2025能源网络安全部分wp

本次介绍的是能源网络安全的一些简单题目，主要针对密码方向

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crypto

NumberTheory

题目源码：

from Crypto.Util.number import *
import hint

flag=b'xxx'
e=65537
p=getPrime(512)
q=getPrime(512)
n=p*q
m=bytes_to_long(flag)
c=pow(m,e,n)
k=getPrime(1024)
assert hint + 233 * k == 233 * k * p
print(n)
print(c)
print(hint)

# 84099006955126261966925371456202769943592466221370095794235167154956697927281125181449320270460637820908574232493978429962263974458426503598700104493216727535451616752760724333653967152401716945549285008242019874215196489846481143398374860288545040874468108191037481101604627874268575884573685952474988256841
# 28098063654079651384124474197746356824080585622155888018279898490747561415908220072536298610509681898119018709183606442183944207485940115624047842734359988590155403601250406116023121958193303908964857108526965815235457652033182982467968474248778435731228104089366239566977364311197776651102290796373095167764
# 411245630228311610573345621334618725748702407327926883063919892785851166202383809662483938501531987094884084543300939673794551515912845363503988032311234800260819110323258416786417746444373651130257247926678135654564298408894174083333804257126735899220917359603430399033328133462456659839525671074605146583034398735379485362144932899212206419889556154825755723979850750847762362288223441051219637465296077020565435562941976546609555729574021362954126496825972439730

思路分析：

由 $hint + 233 * k == 233 * k * p$ 可以推出 $hint= 233 * k * (p-1)$ ，根据费马小定理：

$a^{p-1} \equiv 1 \pmod{p}, \quad \text{其中 } p \text{ 是素数且 } \gcd(a, p) = 1.$ $a^{k(p-1)} \equiv 1 \pmod{p}$ $gcd(a^{k(p-1)}-1,n)=p$

但是有一个难题，即使我们令a=2(最小的素数，一般推荐令a=3或者5)，在python中仍然不容易算出 $a^{k(p-1)}-1$ ，更没法与n求公因子了

所以我们这里先 $t \equiv 2^{k(p-1)} \pmod{n}$ ，以此来降低指数 $k(p-1)$ 里面的k的大小，从而求t-1与n的公因子，即如下：

$t \equiv 2^{k(p-1)} \pmod{n}\\$ $p=gcd(t-1,n)$

exp如下：

from Crypto.Util.number import *
from gmpy2 import gcd

n = 105531804094410236294687082475828411218788143973352026416392542762437103918840861241726193253936370648195682452618343195471719649394086997793137653518966739212122830015579955183805636213883066694989610003565432493653164047938048521354525623612253955387430773546124647105772639376194421783783651686606080214099
c = 6838127295540107402282470465780599628759317234806902778570348919850980664834107227012249617036087381075344016550381137159643512672239826438903241091658619314078921936719784123522758604349399440232971511649918093228288847534685526358191804172060250409498531941883162873696671060909325234109062997554795436940
hint = 225457129615945961139095949356083106510992163176770860368085043522677811094793442173512565115313130227614423196268240217775831118417780318014842280209747426271227826513967791945116378179885000662888744992914390207196310600996050316737090999399962338133222370745589250853315876818226312453376340282748842779666176953455553054310328901299083159029050169950812885486884682347263045764918907196922313892044095742248895091717187372068779768743879411865275203496650858608
e = 65537

t = pow(3, hint // 233, n)
p = gcd(t - 1, n)
q = n // p
phi = (p - 1) * (q - 1)
d = inverse(e, phi)
m = pow(c, d, n)
print(long_to_bytes(m))

# flag{158f26cca09e89b08cbd234c1f9c98c0}

easy_lwe

from Crypto.Util.number import *
from secrets import flag
assert len(flag) == 38

p = getPrime(512)
m = getPrime(512)
while m > p:
    m = getPrime(512)

aa = []
cc = []
bb = []
for i in range(30):
    a = getPrime(512)
    b = getPrime(400)
    c = (a * m + b) % p
    aa.append(a)
    cc.append(c)
    bb.append(b)

enc = pow(m,flag,p)
print(f'p = {p}')
print(f'aa = {aa}')
print(f'cc = {cc}')
print(f'enc = {enc}')


