_&_simpLE_Circulaitng_it_make_it_funnier!!}

## Namura

### Challenge

def encrypt(pubkey, msg):
C = 0
for i in range(n):
C += pubkey[i] * int(msg[i])
return C

flag = flag.lstrip('CCTF{').rstrip('}')
bflag = bin(bytes_to_long(flag.encode('utf-8')))[2:]
n = len(bflag)
u = n - 30

pubkey = keygen((n+1) // 3, n, u)

print('pubkey =', pubkey)
enc = encrypt(pubkey, bflag)
print('enc =', enc)


### Solution

This looks like a knapsack cryptosystem, which are usually solved by lattice reduction algorithms modelling a Shortest Vector Problem (SVP). We noticed that the title of the challenge “Namura” hints at the paper describing this algorithm, Naskao and Murakami’s Knapsack Public-Key Cryptosystem Using Chinese Remainder Theorem. Section 4.2 describes a lattice that can solve low density subset sum problems, and the public key in this challenge has a very low density:

d = len(pubkey) / log(max(pubkey), 2)
print(CDF(d))
0.5625


Since the flag can be assumed to be printable ASCII chars, we can reduce the dimension of the pubkey by removing the corresponding number to the MSB of each character in the plaintext.

new = []
pubkey =  + pubkey
for i in range(len(pubkey)):
if i % 8 == 0:
continue
new.append(pubkey[i])
print('pubkey =', new)


Then we can just solve this like other knapsack problems by bruteforcing the permutation and using the BKZ lattice reduction algorithm to find the SVP.

### Implementation

import re
import random
import multiprocessing as mp
from functools import partial

def check(sol, A, s):
"""Check whether *sol* is a solution to the subset-sum problem.
"""
return sum(x*a for x, a in zip(sol, A)) == s

def solve(a, s, ID=None):
rand = random.Random(x=ID)

mat = []
for idx,val in enumerate(a):
mat.append(*idx +  + *(len(a)-idx-1)+[val])
mat.append(*len(a)+[-s])

# main loop
itr = 0
start_time = cputime()
while True:
itr += 1

# 2. Randomly shuffle
l = mat[::]
shuffle(l, random=rand.random)

# 3. BKZ!!!
m = matrix(l)
t_BKZ = cputime()
l_sol = m.BKZ(block_size=25)
print(f"{itr} runs. BKZ running time: {cputime(t_BKZ):.3f}s")

for line in l_sol:
if all([x >= 0 for x in line[:-1]]):
print(line)
print(check(line,a,s))
print(f"After {itr} runs. FIND SVP!!! {line}\n"
f"Single core time used: {cputime(start_time):.3f}s")
return True

