Solution to picoCTF 2014 problem Low Entropy

low_entropy.md

Low Entropy

Solution

Getting all public key pairs

• There are only 30 primes in the random pool in which p and q was chosen, so there're a maximum of 465 p*q pairs. What's more, p != q, so the number drops down to 435.
• As the number is small, we can put nc in a loop to continously get p*q pairs until we have the required unique pairs:

while true; do nc vuln2014.picoctf.com 51818 | tail -n 1 >> keys.txt; done

Finding the original primes

The faster way

Assuming we have:

x = p * q
y = p * r

We can figure out p using Euclidean algorithm

The longer and slower way (without using GCD)

Assuming we have:

x = p * q
y = q * r
z = r * p

x, y, z is the public key. Our job is to find p, q, r

From x, y, z, we can have t:

t = sqrt(x * y * z) = sqrt(p^2 * q^2 * r^2) = p * q * r

And from t, we can calculate p, q, r:

p = t / y
q = t / z
r = t / x

Getting the public key and the ciphertext

• They can be found easily in the pcap file (use Wireshark and then follow TCP stream in packet no. 7)
• Welcome message should be in packet 7, public key in packet 9 and ciphertext in 11

>: from server to client
<: from client to server

> Welcome to the Thyrin drop box. Please send your public key and message.
< Public key: c20a1d8b3903e1864d14a4d1f32ce57e4665fc5683960d2f7c0f30d5d247f5fa264fa66b49e801943ab68be3d9a4b393ae22963888bf145f07101616e62e0db2b04644524516c966d8923acf12af049a1d9d6fe3e786763613ee9b8f541291dcf8f0ac9dccc5d47565ef332d466bc80dc5763f1b1139f14d3c0bae072725815f
< Message: 49f573321bdb3ad0a78f0e0c7cd4f4aa2a6d5911c90540ddbbaf067c6aabaccde78c8ff70c5a4abe7d4efa19074a5249b2e6525a0168c0c49535bc993efb7e2c221f4f349a014477d4134f03413fd7241303e634499313034dbb4ac96606faed5de01e784f2706e85bf3e814f5f88027b8aeccf18c928821c9d2d830b5050a1e

Decrypting ciphertext

• We now have N, e, p, q (N is the public key above)
• d can be calculatd from p and q (https://crypto.stackexchange.com/questions/19444/rsa-given-q-p-and-e)
• With N, e, d, p, q we can create a RSA key object (https://www.dlitz.net/software/pycrypto/api/current/frames.html) and decrypt the ciphertext
• Decrypting gives us the following message:

Good thing no one can read this! I'd hate for them to know that the flag is make_sure_your_rng_generates_lotsa_primes.