# p = 0x83b05d231fd40ff8ca26b4fb8136dc920754c14412960ce2ec700457861d48fe74f3958fc3a153f77a23fb850ecf0ac1e9722c71b6cc8a104b372cc17bf1528f
# aa = [0xf53f440f2e76b60380e68e96508f5dd716b2c3df2ed8265ced83a93fd61a708eeff31fbee9efa22fa7b441cbc406c210de6273f81eb7d093561d5c6394ef2abd,0xe9ecbd5457dcf1bdbc1a852b625c7a8ae6f530e348c2dc0416afd6a375aeb06d4800cc6471ae7d29681715d0407aa8726c32cd35e54960f56b0d9b47a2eed9ed,0x86a261c1590a774ced6c7db439e53e4068a8dbc0ad111a0e0371e8731fb939a068348d035da04afb9a3914a011574e35cefc4d5c5740f7cbc27459d944f51d15,0xba99a871d8f805e3c0dcd4e04cee66ec5e213a7902a65f2faa8e86368e56c42d09fde536b07471fff8f72db922725a24d6288d1bfe9edc2cd76b756eb464e1eb,0xa99efeb79377db6baa8787b5d2d2ebf123d8ec77e820d1b88644883a07c38498aa08df82cb9802c7fa128a5fa08b66c56a0805f70a78b7cb45b1a74bde095165,0xe6341800304b0a6f2de941432bb253f53a3c73c7b0f0382fd1ec4c882da5fd1b5151c619f9279de3767ae03af387c495c50c8c0ff79fd6c8acb51bd0b16afc23,0xa8670f7142e5d90e781e335d5e870188b94288defd8302c1b183bf20b5a3720be1ae1afe0ad937bc4678727e1daf262194c430086f3810447dfce721c8fb36a5,0xe9d150d4e7a7f82e89d26fcaa003579c8dfc25f4a50794175ae0a4407dd33e87eb5fded328155009548c002d2a3198afa356e1692a6bb820d8ae71ca6e506b35,0xd7ec72dd643449f5d21a32a11e2458d292c524b91b4b1a8515a5f8351717813f51a55c5675aa0eaa5c80a32d26aa385d7425e7d3e1e50936b2744a1534e3c7b7,0xaf5d8062f6bc03875b5fb10d0888d586cb08fa62709910cd7d931201d4d833d31d935003b801cac4fd51ef1e4db3bc13e41d740d7881560c0942faf9acca55d5,0xf63c4a5add0474bf6c4fb6617a2965e2474a2902c59bcc4243d5dcffa0f1aa0a6136e9c9093ea05d84a26888aaad63ec652602200141abdfcdc0d1912d208dbb,0xd65c8ef9b01085819f2f5f2dfbb7641521966600ddd7a03b886320328144ec42ed5585206744c8e1b5beb5502aff6f0cd01934890926bbb5387363b321dbb6bd,0x9bd7a0e9d4940126c9668de7e29e198ac38010d505b90e92ee560307b8a134545bbd0277f14e7651e91c5a9362207097e17888b9a49a889c672f76681f41962f,0xff387c4e94f47cba242936e80d620d3eeb203a1c7f365ae178b33b29ff3d8c2f932733d0605876c23942bae7f096b3ee457d02758796bda01bbcd3bd2e1229a1,0xbd6c24205cf5b9b2acc909296369ca77ef34ceac5f9900742a7ec37c00b94a56418d168d576d33383d9f782386524c4cecebc3a9fa68c81a5a6867de564d07e3,0x8850ac2051b6a7450f228e676abbfd510ce08a43a0b791182aedcfb6b3f1d478e3dc953e59eb99fe370f71d52af3a1625e0700078927f8a5919becbfa60af96b,0xfa474444b5372681c5b7316871cf9a93306b4d6a3eac2492c71780eb6903bd4ff77b1c4a28608dfa10c0c8b8bfc23942fc0a8ec64d3967504621e692eafcf4af,0xfd7f772c719929d33f15ce0122f6efc278b728d75dbd16a343c649e49118a79084d169d6db1b6b859e1f4c82694a850622dcdff23c8fbd7d0a736a409b94f471,0xedf6566ecd7f0a0beaa2f45a2a509358c9537dbf772e6f63ccb0e21b5c5c3dae0939e9e15c9bcdf80f7875e74ebfba16f0660737719c8ad435981b5b8df89a23,0xd7e718dafb753084cb7c1deee139f6e158a0228bbf9b0ae5ed3a8b3d2508ce62e4e32cd43a3135540b31e052e6e3a0dba3ae623c69d74da34387c3429b6f0487,0xbff5ab718c54909a34ba8ab5787cbfb0c9c57394350b71744d5751f577cefda87cc8263b21dd2c21a8f19fff8362c3371fcb054dddb1293df7d6efe2d661e4ab,0xb946ad47cf1f9186c1eeaeed12f21b41a29f9fb74b14f577c97731004b372c5f41023aafc9b93557c984ff87dd263b293aa500c3a2ce1f0815319263cf7c42d5,0x9b93853dc5f4c052fdda339f69c625bc98ffdbde439078458b84278e34542f9110863e5beef166a766f565f7a815ec4a462510b42f81454b65e648622c610b27,0xae050aaa86b9580747efa33561973597d7d7a0684a45219eed3224ebc37fdf067ed61a29ecaa501266c9bcdd16b850c0dd40da3f964a5b03ae60c3967fd55913,0xea57b0482a876d49a3e21bb885bda0aae21f5fcf2521f7e1db771998b6b639d0833c17d7c8c67c12ce9b60e4210e068f98344a8cabc5faa17a048100bab797bb,0x9f866ad27c9ea2fb29168ec6db0f7755221484d0c89bfef0cd70fc2ed14fda15f7b59bbca2e23f40b5effeb9e53ad821e06cf986b34da4407c85bfbf78d2920b,0xab0939605dcff5d1ec8405b6daeb65eccc6e3b5956601dddea95c6310ac32bfeec1bab6b83e85371078a16ea9489050175098d39488aaed2a190c647fe2b1b69,0xe85698e04d8bedbac2c7884c914b7026dbb1dd5134c4a5a8e7541b07c8a94ab3d2f12eb7f1171ccd564054e1dc63d5ac044e5c5552870b419fef35a572199239,0xea94663e103e4354fb7feb80b11d06c7e16feb7265f69ee882180baba70fa075df24e3fe1ad12a99f054cabf4a3f5e4823416d4c4daa02ca51ce3926034186db,0x93662718294005f4ac8f79b0799e240a05eea871ef07d623ea7c68ef818b3b55fb4f9b6f06c399726e59cec03389053b448f187404cddb93cb3c55e3d12cebc9]
# cc = [0x4aae29aaaab89fbc672db400c41d1ad3ffce937e7810065bed552c12101fb778046d22b00c05bbc3f61825b5af3c3e57f1abaafe3d9a58a573a905e2a1cecde7,0x502a1c23ef44f174aaedb9a49705eb72f805bab13e82d599525ea7484cc11f2e7c4475526b4be344390e46bcb8bcdaba2768c6321f8ca5482666171eee498f28,0x546f2276b59b7186ef5b9a04e0ba2691648d005fc780303411a1082ea3b05be127a10e26921a4b84b14d8acc45a6c32a0142ac6eb396415ecb5841a01b775b5d,0x43986eb7208b66dd86c12f953e10b2d3907873151170278393b6a4f7ea518ab5745e2db5f4dbec84087c4817c5df10e743f35ed1190515aba34832b1b274bf2e,0x7b2e448342bc7409ed891cdbf5137014f417866097297302085d8800458495e374fba8398d069f1c1c7792a9f03194e36921c378308de18313fb9b62db45cc0a,0x41ac8f265bc96ae868256fb08caf0a43a547346522b5d90cb9489c87cd5d726447d20354332dd3cb771003eddfa9c4bfa6923ae45ac8c0994f7cce46a3302eea,0x51b59ad39b388fdb2056279e2de02d32d36b52da1cc1fe4f6843964273b4585704e21405e2528e5894bd4fdb436c382a5f7d3849f9c5d7902be74edd2b16a31c,0x4e52d207f16a6f42497b25600e039de3ce49d7945ac2201ac12bf9fa9b3dc136f35328cbe9d3f9d4204d90f29fe1ce209c8904c99de0f85c5c572d05609095a6,0x6485f505cd8296e9f1c9abdc39eebc6e767e98e1587b18878fa8a582a012bb609c2f36daffc2b9c460514bc47533525ce6835e7f8123331a833958fa47f2e40,0x1ff809e71ab0347a8a9ec4356d5b8fbf109ade5881e5b59ac14adaaef2034fc40880e495070442513f42434fa8cbfcd9bce0501574997b35939f201ba1c87872,0x5b3fad73b402fcc1148758d80a61f637257d35f2773c8dc9f22859a01aecaa37a37858232ab5b3e3622f234bf35bb02a6e93ecd5a06182b61e583d0193afaf55,0x640e3556a526209f8528fbe678da3914d912a7e701abc3d4fcd65d84bcdbfc22623dfa3db31f9585a615c6de869b39c040dbfe94bb7eafd91bde15c4b87bd2c0,0x335b0ff42f39541539752faade7510a301a861720d85958ce2890ddca9e0693b342604a5b134a0974ea21dc0dc0a156c01159898e5f87e16b4a56121c2e3bbb5,0x89621740ec8b81065457105685666dfe6e31ee4f0a6efa6901e20ad6e5ea19ba438ea92f632764e52ccfb1ed35639443b5536f19ca69e9c295c6f5287a5e31b,0x1e4162060327045be0f5a0ae5d7b87c60b928e4669450339af64a39ae45b108eb58bbe83e15bef7a5c92243b213adc3e9d3514632249314bf5b588df6202ae10,0x5d10e308a16f8cd5c53b3ecd1081624e6eed4c3bfe522a8b7daf301aa6fc370b0a1cff1db7ea27aa6f5200903b365c53bc6d890ca1167ca57de87f80c5321742,0x434ac82b1ed073a1d73606ffe973e316768368fb522644f6cc76d2a0e6b83f533524c7589113aeb2db35d50c2f9ff64ceb27958a93d47b6875b8eb9b158ebaac,0x7cf3028273ece2a8af9790d3a689875bf7a3894e351a639d88fa13de05dc7f18be20ddeef4414a31b0f3dd65291e5d4d47d098027dbec25d4bc5409c017f0b03,0x5d0eefe7de32360d7950d96d821ce06741b737a1dca016cb003f75e380f5c8e3d9b8505b0eda21b36879a6455d1f640bdac5ad648b97d80eacf406b57bb1d692,0x4e4c6156da7f71123e1b4efde4a436985c6f2cccdc9b182735a75153927c6e7dc94693ab7fdd8646821a9d42eeb76c1be54984f81296a10e689805975185bf2c,0x68db6b837d968569ba944591efb36587f0022b05cefbedaad1e1d7652d0c233d1c0b036364f25e0865c7a1dab8d9d1081f23928a3573b5ec711e5b7f32714a4,0x333db46bc5fb83cbbf68475fd612bcad6becd30f95bf5b9ab6058ed0777fcb78cfe6da5c3386799fe9a1d6616801154ae7f45e35612de08efed7e2e750088b8e,0x70d4dc8986c228e12329af934c08f7c0ca94ce6913b5c641c59b3cf629a1957f82d4f40aaa7faa765c75e4f994a8222a4d08e045fb529da5ee277e1b9540c148,0x20421569b3bc73da4505760f7798a504f2276e6df9c9e48320b23201fb682021d168f6ee657fc2080722eb576ac78bf4b63ad7419983a05196fa0724fc4886d,0x5f523df584f43d7eff229d562793c8e5d4713d9c80bd40b95434c34c982324f7c282ac2fe3ac151cb62435b85bece2ef115c6f4b7a33d47dcbfa8360f89b91ac,0x7cbedea149640064a93cb75b909804ee7a896e17da808579206d1db7523b98bdf15bdc70e267e0a201bf293f980c1b5304337a7a78cc655aca07b9818b7169ab,0x62b34eaac3d3a2b360805cc23147ff8754a9d90788461107a5c8cc7053a0910ca7af45035d333c9a0b7cf6a2c13c9c367ad8eea0ccda2a6d8e089b5ee07a76ef,0x3e1c31f988e238869fad4794beb32164fa3bf3880041d1f9f2a65e2679f951491fcbeff1aa067313bf02100ae15d1af4d87050db05cec934e077c3eec238b72,0x3c056ad33432e1cf5548ae3a6db21ee1471eb70619e0ad542bd38dd80f37b76b571ae6469bacd33f9618b7a61b8e8424fc33cae479375df72b064a5d1b8cd90f,0x4caf081f5b949e65115e18b613ad8dc1fc208d2c5bea710b27b1db11d4a7eecc455c13f2fd92481f8cfea3e6fa75c0a58a154f12b6ce92c66107f617fd7ed7d8]
# enc = 0x191eb43459bd0f2d5ece00ab52c612668bb4c161014641a6e4afb41020465d7b82e9b60a55ab831bb5695f2fd832d08258c752ebf27ba0374b7b11b001b2629a