enc = 154657917005376465967753276253676484467260782425419406781078357515
pubkey = [636730424634282684150787505024636846878192530834301373045417941, 443443736056701854821550045409138156747702906153207509789193893, 4044894679347221866903041471393250783970070284064844489844729640, 2438178506188801348411667154785222321653401060527584288473058029, 1900607069477409243358471897897077706622577630696771143373974126, 4396893130381899655054557551793492148977658211100513328122993482, 2601912276825314189427819328705612999759768062840709690416685851, 849578696430489144601066711846105434868737506048858961510584478, 867634152731852110202428052792503837522496305953184128350918090, 2141949199052254673518707523413310868963934449025085556791898943, 1317724781892829476727276429613649391725697391917627197350586077, 846616203254169113248714620324777288157484807522832537271896727, 1889890413622399357217368964385384275068207071755568870885142697, 4345106754542111105556800435292359436746763182165461814839878219, 1844751943408649439970784234819027788878268101086942786334241578, 4635151867785248653584925319820032342108353583278365090165351369, 1891260029110631153447767958167471428147295737587261835048395769, 3273672699905851794278838098554938393037792687468962414002119644, 4759683391852904086863354069372064775438697384972606618058259428, 2277479715112568474291874878404028785747567257268529120464806983, 712281270914494089486011482407537474741428127403029959878626851, 4663860235979475414650446442104011820603148660069426522253772670, 3570757581386148492619721379754470899316095256123109990599128391, 4609713244848853151872498160877375734335329160891300656414838786, 1431248994688391495017629590719567118297228062918817671705412012, 2225618736576399852718197161416790353023368081178287753385225648, 4782768885432039605448539230699045953181097923357764740448690485, 1025808412433473089433862844337525335386046496037581875356716631, 2850703152833612251035169162871614900872662336925683266673455769, 4686484042664673737330267137247259184248980902553457550045106744, 3316117133062845045327521738264790714934051828005331038083037906, 1411496297445655314847983724570982636448577335114241954690062680, 2542720351620819402979547749565244924618621495731029455602801063, 4197157173419472170084161918188987699514176876278506629655813541, 775178221793495085043576729609381220589053240944436598437451103, 1341597796943613200200560889564801116846969301604051962802959921, 4724275587384586890632093268426638078191399337509178017491641396, 2254966368661541088781913210011063323766242664855534020654216185, 1559111672805843337464695743725374999443380244436636784823457268, 263060461355351726244024949311923372968467484234342136010504498, 4218489168395358789072676527116792449437414059225489587311420630, 2251347608406477583876276692162280889042972229705782250944073182, 1048197300230894759772482326800601949486880189444304544917201349, 4594309375612539584017914006965726879737368434732989117961461158, 1233526648681303204756491942769500757542366936959132748188681389, 3016611933554222534504704995395833948561521013355966057174149640, 685431642960387458833365483769661272653394129002170162343687962, 1252350578439116321952733140441764993245772656606639708501799071, 2004856906093670950398190666612521156885201358615487450361722687, 1725528220657822102510144312466698156124143365979935333948441423, 3301536380780212554033742823735941195638359575128262344444357806, 3361176781081176991336986769591969284375994647164396417238879397, 2555688594908398218735938381172552745926292876995621798813594216, 2149199142659861721027875250011594210747138266017264150889296633, 4654853318545885657451422703700711659405223637471250014707272999, 1783827755250002883819223478577480687561868815037598618999110299, 3876452588731221361888242546888347728419654382142841199604124779, 2283317070521561115970687892255141823986922119608171153201553969, 3015343638915794545630411225203386258523748035633382700837350107, 1308963799032621611684027617168973833892399982687941479751647735, 3363298156592318867171609036073104624481755649616128282937579774, 3543718722328215918394832245155182570535205536855659505934745836, 2955555006922666284454361589146164283232856958998231643765012061, 4193238914021395832431998242809775488323071053203281739810565939, 4863715450542324142694503897491361069694288484386266524822426647, 3583711168144466683911674650704848504341023445180872465082660398, 1433492863048866856968843544774985957106873111077658213115876127, 3622680772935480479302879234497984985614630209532096422962674742, 3887543917518693741822422553185837022122830870466259214366790339, 3010960827639423613523800853443011766752449479334524527050675334, 1675955542074383948970870230135814936799951109866034174734491734, 2568984843336400124353481960719548494069287783874504372170058935, 680042408260630675242336818143271325154353883745350135887078713, 1896768391347692167859873865813768792359296006947445277687988097, 537513148597668578568712785471862479586342936485511258184103046, 4338318157572996055378474172251186724034148657838505626251846209, 4509359887372553408550688030273180923282246069532844476087185588, 1961425576962957081785371096529881351777256192797473186708183898, 4562726127192998241808421239521775685020063730950933119470565151, 3197416476037069116835447572914275965582808251383336065711778098, 4509743379431751154130020063323115916220642219739358425391068150, 1737231313740527925458531852974418083735884963087687882655818328, 4723771434844173636187013002792278070911838005170476297565209636, 4021068815924596682472342770957679146819658388809493963529859273, 4786593367935490268774249574977281762592209022792374805751998882, 1706847947841349067687051586871379604391823960780007506398289654, 2092911436130136930529034363620771320336826052044341129920779847, 2386542753409262049262444109479898339116742017285966198413932291, 2575514997936878781309794857665223684996125674280321049577858392, 2526059212864002845504783002187945419965243527858703947395965701, 2077055376963690862993188737229202782309424513798741527458096967, 1947721666793448806619506886745665574368753315129031773531178573, 2321982120042809240576670901783025887795295409352093643395133004, 4191930348600938505176612143361132888157091847500134549846473180, 1279873852200144323116032749112043797286486924653552312015287694, 3934811009597203954835516432740855968621865146569217009553064951, 804570958275502176779582603101955727481164663345322968855176622, 4755601230261360181533138175300662604366870408130917516343576381, 2016264908613514961521473342929083040444069560476054659007958347, 3857121931198808981033402131835999166260880661479936388701406991, 4787908501772479625441292392638080593307265124479164945134226910, 403228266126326263488043524077179619385866145325037513940941892, 4080757802977772396554968304371742747141072297333640725823656444, 248086288384249359079536769334310714884272887049336400711180125, 1607777042247987295060365154963999272145526955355524894746933487]
solve_n = partial(solve, pubkey, enc)
CPU_CORE_NUM = 8
with mp.Pool(CPU_CORE_NUM) as pool:
reslist = pool.imap_unordered(solve_n, range(CPU_CORE_NUM))
# terminate all processes once one process returns
for res in reslist:
if res:
pool.terminate()
break