output.txt

Read in 435 p*q pairs (435 unique pairs)
FOUND PRIMES:
* e66b30bf47da377f89c43ce1092c3cd0a4483f3524b8d6ebdb8cc5ed6dfe4ed31e5a9384b663eb320c051423544483070446759b9ffb4e1704ca0a67c071783d
* d794faaf3a00bdbf8d23256b92e164b5b79e47d26a4418e3e9f6be1b92efa0f9a449e03e38cbcef2d00cd6417fe3029636d67a9f647ae8e9eb143f4bd1415dcb
* c455e3f0dd50b980ed15ad0005581ad632dd7a455fef4d96a44f6f391264f7f87df3a272b6d8057c1c4fd831aba0066473e61ab4ef4b5261632bf3d53bbeaa25
FOUND PRIMES:
* e00276ace6bb31496fa763a03994d5c8e10dca6ee1bc1181127c9f6e8d71dbef6b70a1c34d4195ec18e71f2993bf63f2148230236c5a0c0bed82c3b499a048fd
* f015c52534b161b0dee27d454a5d56b4faa74d5ae0dfb7eea91702a716f34c502679a709042b43127a842272f24e67742008bbc2ab3dcb7825b8ac11bbb33aef
* dd2b1dd773395efa8c171a5ad068e193d4e8b8e43dba835bc8bc4da5a3045a79bd9c73b8348a56125fed0305278df52ccde0b666c71cd3749028a0acb23ce6b7
FOUND PRIMES:
* e8015bd1c0492ff358d3cb7b5d6c7732af0276a2ea148e5d89f674cbdd1555f03e990d3c7d8941e368b60cc2c10127849431938a8fce13a5b09c6f7e021c9b83
* c5e8e7d069bc608cdbcc042cbda280d306ab3dfed1655ce0c4a5f77eb6e3954a4160dc9e370de858a83f14ad1a965d30ccf5aaf91d66a346af29e6dde25a7847
* f03b1aae26603d7e6337d59555c3e24f7e8bbd2716bc1a8e33276ef4ffe4f674c42aa43344a2ccf1811484859130139eb1aed548f47c3db521b5a13f7e83bab1
FOUND PRIMES:
* c6759a14d892d2e739d3fbcd1f65575714d04b358ef1bd4a491e3bc111378edcd9c9b30a7cdc81c3fe722f22400494c8faa612634f51ecde8dad0af34f645805
* f529b5ec2fa734d9d25820bf63e0a50d750ef031669975d7f785fc854b862593d6b47a72aa9b431a361d21f32c2e494a0b7a243cb00d897ec9d31f0c90ad83b7
* df22c2ec36787e5fccf154a95b1afd0333aabc9cd68aa3750981999ad3149c0f51cb2d2671c648875d1b929961af000bebfa427448b54486d20e1e570ff5b31f
FOUND PRIMES:
* b75c72f11dd3ed5ac999925d78b451c5e309e4dca7d6e11de321f3b8bfe6e6abb952b5d1f2ad409377d3a4bb12fa663974c4c11ab17ad1b499bedd829ba4e6b1
* c8b475594ebf8527c7f7130a24cfcb1668138563df68e0aed90aa77eafbea792332f9cc367b07e78e5c021a5478fa962da28b30b6f8f8c82ec2d6037c162c675
* c90773f2a74e399449af95dd93a85510d900ca2d007b399d5a2eb89099d1cf3ae102799d94dc6d4bd17f1ba3a11c439db61d245d3bfde6d650b5f68d70d5363f
FOUND PRIMES:
* bc753d1c1f258a83a1b6937b37385b78bbb6c71752ef4381b02dccec2ed1bd499c81a5f3c7b388e51a6a30673377a7cafc0cf83dd9d1a1501cbee91324288337
* ce43dec5f6db0ceafca0fa6d3b4d1dc3ed25e61d50562241cca57b31b39d4e0f68520a3687af926959d2d5c0ce0099fb026ac6d2ceb0d4e192eee0aff4a05225