思路分析：

第一部分是HNP问题，第二部分是DLP问题。

$HNP$

HNP，即the Hidden Number Problem，隐藏数问题。本题中的是LHNP，即线性隐藏数问题。定义是从形如 $c_i = r_i \cdot x + e_i \pmod{p}$ 的一组方程组恢复。将单个方程可以转化为 $e_i = c_i - r_i \cdot x + k_i p$ 。

我们可以构造出矩阵：

其中 $K$ 为 $e_i$ 的上界， $k_i$ 为常系数。我们进行LLL算法即可得到该向量，并从中还原。

$DLP$

DLP（离散对数问题），由很多算法求解，对于本题。经过检测我们发现 $p-1$ 有很多小因子，因此我们可以认为 $p-1$ 是光滑的，那么就适用于Pohlig-Hellman算法进行求解，对于 $p-1$ 的最后一个大素数因子，我们采用爆破的方法求解最后几位。

完整exp如下：

from Crypto.Util.number import *
p= 0x83b05d231fd40ff8ca26b4fb8136dc920754c14412960ce2ec700457861d48fe74f3958fc3a153f77a23fb850ecf0ac1e9722c71b6cc8a104b372cc17bf1528f
rs =[0xf53f440f2e76b60380e68e96508f5dd716b2c3df2ed8265ced83a93fd61a708eeff31fbee9efa22fa7b441cbc406c210de6273f81eb7d093561d5c6394ef2abd,0xe9ecbd5457dcf1bdbc1a852b625c7a8ae6f530e348c2dc0416afd6a375aeb06d4800cc6471ae7d29681715d0407aa8726c32cd35e54960f56b0d9b47a2eed9ed,0x86a261c1590a774ced6c7db439e53e4068a8dbc0ad111a0e0371e8731fb939a068348d035da04afb9a3914a011574e35cefc4d5c5740f7cbc27459d944f51d15,0xba99a871d8f805e3c0dcd4e04cee66ec5e213a7902a65f2faa8e86368e56c42d09fde536b07471fff8f72db922725a24d6288d1bfe9edc2cd76b756eb464e1eb,0xa99efeb79377db6baa8787b5d2d2ebf123d8ec77e820d1b88644883a07c38498aa08df82cb9802c7fa128a5fa08b66c56a0805f70a78b7cb45b1a74bde095165,0xe6341800304b0a6f2de941432bb253f53a3c73c7b0f0382fd1ec4c882da5fd1b5151c619f9279de3767ae03af387c495c50c8c0ff79fd6c8acb51bd0b16afc23,0xa8670f7142e5d90e781e335d5e870188b94288defd8302c1b183bf20b5a3720be1ae1afe0ad937bc4678727e1daf262194c430086f3810447dfce721c8fb36a5,0xe9d150d4e7a7f82e89d26fcaa003579c8dfc25f4a50794175ae0a4407dd33e87eb5fded328155009548c002d2a3198afa356e1692a6bb820d8ae71ca6e506b35,0xd7ec72dd643449f5d21a32a11e2458d292c524b91b4b1a8515a5f8351717813f51a55c5675aa0eaa5c80a32d26aa385d7425e7d3e1e50936b2744a1534e3c7b7,0xaf5d8062f6bc03875b5fb10d0888d586cb08fa62709910cd7d931201d4d833d31d935003b801cac4fd51ef1e4db3bc13e41d740d7881560c0942faf9acca55d5,0xf63c4a5add0474bf6c4fb6617a2965e2474a2902c59bcc4243d5dcffa0f1aa0a6136e9c9093ea05d84a26888aaad63ec652602200141abdfcdc0d1912d208dbb,0xd65c8ef9b01085819f2f5f2dfbb7641521966600ddd7a03b886320328144ec42ed5585206744c8e1b5beb5502aff6f0cd01934890926bbb5387363b321dbb6bd,0x9bd7a0e9d4940126c9668de7e29e198ac38010d505b90e92ee560307b8a134545bbd0277f14e7651e91c5a9362207097e17888b9a49a889c672f76681f41962f,0xff387c4e94f47cba242936e80d620d3eeb203a1c7f365ae178b33b29ff3d8c2f932733d0605876c23942bae7f096b3ee457d02758796bda01bbcd3bd2e1229a1,0xbd6c24205cf5b9b2acc909296369ca77ef34ceac5f9900742a7ec37c00b94a56418d168d576d33383d9f782386524c4cecebc3a9fa68c81a5a6867de564d07e3,0x8850ac2051b6a7450f228e676abbfd510ce08a43a0b791182aedcfb6b3f1d478e3dc953e59eb99fe370f71d52af3a1625e0700078927f8a5919becbfa60af96b,0xfa474444b5372681c5b7316871cf9a93306b4d6a3eac2492c71780eb6903bd4ff77b1c4a28608dfa10c0c8b8bfc23942fc0a8ec64d3967504621e692eafcf4af,0xfd7f772c719929d33f15ce0122f6efc278b728d75dbd16a343c649e49118a79084d169d6db1b6b859e1f4c82694a850622dcdff23c8fbd7d0a736a409b94f471,0xedf6566ecd7f0a0beaa2f45a2a509358c9537dbf772e6f63ccb0e21b5c5c3dae0939e9e15c9bcdf80f7875e74ebfba16f0660737719c8ad435981b5b8df89a23,0xd7e718dafb753084cb7c1deee139f6e158a0228bbf9b0ae5ed3a8b3d2508ce62e4e32cd43a3135540b31e052e6e3a0dba3ae623c69d74da34387c3429b6f0487,0xbff5ab718c54909a34ba8ab5787cbfb0c9c57394350b71744d5751f577cefda87cc8263b21dd2c21a8f19fff8362c3371fcb054dddb1293df7d6efe2d661e4ab,0xb946ad47cf1f9186c1eeaeed12f21b41a29f9fb74b14f577c97731004b372c5f41023aafc9b93557c984ff87dd263b293aa500c3a2ce1f0815319263cf7c42d5,0x9b93853dc5f4c052fdda339f69c625bc98ffdbde439078458b84278e34542f9110863e5beef166a766f565f7a815ec4a462510b42f81454b65e648622c610b27,0xae050aaa86b9580747efa33561973597d7d7a0684a45219eed3224ebc37fdf067ed61a29ecaa501266c9bcdd16b850c0dd40da3f964a5b03ae60c3967fd55913,0xea57b0482a876d49a3e21bb885bda0aae21f5fcf2521f7e1db771998b6b639d0833c17d7c8c67c12ce9b60e4210e068f98344a8cabc5faa17a048100bab797bb,0x9f866ad27c9ea2fb29168ec6db0f7755221484d0c89bfef0cd70fc2ed14fda15f7b59bbca2e23f40b5effeb9e53ad821e06cf986b34da4407c85bfbf78d2920b,0xab0939605dcff5d1ec8405b6daeb65eccc6e3b5956601dddea95c6310ac32bfeec1bab6b83e85371078a16ea9489050175098d39488aaed2a190c647fe2b1b69,0xe85698e04d8bedbac2c7884c914b7026dbb1dd5134c4a5a8e7541b07c8a94ab3d2f12eb7f1171ccd564054e1dc63d5ac044e5c5552870b419fef35a572199239,0xea94663e103e4354fb7feb80b11d06c7e16feb7265f69ee882180baba70fa075df24e3fe1ad12a99f054cabf4a3f5e4823416d4c4daa02ca51ce3926034186db,0x93662718294005f4ac8f79b0799e240a05eea871ef07d623ea7c68ef818b3b55fb4f9b6f06c399726e59cec03389053b448f187404cddb93cb3c55e3d12cebc9]