Output:

After 19 runs. FIND SVP!!! (1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0) Single core time used: 643.215s

flag = ''
svp = (1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1,
0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0)
for i in range(0, len(svp), 7):
flag += '0' + ''.join(map(str, svp[i:i+7]))
print(b'CCTF{'+long_to_bytes(int(flag, 2))+b'}')


### Flag

CCTF{MuR4kam1_nA54K0}

## Decent RSA

### Challenge

RSA can be decent as well!

Note Although this task is very decent and solvable with focusing on the module number, you may use any tools, guessing, or whatever you know to solve it!

### Solution

TD;DR

• See that when written in base 11, the modulus is mainly zeros
• Write the modulus as a polynomial in base 11 and factor the polynomial
• Solve RSA

All we are given is a RSA public key

-----BEGIN PUBLIC KEY-----
MIIBIjANBgkqhkiG9w0BAQEFAAOCAQ8AMIIBCgKCAQEA/Ug8rlEPci1UXMsT+UDo
y8DfxbTHX/3BK2oU+FPWiJf+EiUBM2x4ep04qZ1SO9Pmqj/WH9skMrF1J/LXuY3l
fjvJCh0DXa9VUyX2dAJidja9Ior7GpFwwjYdKh+OETNV+2/CcX4RiPvj+8ApmedW
gn4Fxaeivki+f/UwDa+ws1fTUzmI325v8yvcryHhbgeUWiF85EP6HFAavTsVPlxb
LikVMAB1fuzDbqqJvW2u138w6b2FH3WrezYF6tbAyZej2HX46phwDm9C7MXYJ/sU
oS+E8P7S1jMTCWjfwMCOKU3SFGrkWtXuTaoMZ2nZ+HVfJV8xJOjWez1OxQ5P3F1w
GQIDAQAB
-----END PUBLIC KEY-----


and some encrypted data.

From the .pem we get

Algo RSA
Format X.509
ASN1 Dump
RSA Public Key [21:5d:61:5d:7e:ef:d0:58:12:d0:dc:14:bd:7c:e1:69:eb:77:01:f0]
public exponent: 10001


where the modulus is some 2048 bit integer. As we are given a X.509 key, esrever suggested looking at a database of predictable RSA keys, which contains 30k public keys which were insecure. We downloaded these and looked for a common factor between our common modulus and one of these known, weak keys. We didnt have any luck though.

Another idea was that maybe this would be solved with Fermat factorisation, with “Decent RSA” being a pun for the infinite descent method. I let the algorithm run for a while but eventually killed it.

The solution came from looking at the modulus in various bases. My initial hope was that the primes might be Mersenne primes, which would be exposed by looking at the modulus in base 2, but it turns our the right base for the solve is base 11.

sage: N.str(base=11)
'10010000000000000000000000000000020000000000010000000000000000000000000000000000000000000002002000002000000000000000020020004000000000002000000000004040000000000020000000002000000000000000000400000000000000000000000004000000000000000000000800000000000000000000000408000000000000000200000004000000600200000000000000000000000000000400000000000200000000000000000000000000000040000000000000080000000040400000000000000800000000000000000000000000000080000000000000000000000040000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000004000000000000008'


