HackTheBox Binary Basis Writeup

Explore the basics of cybersecurity in the Binary Basis Challenge on Hack The Box. This easy-level Challenge introduces encryption reversal and file handling concepts in a clear and accessible way, perfect for beginners.

Provided Output

output.txt

n = 119493145368134756606524581488517378672235496744508597127639648421685629462649680337305664581985725401892086474314689141024498358715132868441978738105268336335778301627806695859205063825847290995872425532374614756539044160447358903375166525543471922868488499105664355206800751381027260462074705122647672994384365333260862845803962495145681040264675164695690862737006515315087470040757606517140071798300326022127659497206583492859643180260756285101547522674696613261207152550090000611298331939402585491230470733454735317291288404096144773097239721697692471487615973994925335486666529576878075473132154318910180009691
e = 65537
c = 78434014553061170529838401917720357645901208101352296269884472900978844319305951884764695717375322363272928093737865793113104432023794235229497701142692681415679549147532592802702053823324082733182072770779006523720442490771120987890027997070704393610765382574187921750560918055356164458340750383236011537901475045013520557031618820264221766743094536121925226791850895255993453048670745607742469934947653147879259892501050933253400391984499229656112961295969492487005786860006961292676090836472527246673699345029628224872195546099448661501561676622720260514226385260271583307493991971640825413235694692735476316469
treat = 41064122210117590309893920850940851964448573197364702312218652847563074658468550776049623436077059647900426409109763901345955202703985682630778316836716410976908994802298857481157910324003086887604087560484126319093760348302784031959884608690718739939531289130555241671099503557713216197592401241234208735819631180320413760718656876867323406561829316166491696103527500078153837145772310740455509956660694823144830260095551391236403182836214728060291251693690579998717099897785954779748569649102911569556210804834065634957865690729391572538723259866564481741517395331813459603466263827141279911919706281328288050460721861661990446597425850271364217052936489186818846863874578006688813944143685727931104321810713830759839918256937075645849871941798921033683465399959646136621559049887273682054277330663458615787744906360121329675910070999119357809181250170609599790023712764414191551844874675524795782430239120555096062003800058693423405350206617745916914116965357631650740005441388405061692790006508760971107741027556312990780727724371467017609554695371798159449361071175599514947307436760940029734313697931199508407965056916479246854944553994533969739046153764629534835323207379057477612661669942078609127146916601554609679997006901572586628537881511818938391905350113674432833361660748486019032622987316746412638187233454037252369980080224744837380800996358881451638758024280972061536248320550269641404509205922332930938158599125930491114698209185841465863646506532208640

Provided Script

source.py

from Crypto.Util.number import getPrime, bytes_to_long
from math import prod

FLAG = open('flag.txt', 'rb').read()

primes = [getPrime(128) for _ in range(16)]

n = prod(primes)
e = 0x10001
m = bytes_to_long(FLAG)
c = pow(m, e, n)
treat = sum([primes[i]*2**(0x1337-158*(2*i+1)) for i in range(16)])

with open('output.txt', 'w') as f:
   f.write(f'{n = }\n')
   f.write(f'{e = }\n')
   f.write(f'{c = }\n')
   f.write(f'{treat = }\n')

Proof of Concept (PoC)

from math import prod
from Crypto.Util.number import inverse, long_to_bytes

n = 119493145368134756606524581488517378672235496744508597127639648421685629462649680337305664581985725401892086474314689141024498358715132868441978738105268336335778301627806695859205063825847290995872425532374614756539044160447358903375166525543471922868488499105664355206800751381027260462074705122647672994384365333260862845803962495145681040264675164695690862737006515315087470040757606517140071798300326022127659497206583492859643180260756285101547522674696613261207152550090000611298331939402585491230470733454735317291288404096144773097239721697692471487615973994925335486666529576878075473132154318910180009691
e = 65537
c = 78434014553061170529838401917720357645901208101352296269884472900978844319305951884764695717375322363272928093737865793113104432023794235229497701142692681415679549147532592802702053823324082733182072770779006523720442490771120987890027997070704393610765382574187921750560918055356164458340750383236011537901475045013520557031618820264221766743094536121925226791850895255993453048670745607742469934947653147879259892501050933253400391984499229656112961295969492487005786860006961292676090836472527246673699345029628224872195546099448661501561676622720260514226385260271583307493991971640825413235694692735476316469
treat = 41064122210117590309893920850940851964448573197364702312218652847563074658468550776049623436077059647900426409109763901345955202703985682630778316836716410976908994802298857481157910324003086887604087560484126319093760348302784031959884608690718739939531289130555241671099503557713216197592401241234208735819631180320413760718656876867323406561829316166491696103527500078153837145772310740455509956660694823144830260095551391236403182836214728060291251693690579998717099897785954779748569649102911569556210804834065634957865690729391572538723259866564481741517395331813459603466263827141279911919706281328288050460721861661990446597425850271364217052936489186818846863874578006688813944143685727931104321810713830759839918256937075645849871941798921033683465399959646136621559049887273682054277330663458615787744906360121329675910070999119357809181250170609599790023712764414191551844874675524795782430239120555096062003800058693423405350206617745916914116965357631650740005441388405061692790006508760971107741027556312990780727724371467017609554695371798159449361071175599514947307436760940029734313697931199508407965056916479246854944553994533969739046153764629534835323207379057477612661669942078609127146916601554609679997006901572586628537881511818938391905350113674432833361660748486019032622987316746412638187233454037252369980080224744837380800996358881451638758024280972061536248320550269641404509205922332930938158599125930491114698209185841465863646506532208640

exponent_0 = 4919 - 158 * (2 * 0 + 1)  # exponent_0 = 4761
treat_bitlen = treat.bit_length()  # treat_bitlen = 4922
prime_bits = treat_bitlen - exponent_0  # prime_bits = 161

primes = []
for i in range(16):
    exponent_i = 4919 - 158 * (2 * i + 1)
    prime_i = (treat >> exponent_i) & ((1 << prime_bits) - 1)
    primes.append(prime_i)

n1 = prod(primes)
assert n1 == n

phi = prod([p - 1 for p in primes])
d = inverse(e, phi)
m = pow(c, d, n)
flag = long_to_bytes(m)
print(flag)

Summary

Binary Basis on Hack The Box involves recovering the private RSA key by reconstructing the prime factors of a large modulus (n) from encoded values. A “treat” variable contains encoded primes, each at specific bit-shifted positions. The solution decodes these positions, calculates φ(n), derives the decryption key d, and decrypts c to recover the flag. This challenge demonstrates RSA vulnerability exploitation via modulus reconstruction and bitwise operations.