cs =[0x4aae29aaaab89fbc672db400c41d1ad3ffce937e7810065bed552c12101fb778046d22b00c05bbc3f61825b5af3c3e57f1abaafe3d9a58a573a905e2a1cecde7,0x502a1c23ef44f174aaedb9a49705eb72f805bab13e82d599525ea7484cc11f2e7c4475526b4be344390e46bcb8bcdaba2768c6321f8ca5482666171eee498f28,0x546f2276b59b7186ef5b9a04e0ba2691648d005fc780303411a1082ea3b05be127a10e26921a4b84b14d8acc45a6c32a0142ac6eb396415ecb5841a01b775b5d,0x43986eb7208b66dd86c12f953e10b2d3907873151170278393b6a4f7ea518ab5745e2db5f4dbec84087c4817c5df10e743f35ed1190515aba34832b1b274bf2e,0x7b2e448342bc7409ed891cdbf5137014f417866097297302085d8800458495e374fba8398d069f1c1c7792a9f03194e36921c378308de18313fb9b62db45cc0a,0x41ac8f265bc96ae868256fb08caf0a43a547346522b5d90cb9489c87cd5d726447d20354332dd3cb771003eddfa9c4bfa6923ae45ac8c0994f7cce46a3302eea,0x51b59ad39b388fdb2056279e2de02d32d36b52da1cc1fe4f6843964273b4585704e21405e2528e5894bd4fdb436c382a5f7d3849f9c5d7902be74edd2b16a31c,0x4e52d207f16a6f42497b25600e039de3ce49d7945ac2201ac12bf9fa9b3dc136f35328cbe9d3f9d4204d90f29fe1ce209c8904c99de0f85c5c572d05609095a6,0x6485f505cd8296e9f1c9abdc39eebc6e767e98e1587b18878fa8a582a012bb609c2f36daffc2b9c460514bc47533525ce6835e7f8123331a833958fa47f2e40,0x1ff809e71ab0347a8a9ec4356d5b8fbf109ade5881e5b59ac14adaaef2034fc40880e495070442513f42434fa8cbfcd9bce0501574997b35939f201ba1c87872,0x5b3fad73b402fcc1148758d80a61f637257d35f2773c8dc9f22859a01aecaa37a37858232ab5b3e3622f234bf35bb02a6e93ecd5a06182b61e583d0193afaf55,0x640e3556a526209f8528fbe678da3914d912a7e701abc3d4fcd65d84bcdbfc22623dfa3db31f9585a615c6de869b39c040dbfe94bb7eafd91bde15c4b87bd2c0,0x335b0ff42f39541539752faade7510a301a861720d85958ce2890ddca9e0693b342604a5b134a0974ea21dc0dc0a156c01159898e5f87e16b4a56121c2e3bbb5,0x89621740ec8b81065457105685666dfe6e31ee4f0a6efa6901e20ad6e5ea19ba438ea92f632764e52ccfb1ed35639443b5536f19ca69e9c295c6f5287a5e31b,0x1e4162060327045be0f5a0ae5d7b87c60b928e4669450339af64a39ae45b108eb58bbe83e15bef7a5c92243b213adc3e9d3514632249314bf5b588df6202ae10,0x5d10e308a16f8cd5c53b3ecd1081624e6eed4c3bfe522a8b7daf301aa6fc370b0a1cff1db7ea27aa6f5200903b365c53bc6d890ca1167ca57de87f80c5321742,0x434ac82b1ed073a1d73606ffe973e316768368fb522644f6cc76d2a0e6b83f533524c7589113aeb2db35d50c2f9ff64ceb27958a93d47b6875b8eb9b158ebaac,0x7cf3028273ece2a8af9790d3a689875bf7a3894e351a639d88fa13de05dc7f18be20ddeef4414a31b0f3dd65291e5d4d47d098027dbec25d4bc5409c017f0b03,0x5d0eefe7de32360d7950d96d821ce06741b737a1dca016cb003f75e380f5c8e3d9b8505b0eda21b36879a6455d1f640bdac5ad648b97d80eacf406b57bb1d692,0x4e4c6156da7f71123e1b4efde4a436985c6f2cccdc9b182735a75153927c6e7dc94693ab7fdd8646821a9d42eeb76c1be54984f81296a10e689805975185bf2c,0x68db6b837d968569ba944591efb36587f0022b05cefbedaad1e1d7652d0c233d1c0b036364f25e0865c7a1dab8d9d1081f23928a3573b5ec711e5b7f32714a4,0x333db46bc5fb83cbbf68475fd612bcad6becd30f95bf5b9ab6058ed0777fcb78cfe6da5c3386799fe9a1d6616801154ae7f45e35612de08efed7e2e750088b8e,0x70d4dc8986c228e12329af934c08f7c0ca94ce6913b5c641c59b3cf629a1957f82d4f40aaa7faa765c75e4f994a8222a4d08e045fb529da5ee277e1b9540c148,0x20421569b3bc73da4505760f7798a504f2276e6df9c9e48320b23201fb682021d168f6ee657fc2080722eb576ac78bf4b63ad7419983a05196fa0724fc4886d,0x5f523df584f43d7eff229d562793c8e5d4713d9c80bd40b95434c34c982324f7c282ac2fe3ac151cb62435b85bece2ef115c6f4b7a33d47dcbfa8360f89b91ac,0x7cbedea149640064a93cb75b909804ee7a896e17da808579206d1db7523b98bdf15bdc70e267e0a201bf293f980c1b5304337a7a78cc655aca07b9818b7169ab,0x62b34eaac3d3a2b360805cc23147ff8754a9d90788461107a5c8cc7053a0910ca7af45035d333c9a0b7cf6a2c13c9c367ad8eea0ccda2a6d8e089b5ee07a76ef,0x3e1c31f988e238869fad4794beb32164fa3bf3880041d1f9f2a65e2679f951491fcbeff1aa067313bf02100ae15d1af4d87050db05cec934e077c3eec238b72,0x3c056ad33432e1cf5548ae3a6db21ee1471eb70619e0ad542bd38dd80f37b76b571ae6469bacd33f9618b7a61b8e8424fc33cae479375df72b064a5d1b8cd90f,0x4caf081f5b949e65115e18b613ad8dc1fc208d2c5bea710b27b1db11d4a7eecc455c13f2fd92481f8cfea3e6fa75c0a58a154f12b6ce92c66107f617fd7ed7d8]
enc=0x191eb43459bd0f2d5ece00ab52c612668bb4c161014641a6e4afb41020465d7b82e9b60a55ab831bb5695f2fd832d08258c752ebf27ba0374b7b11b001b2629a
#第一部分
t = len(rs)
kbits = 400
K = 2 ** kbits
P = identity_matrix(t) * p
RC = matrix([[-1, 0], [0, 1]]) * matrix([rs, cs])
KP = matrix([[K / p, 0], [0, K]])
M = block_matrix([[P, 0], [RC, KP]], subdivide=False)
shortest_vector = M.LLL()
x = shortest_vector[1, -2] / K * p % p
print(x)