We can then write N as a polynomial in the following way

sage: poly = sum(e * x^i for i,e in enumerate(N.digits(11)))
sage: poly
x^592 + x^589 + 2*x^559 + x^547 + 2*x^501 + 2*x^498 + 2*x^492 + 2*x^475 + 2*x^472 + 4*x^468 + 2*x^456 + 4*x^444 + 4*x^442 + 2*x^430 + 2*x^420 + 4*x^401 + 4*x^375 + 8*x^353 + 4*x^329 + 8*x^327 + 2*x^311 + 4*x^303 + 6*x^296 + 2*x^293 + 4*x^263 + 2*x^251 + 4*x^220 + 8*x^205 + 4*x^196 + 4*x^194 + 8*x^179 + 8*x^148 + 4*x^124 + 4*x^15 + 8


Which sage very quickly can factor

sage: poly.factor()
(x^296 + x^293 + 2*x^263 + x^251 + 2*x^196 + 4*x^148 + 2*x^124 + 2*x^15 + 4)*(x^296 + 2*x^205 + 2*x^179 + 2)


Setting x = 11 will return the primes p*q = N, solving the challenge.

### Implementation

from Crypto.PublicKey import RSA
from Crypto.Util.number import long_to_bytes, bytes_to_long

n = Integer(key.n)

poly = sum(e * x^i for i,e in enumerate(Integer(key.n).digits(11)))
(p, _), (q, _) = poly.factor_list()
p, q = p(x=11), q(x=11)
assert p*q == n

d = inverse_mod(key.e, (p-1)*(q-1))
print(long_to_bytes(pow(flag, d, n)))


### Flag

CCTF{___RSA___1n_D3cEn7_W0rLd_cRyPtO5!!!}

## GenGol

### Challenge

We updated the cryptosystem to encrypt large messages. Try to break it!

We can connect to a server and read a snippet of source code:

def genkey(k, nbit):
assert 2*k <= nbit
while True:
p = random.randint(1, 2**(nbit - k)) * 2**k + 1
if isPrime(p):
q = getPrime(nbit)
n = p * q
while True:
y = random.randint(1, n)
jp, jq = pow(y, (p-1)/2, p), pow(y, (q-1)/2, q)
if (jp + jq == p + q - 2):
d = random.randint(int(exp(0.272 * log(n))), int(exp(0.292 * log(n)))) | 1
e = inverse(d, (p - 1)*(n/p - 1))
if e * d % ((p - 1)*(n/p - 1)) == 1:
return (n, y, k, e), (p, d)

def encrypt(msg, pkey):
n, y, k, _ = pkey
m = bytes_to_long(msg)
assert m <= 2**k - 1
x = random.randint(1, n)
c = pow(y, m, n) * pow(x, 4**k, n) % n
return c


We can also request public keys and encrypted ciphertexts.

### Solution

This is a strange cryptosystem with elements of RSA and obviously faulty key generation. The private exponent d is not directly used in the decryption, but d would give us the factorisation of the modulus which looks to be relevant in the decryption. Furthermore, the lines about jp and jq ensure that that y is a quadratic non-residue (i.e. not a square root) modulo p and q. The use of quadratic residuosuity reminds us of the probabilistic Goldwasser-Micali cryptosystem.

Immediately, we noted that the upper bounds on d (N0.292) is the upper bound of the Boneh-Durfee attack (B-D). B-D, an extension of Coppersmith’s Method, is able to recover d from the modulus if d is small enough.

We tried using the “off-the-shelf” implementation of B-D provided by David Wong here, however it didn’t work even after tweaking the delta and m parameters.

In the meantime, we needed to figure out how to decrypt ciphertexts after recovering the factorisation of the modulus. We found the paper Efficient Cryptosystems From 2k-th Power Residue Symbols which describes the cryptosystem we see here, a sort of “Generalized Goldwasser-Micali”, thus the challenge name “GenGol”. Section 3.2 contains the decryption algorithm.

### Factoring the modulus

We were able to solve the challenge when pcback had a clever idea to alter the B-D attack, using the fact that the prime $p$ had an unusal form. Let $p_0$ be the top 512 bits of $p$, and $p_1$ the bottom 512 bits. Then $p = 2^{512} p_0 + p_1$, and $p_1 = 1$. Similarly, let $q = 2^{512} q_0 + q_1$. Then $N = 2^{1024} p_0 q_0 + 2^{512} p_0 q_1 + 2^{512} q_0 + q_1$. Now we are ready to derive a modified B-D attack:

• $k \phi(N) + 1 = ed$
• $k (p - 1) (q - 1) + 1 = 0 \pmod e$
• $k 2^{512} p_0 (2^{512} q_0 + q_1 - 1) + 1 = 0 \pmod e$
• $k (2^{512} p_0 q_0 + p_0 q_1 - p_0) + \frac{1}{2^{512}} = 0 \pmod e$
• Let $A = \frac{N - N \pmod {512}}{2^{512}} = 2^{512} p_0 q_0 + p_0 q_1 + q_0$ (equivalent to N >> 512)
• $k (A - q_0 - p_0) + \frac{1}{2^{512}} = 0 \pmod e$
• $k (A - s) + \frac{1}{2^{512}} = 0 \pmod e$ with $s = p_0 + q_0$

This results in a modified equation, similar to the original B-D equation: $k (A + s) + 1 = 0 \pmod e$ with $A = \frac{N + 1}{2}$ and $s = -\frac{p + q}{2}$. There is one crucial difference: $A$ and $s$ are ~512 bits smaller than in the original equation. This results in a faster computation, with a smaller lattice size.

Now David Wong’s script can be upgraded by changing the polynomial:

P.<x,y> = PolynomialRing(ZZ)
# [-] A = int((N+1)/2)
# [-] pol = 1 + x * (A + y)
A = (N - int(N%(2**512))) // 2**512
pol = inverse_mod(2**512, e) + x*(A - y)


We can then recover $p$ and $q$ from $k$. Running pcback’s modified script, we get the following output

=== solution found ===
x: 26831241432833160806520729053166227091993916416483668950238588764594692681480769707324016554574698948603027409300338968573544143950532532690604344744985338216091633964853639094807
y: 12067175061535056740866328614179238011084703807935942170222332594324074913836918898452931825405156276243520869293655223677329902044037479631524145496590233
p: 68051654200085157207332136111091263403260677985149014940942748721163833107639364882355808718200151519497304188154695981671131677635619224129781969654824622900517674305201415050592661606881387625228654411615892410789900017222366193872604384511674439510956828703775115042284171259024209520418324094905850265601
q: 93742711281970123687375274125008994006083293818248949034193069509248449466340860308831673133907193988724549474397518306519671830205855974368567497715523859186789480391221672901717451785259798844945176989467876128830079144227608792557456138362303322064034291267314238570313761147918058776114912984468505970527
=== 3.14493989944458 seconds ===


### Decrypting the flag

While pcback was learning how to solve the modified B-D script, UnblvR was working on implementing the generalised Goldwasser-Micali protocol, which he implemented using python

from math import log, exp
import random
from Crypto.Util.number import getPrime, isPrime, inverse, bytes_to_long, long_to_bytes

def legendre_symbol(a, p):
ls = pow(a, (p - 1) // 2, p)
return -1 if ls == p - 1 else ls

def genkey(k, nbit):
assert 2*k <= nbit
while True:
p = random.randint(1, 2**(nbit - k)) * 2**k + 1
if isPrime(p):
q = getPrime(nbit)
n = p * q
while True:
y = random.randint(1, n)
jp, jq = pow(y, (p-1)/2, p), pow(y, (q-1)/2, q)
if (jp + jq == p + q - 2):
d = random.randint(int(exp(0.272 * log(n))), int(exp(0.292 * log(n)))) | 1
e = inverse(d, (p - 1)*(n/p - 1))
if e * d % ((p - 1)*(n/p - 1)) == 1:
return (n, y, k, e), (p, d)

def encrypt(msg, pub):
n, y, k, _ = pub
m = bytes_to_long(msg)
assert m <= 2**k - 1
x = random.randint(1, n)
c = pow(y, m, n) * pow(x, 4**k, n) % n
return c

def decrypt(c, pub, priv):
n, y, k, e = pub
p, _ = priv
m = 0
B = 1
D = pow(y, inverse((p-1), p) // (2**k), p)
C = pow(c, (p-1) // (2**k), p)
for j in range(1, k):
z = pow(C, 2**(k-j), p)
if z != 1:
m += B
C = (C * D) % p
B *= 2
D = pow(D, 2, p)
if C != 1:
m += B

m -= (1<<512)
return abs(m) % (2**k)

pub, priv = genkey(512, 1024)
msg = "CCTF{this_is_a_test_message}"
c = encrypt(msg, pub)
m = decrypt(c, pub, priv)
assert long_to_bytes(m) == msg