* d4e554b87c8e7e1cf3da7f386cdbaafd03f212fd1243d8a1b6d28ed2e8274c2b2937e201d5bf28404821f3970dd6a1192966b047db351d2ed1117854ad505aa9
FOUND PRIMES:
* c39a4d7985888f9b8dae33d2caca1385b4e73f4f07e57fea197d80a171a84e74b008a46a3c1d92d18b7cf513dadb4e756c130c6bf8b3b9a64f72e0208a22b265
* ef3b053e3488834d747ab1ab2c228fd8a58eef1f0dd6cfbd7d431ba8723580223dc31c138d5cd0879abc498a509bac826279f48e27f2fdcc3f49e1b75bd57a7d
* d48865f3c326a808e57f95cf5b8c7d019d6026d3818e6dd9d3f10ef8f1c9eb60b523ce08d2a8b166f8ed4613f710717f58192f2bcf996105d46291bc8efd1a6d
FOUND PRIMES:
* c3cd9e03bdb68392b8c35e80977e4f23d397367084b89ee0cca572147ab8515ca2697d5e06af6587ac4f0efe83c4e5e6d16986bda89cb410b507411db995ea5d
* b746f5ba6887c5a0e031481b6b37cb3abe9e871bfe224cae9b70271472cc8d7936d33b15c5f0bcc0a4da346881713b868d9f35dbc4f77818138ea16171d4d5d9
* d0325f5d7d01f7f778127ce4723a5ba034e02da9591e390fcad38f6662fd516965b543b49a150242ac552970442756e43c81f99ddbad8a18128bcb8f3959ad43
FOUND PRIMES:
* e16b43f9c9c5838b64d131de2b32fbc43ecf1dde2f1e925b38b5269a57dd5f51f1e0d0e625e8b0700aa33ba99b5ac794f7017d766373bff465266aa1f6c0c075
* e03bcd6143e8b8bfb923c076434ef07dd0fceed005f0e3f546eb75d8e0229e770e271da35cb990ce5d4ecb79cf3d5d7f8becfd201b6f12b9043a83168f9a515f
* fac1fbf978bfa5d5c9e4d567af346c8f651c1a7cad59805bf1a28cf613c86ad6672d5956717729c96f3fffcfa297856aab1e42c0776a56915ed7678327c72385
FOUND PRIMES:
* cc0a359833f081005db371e859dca7739fa420a3aee15887235725cd61957eca75246f53063590ff2c7a9b6f5fcf4828abfa780263594c98980a8f6b40e490e9
* ffe476c461231be2f94d426361deecddccf47d1c3df88904d51300e0c95f2fcb0130f5ac1a307b7bc6725bb32e815c25a90e214275cd0a6b0e003d151cfd2791
* e0ec06df35f689a30244d1a1b3885b83820e8183f71bca3c93247e461aea4174a4dbd452d5ce60bf4c79c80cb47bdc63760020f8520f35ee859056aa569b6d17
Decrypting message...
FOUND p, q: 
p = e66b30bf47da377f89c43ce1092c3cd0a4483f3524b8d6ebdb8cc5ed6dfe4ed31e5a9384b663eb320c051423544483070446759b9ffb4e1704ca0a67c071783d
q = d794faaf3a00bdbf8d23256b92e164b5b79e47d26a4418e3e9f6be1b92efa0f9a449e03e38cbcef2d00cd6417fe3029636d67a9f647ae8e9eb143f4bd1415dcb
FOUND d: 
d = -5e3012cf35c9d07b248754ca24c43251ed522bfec2b8df85a12eae81dc24da1da6e31ae1116af0f946d6bd68c888ce326d609492b4ae2bf144b866408f52e38a74cfd9dd1533c173daea57565cc7521490ff7c611909089838907b46d4acff8de70d70bbacbbe3e21c49b1bebad8af777970887cf1354105302508821a15235f
DECRYPTED MESSAGE: 
Good thing no one can read this! I'd hate for them to know that the flag is make_sure_your_rng_generates_lotsa_primes.