#第二部分
G=GF(p)
factors, exps = zip(*factor(p - 1))
primes = [factors[i] ^ exps[i] for i in range(len(factors))]
print(primes)
dlogs = []
for fac in primes[:-1]:
    t = (p - 1) // fac
    dlog = discrete_log(G(pow(enc, t, p)), G(pow(x, t, p)))
    dlogs += [dlog]
s = (p - 1) // primes[-1]
print(s)
res = crt(dlogs, primes[:-1])
for i in range(100):
    if b'flag{'in long_to_bytes(res + i * s):
        print(long_to_bytes(res + i * s))
        break
        
#flag{70b1b709ce431682addb581596320007}

web

这网页怪怪的

第一部分是重定向，burp抓包发现隐含文件yunnuuu.php，然后tmp1利用php的弱比较绕过，tmp2使用伪协议读取flaaaaaaag.php文件内容为base64，然后解码即可

下面是网页源码：

# yunnuuu.php
<?php
include("flaaaaaaag.php");
highlight_file(__FILE__);
$tmp1=$_POST['tmp1'];
$tmp2=$_GET['tmp2'];
$secret=289114;
if(is_numeric($tmp1)){
    exit('Too early');}
else{

    if($tmp1==$secret){
        include($tmp2);
    }else{
        echo('you are close');
    }
}

misc

knn

根据题目描述KNN，使用KNN分类模型进行预测。使用Python机器学习模块 - sklearn - 对数据结果进行分类；

经过测试，发现需要对A车、B车要转换为二进制0和1再进行ascii解码。~~（没学过，只能照抄大佬的了）~~，不过只要调试好knn中n_neighbors的参数，就能得到最准确的flag

import pandas as pd
from sklearn.neighbors import KNeighborsClassifier
from sklearn.model_selection import train_test_split
from sklearn.metrics import accuracy_score

# 加载历史评分数据，指定编码为GBK
history_data = pd.read_csv(
    "新能源汽车检测数据.csv", encoding="GBK"
)

# 读取待检测新能源车数据
test_data = pd.read_csv(
    "待检测新能源车.csv",
    encoding="GBK",
)

# 提取特征和目标变量
X = history_data[["防碰撞评分", "电池容量评分", "智能驾驶能力", "智能座舱评分"]]
y = history_data["名称"]

# 划分训练集和测试集
X_train, X_test, y_train, y_test = train_test_split(
    X, y, test_size=0.2, random_state=42
)

# 创建KNN分类器，这里假设K取5，可根据实际情况调整
knn = KNeighborsClassifier(n_neighbors=300)

# 在训练集上训练模型
knn.fit(X_train, y_train)

# 在测试集上进行预测
y_pred = knn.predict(X_test)

# 计算模型的准确率
accuracy = accuracy_score(y_test, y_pred)
print(f"模型准确率:{accuracy}")

# 提取待检测新能源车数据的特征
test_X = test_data[["防碰撞评分", "电池容量评分", "智能驾驶能力", "智能座舱评分"]]

# 使用训练好的模型进行预测
test_pred = knn.predict(test_X)

# 将预测结果添加到待检测新能源车数据中
test_data["预测车型"] = test_pred

# 将结果保存为csv文件
csv_path = "待检测新能源车_预测结果.csv"
test_data.to_csv(csv_path, index=False, encoding="utf-8")

df = pd.read_csv(csv_path, encoding="utf-8")