All that remained was to combine their work and grab the flag. We can correctly identify the primes p,q while solving to ensure we supply the right prime for the Goldwasser protocol:

if solx > 0:
print ("=== solution found ===")
if True:
print ("x:", solx)
print ("y:", soly)

k = solx
s = (N + 1 + inverse_mod(Integer(k), e)) % e
P.<x> = PolynomialRing(ZZ)
pol = x**2 - s*x + N
pf = pol.roots()
if (pf - 1) % 2^512 == 0:
print("p: ", pf)
print("q: ", N / pf)
else:
print("p: ", N / pf)
print("q: ", pf)


### Implementation

from math import log, exp
import random
from Crypto.Util.number import getPrime, isPrime, inverse, bytes_to_long, long_to_bytes

# Challenge data
n = 6379346571939052419405006365144325711659387692572782917521683850800146277195077976472486871485207229367019052490145665163297786566841633505558063871407960795788255964350193702239693210208145457308480545438217518577295330054116121151347373114755560782091795900164067254792640967719557771138249404058566890490954215416904510736083128047275573547308653508508251678297758665771062241007774510795971063027233829414573035463227197757331376462301069384355548431423078078671509117155387034876199982416794024112992052740183312425799530813901896580055307267728332744004833470059699636327292472801239570572097698759537227941727
k = 512
e = 4000705832641304831719820890641730711910595477856628944800595532469774487124283398595587403418104883916158926323730475515283033807566520890825290575737835755837019474596405167016523680812524942147799183649264128776713591855426323904428490647193012502206272373739479178297498877035678377912075921414938337936441439527485344460693469140843705290948854072709663611286438938424604142221724677897600208962741594421412113507006294275515274270872733834184850544011087358368882662400066213073468442612453749942502454640340800031956110689070630604631216896059360054567160909689614639022142186132301488748492678008324441165927
c = 2076236531012576199455804061453889820064642138786354892007120685667610037384949798864855497275350216543927355263467568914282559982665624697899244269603116820671588426350142943143919069494292292917499297783442122092319467511971155786587154474978936572036643051684240363366549216293871321423431247376852947896978619912748318233407621803493738044334513430337745241122514134334555592413303034448331818630278695850858212374697195140803112644060181702284649554514396789034431504588176012105166481555153278186747662435177630254069907052265989634607771522027224143282081835102477403756195979837266124386568855830451903339893

# From modified B-D script
p = 68051654200085157207332136111091263403260677985149014940942748721163833107639364882355808718200151519497304188154695981671131677635619224129781969654824622900517674305201415050592661606881387625228654411615892410789900017222366193872604384511674439510956828703775115042284171259024209520418324094905850265601
q = 93742711281970123687375274125008994006083293818248949034193069509248449466340860308831673133907193988724549474397518306519671830205855974368567497715523859186789480391221672901717451785259798844945176989467876128830079144227608792557456138362303322064034291267314238570313761147918058776114912984468505970527
y = 6255502986116389228034984139242857374780790206304260235834031195913508784283825002169624192827940892131883506676728076842604216458764403070962446250597366611280772436223737779107366455847534672514863767917991550218275567549715295673378289958318905912165633828914581687895303874971235130850309582018888579000769679146004573091024149780380179513271132597140774030458709765002293555647353152272279119678905532272007948898539648658232027312618546750301948096187377831661685014650055118190776205003734702444247745395809759638757742466287334983047217540475915861943863479573235864782496267138830878462336954417976083797888
assert n == p*q

pub = (n, y, k, e)
assert n == p*q

def decrypt(c, pub, p):
_, y, k, _ = pub

m = 0
B = 1
D = pow(y, inverse((p-1), p) // (2**k), p)
C = pow(c, (p-1) // (2**k), p)
for j in range(1, k):
z = pow(C, 2**(k-j), p)
if z != 1:
m += B
C = (C * D) % p
B *= 2
D = pow(D, 2, p)
if C != 1:
m += B

m -= (1<<512)
return abs(m) % (2**k)

m = decrypt(c, pub, p)
print(long_to_bytes(m))


### Flag

CCTF{9en3r4liZed_adDi7iv3lY_h0mOmorphiC_Goldwa5ser_Mic4Li}