Rating:

Challenge is basically multi-prime RSA with a poor key generation. Solution is based on the search for primes in vicinity of 3rd root of n based on the flawed prime_gen() key generation function.

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solve.py:

py
#!/usr/bin/env python2

import sys

try:
from sympy import integer_nthroot, mod_inverse, igcd, isprime
except ImportError:
exit("[x] sudo pip install sympy")

def find_next_prime(n):
if n <= 1:
return 2
elif n == 2:
return 3
else:
if n % 2 == 0:
n += 1
else:
n += 2
while not isprime(n):
n += 2
return n

n = 2739699434633097765008468371124644741923408864896396205946954196101304653772173210372608955799251139999322976228678445908704975780068946332615022064030241384638601426716056067126300711933438732265846838735860353259574129074615298010047322960704972157930663061480726566962254887144927753449042590678730779046154516549667611603792754880414526688217305247008627664864637891883902537649625488225238118503996674292057904635593729208703096877231276911845233833770015093213639131244386867600956112884383105437861665666273910566732634878464610789895607273567372933766243229798663389032807187003756226177111720510187664096691560511459141773632683383938152396711991246874813205614169161561906148974478519987935950318569760474249427787310865749167740917232799538099494710964837536211535351200520324575676987080484141561336505103872809932354748531675934527453231255132361489570816639925234935907741385330442961877410196615649696508210921
e = 65537
c = 2082926013138674164997791605512226759362824531322433048281306983526001801581956788909408046338065370689701410862433705395338736589120086871506362760060657440410056869674907314204346790554619655855805666327905912762300412323371126871463045993946331927129882715778396764969311565407104426500284824495461252591576672989633930916837016411523983491364869137945678029616541477271287052575817523864089061675401543733151180624855361245733039022140321494471318934716652758163593956711915212195328671373739342124211743835858897895276513396783328942978903764790088495033176253777832808572717335076829539988337505582696026111326821783912902713222712310343791755341823415393931813610365987465739339849380173805882522026704474308541271732478035913770922189429089852921985416202844838873352090355685075965831663443962706473737852392107876993485163981653038588544562512597409585410384189546449890975409183661424334789750460016306977673969147

primes = []
current = integer_nthroot(n, 3)[0] - 500 # Note: arbitrary value used to cover the primes in vicinity of 3rd root of n

print "[i] Searching for candidates in vicinity of nthroot(n, 3): ",

while len(primes) < 2: # Note: searching at least two divisor primes in vicinity
current = find_next_prime(current)
candidate = igcd(current, n)
if candidate > 1:
primes.append(candidate)
else:
sys.stdout.write('.')
sys.stdout.flush()

primes.append(n // primes[0] // primes[1])
p, q, r = sorted(primes)

print "\n[i] Found p, q, r: %d, %d, %d" % (p, q, r)

d = mod_inverse(e, (p - 1) * (q - 1) * (r - 1))

print "[i] Plaintext:", hex(pow(c, d, n))[2:-1].decode("hex")


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Example output:


\$ python find_primes.py
[i] Searching for candidates in vicinity of nthroot(n, 3): ..............................................................................................................................................................................
[i] Found p, q, r: 139926822890670655977195962770726941986198973494425759476822219188316377933161673759394901805855617939978281385708941597117531007973713846772205166659227214187622925135931456526921198848312215276630974951050306344412865900075089120689559331322162952820292429725303619113876104177529039691490258588465409208581, 139926822890670655977195962770726941986198973494425759476822219188316377933161673759394901805855617939978281385708941597117531007973713846772205166659227214187622925135931456526921198848312215276630974951050306344412865900075089120689559331322162952820292429725303619113876104177529039691490258588465409397803, 139926822890670655977195962770726941986198973494425759476822219188316377933161673759394901805855617939978281385708941597117531007973713846772205166659227214187622925135931456526921198848312215276630974951050306344412865900075089120689559331322162952820292429725303619113876104177529039691490258588465409494847
[i] Plaintext: flag{pr1m3_pr0x1m1ty_c4n_b3_v3ry_d4ng3r0u5}