pubkeys.txt

8345e972c84bb95a3b26a545ae814d0de7662ee128f43c0d5afab7d5508ca1330cd083d257a49f5bfedfd2bda98d26310f8e82a95fac210e4e18f4046988ae5d1bd50889402ac1b7edcc45a96c3744ff0a4fa5ca11cba1fc1b334cb7708ce33905e32397d9bad29ed103945562c6e2494192fa59da03d566ffb2d7aa4e4cd109
86ec0ba33d923ea8d441b85bf83f048766b7949ca889d977e6568cbfc7aba24cf5e4324b4899397b768ea04e4f43f1cd8594a77a7b1c862b0daccec84da8fee7378eb95f65997a210f29a417de381429992f3573e20967425ae3df345731f1df586cfe2f8734e7ca5d3c1e4ca991f87af1a4c86fc188f729b7f1a09e4c0ffc9f
86fbdd6ec5e96022d6d6311fae1d65da6f3cdd13a0833eed2094dff3003ac8a9acee68d4e9de9049cc752057f29123792818e5cc3915e34e5e2af9ac17ce368a930dd3f42bee0cc0c1d2ae7a07166cbec28250f452537fda4ea018ccc7e72b70891090062b630b07cbcfb5789c67e7b390a3b881ad639bbbe439270cc2222307
8c09855427cf25ef1028fac67d4dc196fc2e52bfe4c9215fe563fe3984c5f5a71f93c327d77e94a9bf27d5cd059eea23249a82507232e1adbfe72ed0e3e2b80a5c05f1bcf16cbe4c3f10512ef7ea88d35fc88a340c0042e6f616747e197a2888f934565eb6c42221c400b4fd747d03e13b01d0d540b6e54c8123e143467b409d
8c19f0a8a7ae10e92afcab3bc0f29396fab7549bcaa0fac292b0d884470d82c299f924c3d4257f6e8cf2e9f11bcad00b06e10e6bef68e7253b9681c5ffaf229236a7d43d70268978575bf13ca26fbe4c0fb766dc93f2007fd59da7d6c9b6220a5888e5104a7e9a5fb5631698c1a75db94aa471ada96748d249db7b221df815d5
8c2e42203d351c057e59a1334f72bb4c7e8815086c972331c2edb678fc1562b7a3cd319a803c6f629ce95f8aa928ade776458291bb5bdaca8e4fe712baa35324f3eb3c47aaa6a3f641cf3068e9221a9626e5c7b4e53d0ab6872289a4044ef062ded798db763b734a31e465014bed25f2b5df20c346ffa6c808a514722e1709d5
8c3eb1c371b574907f8f375aafabfdfa5022f2accac23569960135ddd4d023287087ad7a12e96493a722e52b04f240b32cfd83c71ae1d811b22cec7a9c5e80895a83e021c3cc2acb8aad1b39e2717765e77666a3a4b4cc573d4b3bbf18a61b1fca405566914ae68ea87a1a56545d24578415126ffe5fa64c9466af7473ca984d
8c8fd1e2abc95ef378dfc22288cc9c24ecf3e75097b4a59e403a8fa47783316cb543abf7074e7eaabb777406c93b8dac981591434ddab3c61a5017680723e30037c43ecb27760013fb296049078bb8a7e50a5b38d72607b3d1f62c286dfcc9d5eb55214720ea97f7e7db7a71b4c1efdcb2abf9d57adb775159dc7a7983d3025d
8ca04cf6440159926cd7a41417009817b065ed7fcf3085346111ed373c0eb14bedf4af65391fa77267b5b8eab5f0eed937dad6719c35e1bec36436a6363020d0281db4459dbb24d9f687ee5637e21f0f31596480eca93dae96772e91f504c4d3a5ad1a4d514a60355b73ae6db14ffbbe80aab59ae8e8a24bb266390f0264e195
8db0595d607509ecb60a56d0b0b003470169e348bfbe38b3e27a38eb5ab0337a1c695246752f20adc08f5fa494d1e916cf6ca89bd03ce349c8df2783a46da25f5b7587923e45664c73b05079a707f1e45c3eacf6116bbf3d62318f31700e5ca6f205eb711e63b040b07091a4b0dbd8f10f7b590905049d751e92eb54168f072f
8dc0f6456907ebb94c2cf881a0e089bc335bc242d851bdf67da59d7938fe0be3c1fa515151b5100419a69ab9739e1f04037c9747889c9459bf81ef0145f82ac570c816867483ede8c7412d63b9efa193bdb25eba273cda1685a914f5dc5ab7ffb08108d344cea484afa663b663476a7c027da0bf712e5de7bf0e209cd318f317
8e1513cc17751b24c3ee2d54ef5f184bc159ae947e9b3cc803ee7f0101664c961bd08c0d1e8a9ce46811bb41821159b2c709954b7431dc29524a13698d75ca6afd5be03d6882738940524c2a60bbc97959320988697a701f3175831632bb282c9844b5c94ba8b700538b597ba749f7b58c5d0c4fab29e971eb7a4224e26ec53d
8e25bc8390c1bcfbf690d31309867fbf6a61ddb05112438a5cbb66a53aaf7410d9c6504e75207a70ebd9b8f721b554cd69ab6b21a18a07442543012e8fb174ff4e95b409fd6b23690f5cd29e6105633cbc60df6d4bcbfc631ed8a807c3b3c4f7e0ad2bb47446f05e280f1fcde40f4185eec29933f5f6ba560312ccab75a95975
8fb0a1e1c715ed95835a0db5094d1f4096f53c97247f5f41223ac9f7bd8c42a4504adf5bac0e05b89393fd7658361112c245d2d83fbf092948d2c8f3bf8b3150d37219a6f67021c010518089dfa152ac3fcf898e9ce2e9f064b559c4594f384786004d485633a533fbd6824f86f265a6d58f59142d6b38c4673079e519bd922d