# 将预测车型列中的A车替换为0，B车替换为1
df["预测车型"] = df["预测车型"].map({"A车": "0", "B车": "1"})

# 将预测车型列的所有值连接成一个字符串
binary_string = "".join(df["预测车型"])

# 按每8位一组分割二进制字符串
binary_chunks = [binary_string[i : i + 8] for i in range(0, len(binary_string), 8)]

# 将每个8位二进制字符串转换为ASCII字符
ascii_string = "".join([chr(int(chunk, 2)) for chunk in binary_chunks])

# 输出结果
print("转换后的ASCII码字符串为:", ascii_string)

其他组

crypto

simpleSignin

题目：

from Crypto.Util.number import *
from gmpy2 import *
import os

flag = b'xxx'
p = next_prime(bytes_to_long(os.urandom(128)))
q = next_prime(bytes_to_long(os.urandom(128)))
r = next_prime(q)
n = p * q * r
e = 0x10001
print(f"n = {n}")
print(f"c = {pow(bytes_to_long(flag), e, n)}")
print(f"gift1 = {p % (2**10)}")
print(f"gift2 = {(p >> 20) % 2 ** 800}")
# n = 91717197306065801430692774296739087369692505805873730729014813677164858033475119219496549179322145782790263228034134781592967028480301579462111507372893508636592832600206391905790511488678949157112322777098684707325311891056750963286494634489093620270797637437274546909400418496263799669541769586017282231886023275686719495040493703402244867906367008837217453500300992995258096509545406775279177918160331853363991834113918051468978309081085686108283547874975768959542753094631595260890420558364636303078263220001513817844063960023424429484568985727987064710176511050208253838039386390968276801658300795687804601169987051671314061987254212363117325786734328360418591971610392966867659045907550755979167652038093091970078722854251659581538266806207906127491377972897441916942048136016416739633568604447564119372465662628724153812001753748410162478969725179843125714619352895967577899670208386148053595763674920185320834513587
# c = 53725206995000716522893276595058419071990290621803579636161714383330892673055811337947487241701642126496591685585109862245420917684160662867863785840324861826954623093740844326627026833476771622577100464186879804184565843869885634004202583123814660253474988365767802371596929119773493402264073966760818738577604694066757843772483509464516822006312737285138313587227005339175914335841617310097530993158296079585719772401849963001042345007495440110071538584261056055469925721208755935971137618873034799819342505088130217626174789908762309465751064851354313099531229991764250968313733026934453793364342598912519963653648573385780950148182927905527658001218917373163825532068287661941387464083148002185635404194796699532320249403305428191157038659625440168989244227207215946636349083290983113055351713899941081154804665455470002453942340935361232121886299764516109079270616076269084847241248414373185392810503507977061708566426
# gift1 = 513
# gift2 = 5077110719426498428662246006638349628986894614097694065336047422264042823893900747327210766546701290926253205743419412459378571920759093322149140413682875156857171051511499793127787270654329155934268596972449238336868326196360992252498463385

思路分析：

p的位数是128*8=1024位，gift1是p的低10位，gift2是p的低20位到820位，所以我们需要爆破低10位到20位，得到p的低820位后，使用coppersmith定理即可求出高104位，cooper是根据已知部分位数利用小根来求解未知位数，使用前提是

$unknown\_bits÷p\_bits≤0.44$ $known\_bits÷p\_bits≥0.56$

这里显然是满足的，所以使用copper即可求出高204位，求出p后，直接求e在p-1下的逆元，无需求q和r

from Crypto.Util.number import *
import gmpy2
from tqdm import *
n = 91717197306065801430692774296739087369692505805873730729014813677164858033475119219496549179322145782790263228034134781592967028480301579462111507372893508636592832600206391905790511488678949157112322777098684707325311891056750963286494634489093620270797637437274546909400418496263799669541769586017282231886023275686719495040493703402244867906367008837217453500300992995258096509545406775279177918160331853363991834113918051468978309081085686108283547874975768959542753094631595260890420558364636303078263220001513817844063960023424429484568985727987064710176511050208253838039386390968276801658300795687804601169987051671314061987254212363117325786734328360418591971610392966867659045907550755979167652038093091970078722854251659581538266806207906127491377972897441916942048136016416739633568604447564119372465662628724153812001753748410162478969725179843125714619352895967577899670208386148053595763674920185320834513587
c = 53725206995000716522893276595058419071990290621803579636161714383330892673055811337947487241701642126496591685585109862245420917684160662867863785840324861826954623093740844326627026833476771622577100464186879804184565843869885634004202583123814660253474988365767802371596929119773493402264073966760818738577604694066757843772483509464516822006312737285138313587227005339175914335841617310097530993158296079585719772401849963001042345007495440110071538584261056055469925721208755935971137618873034799819342505088130217626174789908762309465751064851354313099531229991764250968313733026934453793364342598912519963653648573385780950148182927905527658001218917373163825532068287661941387464083148002185635404194796699532320249403305428191157038659625440168989244227207215946636349083290983113055351713899941081154804665455470002453942340935361232121886299764516109079270616076269084847241248414373185392810503507977061708566426
leak = 5077110719426498428662246006638349628986894614097694065336047422264042823893900747327210766546701290926253205743419412459378571920759093322149140413682875156857171051511499793127787270654329155934268596972449238336868326196360992252498463385
e = 65537
PR.<x> = PolynomialRing(Zmod(n))
for i in trange(2**10):
    p_low = (leak<<20)+(i<<10)+513
    f = x*2**820+p_low
    root = f.monic().small_roots(X=2^204,beta=0.33)
    if root:
        p = int(root[0]*2**820+p_low)
        if n%p==0:
            phi = p-1
            d = inverse_mod(e,phi)
            m = pow(c,d,p)
            flag = bytes.fromhex(hex(m)[2:])
            print(flag)
            break

eazy_crypto

from secret import flag
from Crypto.Util.number import getPrime

flag = bin(int.from_bytes(flag, 'big'))[2:]

private_key = []
g = getPrime(10)
private_key.append(g)
for i inrange(len(flag) - 1):
    g = g * 2
    private_key.append(g)

a = getPrime(20)
b = getPrime(len(flag) + 20)
public_key = []
for i in private_key:
    public_key.append((a * i) % b)
print(public_key)

c = 0
for i inrange(len(flag)):
    c += int(str(flag)[i])*public_key[i]
print(c)

