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TokyoWesterns CTF 2019 - happy! Solver & simple writeup
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require_relative './happy' # rename happy -> happy.rb | |
q = 180754955592872777770305021165949447837520820408608437544593001477325680227199967219036800612237524673886247520794601572290402702594122131782305474875236404413820478308317235725623037177247985490515533618988964977186476558003216903 | |
p = 166878663790065040663149504970052368124427462024107500159158464138407657299730521908976684364578086644682045134207945137293534705688910696520830729908263578233018529387676221035298300775812585471932551347478303730822844748034186479 | |
k = 2 | |
e = 65537 | |
d1 = e.pow((p - 1) / 2 - 2, (p - 1)) | |
d2 = e.pow(((q - 1) / 2 - 1) * (q - 1) * (k > 1 ? q ** (k - 2) : 1) - 1, q ** (k - 1) * (q - 1)) | |
cf = p.pow(q ** (k - 1) * (q - 1) - 1, q ** k) | |
key = Key.new({ | |
n: p * q ** k, | |
e: e, | |
p: p, | |
q: q ** k, | |
d1: d1, | |
d2: d2, | |
cf: cf, | |
}) | |
File.binwrite("flag.txt", key.private_decrypt(File.binread("flag.enc"))) | |
=begin | |
Sun Sep 1 18:59:16 JST 2019 ~/Downloads/twctf/happy 100% | |
> ruby solve.rb && cat flag.txt | |
TWCTF{I_m_not_sad__I_m_happy_always} | |
=end |
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from sage.all import * | |
n = 5452318773620154613572502669913080727339917760196646730652258556145398937256752632887555812737783373177353194432136071770417979324393263857781686277601413222025718171529583036919918011865659343346014570936822522629937049429335236497295742667600448744568785484756006127827416640477334307947919462834229613581880109765730148235236895292544500644206990455843770003104212381715712438639535055758354549980537386992998458659247267900481624843632733660905364361623292713318244751154245275273626636275353542053068704371642619745495065026372136566314951936609049754720223393857083115230045986813313700617859091898623345607326632849260775745046701800076472162843326078037832455202509171395600120638911 | |
e = 65537 | |
cf = 25895436290109491245101531425889639027975222438101136560069483392652360882638128551753089068088836092997653443539010850513513345731351755050869585867372758989503310550889044437562615852831901962404615732967948739458458871809980240507942550191679140865230350818204637158480970417486015745968144497190368319745738055768539323638032585508830680271618024843807412695197298088154193030964621282487334463994562290990124211491040392961841681386221639304429670174693151 | |
F = Zmod(n) | |
PR.<x> = PolynomialRing(F) | |
pol = x * cf - 1 | |
pol = pol.monic() | |
# I guessed X and beta from some experiments... | |
x0 = pol.small_roots(X=2^800, beta=0.2)[0] | |
qk = ZZ(gcd(x0 * cf - 1, n)) | |
q, k = is_prime_power(qk, get_data=True) | |
p = ZZ(n / qk) | |
assert is_prime(p) | |
assert is_prime(q) | |
print "[+] q, k = %d, %d" % (q, k) | |
print "[+] p = %d" % p | |
""" | |
Sun Sep 1 18:58:48 JST 2019 ~/Downloads/twctf/happy 100% | |
> time sage solve.sage | |
[+] q, k = 180754955592872777770305021165949447837520820408608437544593001477325680227199967219036800612237524673886247520794601572290402702594122131782305474875236404413820478308317235725623037177247985490515533618988964977186476558003216903, 2 | |
[+] p = 166878663790065040663149504970052368124427462024107500159158464138407657299730521908976684364578086644682045134207945137293534705688910696520830729908263578233018529387676221035298300775812585471932551347478303730822844748034186479 | |
real 0m8.301s | |
user 0m7.995s | |
sys 0m0.265s | |
""" |
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Nice! I overkilled it, by finding small roots of a 3-variate linear polynomial, but over integers.