8fc17ada68ea0b45ca802ce111669e1255868762a3a4eeae5148c3d66eeb9288c52d85752e07600008dc80d65e52252825eb23d1f38c815610fe83b71fd165595529e55f30fe5600a90748f960c264bc3cdff54c15aa4cb7403aa75ba4a21fb5e7371db7445f4a769b8f889f3e71c08ab65b38fb50a82a7b26f218256d8c54e5
8fec0ce2ad69c831f506d6f475e6cfa7b2071f78f92b6f214d7bd176cec00cba2acb18f7ba9f748e60dbeec99ec58d5c970a41963947ce684fbb675e218f90cb3c7a85be4fbe056a5025bf165c9668e925197269ccb7da2a8c7ece9b5446308f7b08b5071e930777f975c4ac347caa505723a5973d6d8c17082a6727c6096667
8ffcecd2c9df84e04778d943c5985007ead6cc3b54e240a5ae1698b34adfe7eb980f9fbc612d95a39f85ab052634112a7ffafb5cbb5bc747769c485d7399526e599b7924ad76f853f82778ecb581f9fc67185a61f0b4e67c188d3ae8d21337d53af8ea1c4c2def4baa8003e0502361eec8d0fbc6a3fd14bc251bda9777831b8f
8ffee51be7bd1d80ae4ffcda48c9ec40852675c3ad3ab2b8e0ded7f0693056b3971f9a37eb99a74b09b9d067869dbf4b96879e20923794e411c135c71919199343370501c6a85e7442cb4ebeb551b1db5be0e76b838005d34f3e99dcd35275b8a264d0805ae1b9e89eec15141f9bff870d0ec5bb650a64d1bac86eae828602b3
9024abc177fe8672eea79f9da38bc37ecfbc9be9f16b798c5195f1531d30c4db8daa4aa955713fd505355e140028b32cce12d245691eab632a3b5607ea1cb20dce91620051c5a3fbbaad08baefea06d587661aa65b511bd8a1267bb46b155891a602471ba25e43a8f30812f95fbbdde4cc9aac8ff6354a984b1451a348aaf0fb
9088fd841834f273448691c2b81bffe38bd65613f0a6896651fefbf19bbc4b7a6991ab7b46f7d5a462ce7d85fea27354929568adcbe7be3a4215813d2a2e559759509bff8f080e8d90c79877095b73e9f2a98bae4add09df876efb61547f9434760203a633b8c19663822867028a816a29868d5e074704b62d67f3d930cf7cf3
91b1aced3cb0e7ce137d1709f4cf3d7999382448b4702801c5db439abed62eab24f9e090a24dfef0c6c712c2281fcd9f7188e363b416ae2b1f3e8066fd7400d8226c8614d6ab704e26298ba52653aa8142e05c7028f031637a7cedfdb757237556033750b5f9022d6ac172353b0ae30bc3e1e4030dd4b9ce8685c7d1b4142c41
9213daf4ccc516f19df4e715ee313b3d32026c30c851fd78b70531838b8dedff595d50dd9b68425c4e3494d529b74c48da62c92f371bf7da6a954aea379e36a15dad5c3a73e8ffc8eb999ae5ecf38e5948d33e8221e3dd1906f932a88c88065399ba78ac708e704b6038eede254ad30e188f0ba97e9304bbc8b9e1700844b281
921940468d4bf7e81a4ceee2b60c0d22da8db514cbab34859d77fda3003e9628b2dcfc8ef2aee558b1044a9b7dadba43a1bca808e28289d844479832a075e8f28ad2bcb6040fe590ff97bd368e6094d21347f27b7410d21e1e9e91effd901e86ccb797b2e8fdd1c6f68c7dcba4e3c5a2b96a07d109610edf023b7553dc617813
9224fb97c913e3b9d830c6424983a1d95c0e99a5bc317dfabad7e569e9b310d5618526f8162dce4c8f29f5f7c4ea825bde44c53b9019963cd67f8b919c21b4034ed8ba51199c9d243a73a61e99625b2e49801b4097b161e85bb6883a55030f0be2878f853cfaa4410f6429225e5256c0509e5af6a2fc2643804036f1207d8719
93abb0cb93545ebd41874302565fc45bc6b80dc0f6eb3323087f6b3cc9ecdfb443da1ba67b6522b5b8df7e01540fe5204c070892a04600c4adf1a54be3ec0cd3c9fcd2f407beaa584abd5a8d5b96d3ee812d06c624f091b7843ec9e891c806f0294be5d8005beb26f5f3aa5850e8f29a0e932885757caf78de0e99bb1be26a5d
93bd014011bc9eaa94ab8f210812d590b942ec85e064746cb2ef96273013c3edafb4945e28e5d6d097723ed50064ac95ccb290a517df058a8d9f8feae0469eb01f6a045cdd1d974879751fa3db281b2941dcce88bf3c7457fa4bd7be4004b8e3fc60733c016bf7cf8082ef7499dfafb77a624bfd1b7bf3ad63a6dfab335a0995
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f02481699c2cd254eaf209a203a7f26da8a9a3f6144ff105c3053b409d29428fb855f571496fce700ecf324fbb5a0c9cf5f72a0808f6354a9d395c3d079161911dd6e7fc4e9b5ef2def578c4015d084c5ed21f8121965a8f19c75dc6dccd3e2bdff5099faf5d4eacafc0f597c6eab0867e3fc4e743a40fb06e36c6e627687313
f50f571994a292b2ba00ee5350544c40d230ffc7e7fe51ab21d39736fdce0cfcf4583ab9a734080ff92498f419068722ec3a978ae62baf7c614878e5d3dc703691e81ef10cbeecda77f610cd420c3b24ca6a2557a5a527c30e6e668ac714181c831e951ff3f948cb9843748de52714b80c34730770d4f9ccc22bdf0e91337ba7
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solve.py