'''
[806186273, 1612372546, 3224745092, 6449490184, 12898980368, 25797960736, 51595921472, 103191842944, 206383685888, 412767371776, 825534743552, 1651069487104, 3302138974208, 6604277948416, 13208555896832, 26417111793664, 52834223587328, 105668447174656, 211336894349312, 422673788698624, 845347577397248, 1690695154794496, 3381390309588992, 6762780619177984, 13525561238355968, 27051122476711936, 54102244953423872, 108204489906847744, 216408979813695488, 432817959627390976, 865635919254781952, 1731271838509563904, 3462543677019127808, 6925087354038255616, 13850174708076511232, 27700349416153022464, 55400698832306044928, 110801397664612089856, 221602795329224179712, 443205590658448359424, 886411181316896718848, 1772822362633793437696, 3545644725267586875392, 7091289450535173750784, 14182578901070347501568, 28365157802140695003136, 56730315604281390006272, 113460631208562780012544, 226921262417125560025088, 453842524834251120050176, 907685049668502240100352, 1815370099337004480200704, 3630740198674008960401408, 7261480397348017920802816, 14522960794696035841605632, 29045921589392071683211264, 58091843178784143366422528, 116183686357568286732845056, 232367372715136573465690112, 464734745430273146931380224, 929469490860546293862760448, 1858938981721092587725520896, 3717877963442185175451041792, 7435755926884370350902083584, 14871511853768740701804167168, 29743023707537481403608334336, 59486047415074962807216668672, 118972094830149925614433337344, 237944189660299851228866674688, 475888379320599702457733349376, 951776758641199404915466698752, 1903553517282398809830933397504, 3807107034564797619661866795008, 7614214069129595239323733590016, 15228428138259190478647467180032, 30456856276518380957294934360064, 60913712553036761914589868720128, 121827425106073523829179737440256, 243654850212147047658359474880512, 487309700424294095316718949761024, 974619400848588190633437899522048, 1949238801697176381266875799044096, 3898477603394352762533751598088192, 7796955206788705525067503196176384, 15593910413577411050135006392352768, 31187820827154822100270012784705536, 62375641654309644200540025569411072, 124751283308619288401080051138822144, 249502566617238576802160102277644288, 499005133234477153604320204555288576, 998010266468954307208640409110577152, 1996020532937908614417280818221154304, 3992041065875817228834561636442308608, 7984082131751634457669123272884617216, 15968164263503268915338246545769234432, 31936328527006537830676493091538468864, 63872657054013075661352986183076937728, 127745314108026151322705972366153875456, 255490628216052302645411944732307750912, 510981256432104605290823889464615501824, 1021962512864209210581647778929231003648, 2043925025728418421163295557858462007296, 4087850051456836842326591115716924014592, 8175700102913673684653182231433848029184, 16351400205827347369306364462867696058368, 32702800411654694738612728925735392116736, 65405600823309389477225457851470784233472, 130811201646618778954450915702941568466944, 261622403293237557908901831405883136933888, 523244806586475115817803662811766273867776, 1046489613172950231635607325623532547735552, 2092979226345900463271214651247065095471104, 4185958452691800926542429302494130190942208, 8371916905383601853084858604988260381884416, 16743833810767203706169717209976520763768832, 33487667621534407412339434419953041527537664, 66975335243068814824678868839906083055075328, 133950670486137629649357737679812166110150656, 267901340972275259298715475359624332220301312, 535802681944550518597430950719248664440602624, 1071605363889101037194861901438497328881205248, 2143210727778202074389723802876994657762410496, 4286421455556404148779447605753989315524820992, 8572842911112808297558895211507978631049641984, 17145685822225616595117790423015957262099283968, 34291371644451233190235580846031914524198567936, 68582743288902466380471161692063829048397135872, 137165486577804932760942323384127658096794271744, 274330973155609865521884646768255316193588543488, 548661946311219731043769293536510632387177086976, 1097323892622439462087538587073021264774354173952, 2194647785244878924175077174146042529548708347904, 4389295570489757848350154348292085059097416695808, 8778591140979515696700308696584170118194833391616, 17557182281959031393400617393168340236389666783232, 35114364563918062786801234786336680472779333566464, 70228729127836125573602469572673360945558667132928, 140457458255672251147204939145346721891117334265856, 280914916511344502294409878290693443782234668531712, 561829833022689004588819756581386887564469337063424, 1123659666045378009177639513162773775128938674126848, 2247319332090756018355279026325547550257877348253696, 4494638664181512036710558052651095100515754696507392, 8989277328363024073421116105302190201031509393014784, 17978554656726048146842232210604380402063018786029568, 35957109313452096293684464421208760804126037572059136, 71914218626904192587368928842417521608252075144118272, 143828437253808385174737857684835043216504150288236544, 287656874507616770349475715369670086433008300576473088, 575313749015233540698951430739340172866016601152946176, 1150627498030467081397902861478680345732033202305892352, 2301254996060934162795805722957360691464066404611784704, 4602509992121868325591611445914721382928132809223569408, 9205019984243736651183222891829442765856265618447138816, 18410039968487473302366445783658885531712531236894277632, 36820079936974946604732891567317771063425062473788555264, 73640159873949893209465783134635542126850124947577110528, 147280319747899786418931566269271084253700249895154221056, 294560639495799572837863132538542168507400499790308442112, 589121278991599145675726265077084337014800999580616884224, 1178242557983198291351452530154168674029601999161233768448, 2356485115966396582702905060308337348059203998322467536896, 4712970231932793165405810120616674696118407996644935073792, 9425940463865586330811620241233349392236815993289870147584, 18851880927731172661623240482466698784473631986579740295168, 37703761855462345323246480964933397568947263973159480590336, 75407523710924690646492961929866795137894527946318961180672, 150815047421849381292985923859733590275789055892637922361344, 301630094843698762585971847719467180551578111785275844722688, 603260189687397525171943695438934361103156223570551689445376, 1206520379374795050343887390877868722206312447141103378890752, 2413040758749590100687774781755737444412624894282206757781504, 4826081517499180201375549563511474888825249788564413515563008, 9652163034998360402751099127022949777650499577128827031126016, 19304326069996720805502198254045899555300999154257654062252032, 38608652139993441611004396508091799110601998308515308124504064, 77217304279986883222008793016183598221203996617030616249008128, 154434608559973766444017586032367196442407993234061232498016256, 308869217119947532888035172064734392884815986468122464996032512, 617738434239895065776070344129468785769631972936244929992065024, 1235476868479790131552140688258937571539263945872489859984130048, 2470953736959580263104281376517875143078527891744979719968260096, 2271772865869042562343933210469277026878003414201539042867658625, 1873411123687967160823236878372080794476954459114657688666455683, 1076687639325816357781844214177688329674856548940894980264049799, 2153375278651632715563688428355376659349713097881789960528099598, 1636615949253147467262747314144280059420373826475159523987337629, 603097290456176970660865085722086859561695283661898650905813691, 1206194580912353941321730171444173719123390567323797301811627382, 2412389161824707882643460342888347438246781134647594603623254764, 2154643715599297801422291143210221617214509900006768810177647961]
9300864384064097558471271281235466391616213967975393508307623247

1	c += int(str(flag)[i])*public_key[i]

思路分析：

经典的背包加密方案，先生成私钥，每一位都是前一位的2倍，私钥个数是flag的二进制长度；然后使用LCG生成公钥，flag个数为二进制长度；然后公钥依次和flag二进制位相乘再求和

可以看出刚开始的LCG中 $a * i<<b$ ，当i逐渐变大，直到最后i的二进制位数是10+len(flag)， $a*i$ 的位数就是30+len(flag)>20+len(flag)=b的位数，所以public_key的后十个数据都进行了一次有效的模b，其他的数mod b还是其本身；

以最后两个数据为例：

x=2412389161824707882643460342888347438246781134647594603623254764，y=2154643715599297801422291143210221617214509900006768810177647961