#!/usr/bin/env python2

from Crypto.PublicKey import RSA
import math
import decimal
import binascii

def isqrt(n):
    # http://stackoverflow.com/a/15391357
    nd = decimal.Decimal(n)
    with decimal.localcontext() as ctx:
        ctx.prec = n.bit_length()
        i = int(nd.sqrt())
    if i**2 != n:
        raise ValueError('input was not a perfect square')
    return i

def egcd(a, b):
    # http://crypto.stackexchange.com/a/19530
    x,y,u,v = 0L,1L,1L,0L
    while a != 0L:
        q, r = b//a, b%a
        m, n = x-u*q, y-v*q
        b,a, x,y, u,v = a,r, u,v, m,n
        gcd = b
    return x
 
def read_pubkeys():
    with open("pubkeys.txt", "r") as f:
        keys = [int(key.strip(), 16) for key in f.readlines()]
    pubkeys_set = set(keys)
    print("Read in {} p*q pairs ({} unique pairs)".format(len(keys), len(pubkeys_set)))
    return pubkeys_set

def solve(pubkeys):
    # x = p * q
    for x in pubkeys:
        # y = q * r
        for y in pubkeys - set([x]):
            # z = r * p
            for z in pubkeys - set([x, y]):
                # t = sqrt(x * y * z) = p * q * r
                try:
                    t = isqrt(x * y * z)
                except ValueError:
                    continue;            
                                    
                # p = t / y
                if t % y != 0:
                    continue
                p = t // y
                
                # q = t / z
                if t % z != 0:
                    continue
                q = t // z
                
                # r = t / x
                if t % x != 0:
                    continue
                r = t // x
                
                return (p, r, q)
                    
            
pubkeys = read_pubkeys()

original_primes = []

while len(pubkeys) != 0:
    p, q, r = solve(pubkeys)
    original_primes.extend([p, q, r])
    
    print("FOUND PRIMES:")
    print("* {:0128x}".format(p))
    print("* {:0128x}".format(q))
    print("* {:0128x}".format(r))
    
    tmp = set()
    
    for key in pubkeys:
        if (key % p == 0) or (key % q == 0) or (key % r == 0):
            tmp.add(key)
            
    pubkeys -= tmp
    
print("Decrypting message...")

N = 0xc20a1d8b3903e1864d14a4d1f32ce57e4665fc5683960d2f7c0f30d5d247f5fa264fa66b49e801943ab68be3d9a4b393ae22963888bf145f07101616e62e0db2b04644524516c966d8923acf12af049a1d9d6fe3e786763613ee9b8f541291dcf8f0ac9dccc5d47565ef332d466bc80dc5763f1b1139f14d3c0bae072725815fL
e = 65537L
message = "49f573321bdb3ad0a78f0e0c7cd4f4aa2a6d5911c90540ddbbaf067c6aabaccde78c8ff70c5a4abe7d4efa19074a5249b2e6525a0168c0c49535bc993efb7e2c221f4f349a014477d4134f03413fd7241303e634499313034dbb4ac96606faed5de01e784f2706e85bf3e814f5f88027b8aeccf18c928821c9d2d830b5050a1e"
message = binascii.unhexlify(message)

for prime in original_primes:
    if (N % prime == 0):
        p = prime
        q = N // prime
        
        print("FOUND p, q: ")
        print("p = {:0128x}".format(p))
        print("q = {:0128x}".format(q))
        
        phi = (p - 1) * (q - 1)
        d = egcd(e, phi)
        
        print("FOUND d: ")
        print("d = {:0128x}".format(d))
        
        rsa_key = RSA.construct((N, e, d, p, q))
        
        print("DECRYPTED MESSAGE: ")
        print("".join(rsa_key.decrypt(message)))
        
        exit()