$b=2^{len(flag) + 20},大概就是这个长度\\$ $x=2^{(30+len(flag)-1)}\pmod{b}$ $y=2^{(30+len(flag)}\pmod{b}$ $y=2*2^{30+len(flag)-1}\pmod{b}=2x\pmod{b}$ $所以,2x模一次b就是y（这里根据x和b的大小关系能够看出来，x就是b的一倍多大）$ $即，y=2x-b，b=2x-y$

这里x和y的取值只要是public_key的后十个数据中任意相邻的两个即可；然后求a，因为当i是private_key中第一个数据g时， $a*g<<b$ ，所以直接对publick_key中第一个数进行分解，分解得到列表的第二个就是a（因为a>g），得到a和g后，按照源代码正向逻辑生成 $private\_key$

还有一种反向方法得到private_key

$因为，public\_key=a*i\pmod{b}\\$ $所以，i=public\_key*a^{−1}\pmod{b}$

1
2
3

private_key = []
for i in public_key:
    private_key.append(int(i*pow(a,-1,b)%b))

得到private_key后，只需要解决背包加密即可：

$已知，c += int(str(flag)[i])*public\_key[i]\pmod{b}\\$ $因为public\_key=a * private\_key\pmod{b})$ $所以，c+=int(str(flag)[i])*a * private\_key\pmod{b})$ $c*a^{-1}=int(str(flag)[i])*private\_key\pmod{b}$

所以我们只需要比较 $c*a^{-1}和private\_key$ 的大小，如果前者大于后者，说明flag的最高位是1，同时 $令c=c-private\_key$ ，反之flag的最高位是0（这个判断条件是在 $private\_Key$ 的逆循环中进行的）,即可得到最终flag

最终exp如下：

import gmpy2
from sympy import isprime, factor
from sympy import factorint

public_key=[806186273, 1612372546, 3224745092, 6449490184, 12898980368, 25797960736, 51595921472, 103191842944, 206383685888, 412767371776, 825534743552, 1651069487104, 3302138974208, 6604277948416, 13208555896832, 26417111793664, 52834223587328, 105668447174656, 211336894349312, 422673788698624, 845347577397248, 1690695154794496, 3381390309588992, 6762780619177984, 13525561238355968, 27051122476711936, 54102244953423872, 108204489906847744, 216408979813695488, 432817959627390976, 865635919254781952, 1731271838509563904, 3462543677019127808, 6925087354038255616, 13850174708076511232, 27700349416153022464, 55400698832306044928, 110801397664612089856, 221602795329224179712, 443205590658448359424, 886411181316896718848, 1772822362633793437696, 3545644725267586875392, 7091289450535173750784, 14182578901070347501568, 28365157802140695003136, 56730315604281390006272, 113460631208562780012544, 226921262417125560025088, 453842524834251120050176, 907685049668502240100352, 1815370099337004480200704, 3630740198674008960401408, 7261480397348017920802816, 14522960794696035841605632, 29045921589392071683211264, 58091843178784143366422528, 116183686357568286732845056, 232367372715136573465690112, 464734745430273146931380224, 929469490860546293862760448, 1858938981721092587725520896, 3717877963442185175451041792, 7435755926884370350902083584, 14871511853768740701804167168, 29743023707537481403608334336, 59486047415074962807216668672, 118972094830149925614433337344, 237944189660299851228866674688, 475888379320599702457733349376, 951776758641199404915466698752, 1903553517282398809830933397504, 3807107034564797619661866795008, 7614214069129595239323733590016, 15228428138259190478647467180032, 30456856276518380957294934360064, 60913712553036761914589868720128, 121827425106073523829179737440256, 243654850212147047658359474880512, 487309700424294095316718949761024, 974619400848588190633437899522048, 1949238801697176381266875799044096, 3898477603394352762533751598088192, 7796955206788705525067503196176384, 15593910413577411050135006392352768, 31187820827154822100270012784705536, 62375641654309644200540025569411072, 124751283308619288401080051138822144, 249502566617238576802160102277644288, 499005133234477153604320204555288576, 998010266468954307208640409110577152, 1996020532937908614417280818221154304, 3992041065875817228834561636442308608, 7984082131751634457669123272884617216, 15968164263503268915338246545769234432, 31936328527006537830676493091538468864, 63872657054013075661352986183076937728, 127745314108026151322705972366153875456, 255490628216052302645411944732307750912, 510981256432104605290823889464615501824, 1021962512864209210581647778929231003648, 2043925025728418421163295557858462007296, 4087850051456836842326591115716924014592, 8175700102913673684653182231433848029184, 16351400205827347369306364462867696058368, 32702800411654694738612728925735392116736, 65405600823309389477225457851470784233472, 130811201646618778954450915702941568466944, 261622403293237557908901831405883136933888, 523244806586475115817803662811766273867776, 1046489613172950231635607325623532547735552, 2092979226345900463271214651247065095471104, 4185958452691800926542429302494130190942208, 8371916905383601853084858604988260381884416, 16743833810767203706169717209976520763768832, 33487667621534407412339434419953041527537664, 66975335243068814824678868839906083055075328, 133950670486137629649357737679812166110150656, 267901340972275259298715475359624332220301312, 535802681944550518597430950719248664440602624, 1071605363889101037194861901438497328881205248, 2143210727778202074389723802876994657762410496, 4286421455556404148779447605753989315524820992, 8572842911112808297558895211507978631049641984, 17145685822225616595117790423015957262099283968, 34291371644451233190235580846031914524198567936, 68582743288902466380471161692063829048397135872, 137165486577804932760942323384127658096794271744, 274330973155609865521884646768255316193588543488, 548661946311219731043769293536510632387177086976, 1097323892622439462087538587073021264774354173952, 2194647785244878924175077174146042529548708347904, 4389295570489757848350154348292085059097416695808, 8778591140979515696700308696584170118194833391616, 17557182281959031393400617393168340236389666783232, 35114364563918062786801234786336680472779333566464, 70228729127836125573602469572673360945558667132928, 140457458255672251147204939145346721891117334265856, 280914916511344502294409878290693443782234668531712, 561829833022689004588819756581386887564469337063424, 1123659666045378009177639513162773775128938674126848, 2247319332090756018355279026325547550257877348253696, 4494638664181512036710558052651095100515754696507392, 8989277328363024073421116105302190201031509393014784, 17978554656726048146842232210604380402063018786029568, 35957109313452096293684464421208760804126037572059136, 71914218626904192587368928842417521608252075144118272, 143828437253808385174737857684835043216504150288236544, 287656874507616770349475715369670086433008300576473088, 575313749015233540698951430739340172866016601152946176, 1150627498030467081397902861478680345732033202305892352, 2301254996060934162795805722957360691464066404611784704, 4602509992121868325591611445914721382928132809223569408, 9205019984243736651183222891829442765856265618447138816, 18410039968487473302366445783658885531712531236894277632, 36820079936974946604732891567317771063425062473788555264, 73640159873949893209465783134635542126850124947577110528, 147280319747899786418931566269271084253700249895154221056, 294560639495799572837863132538542168507400499790308442112, 589121278991599145675726265077084337014800999580616884224, 1178242557983198291351452530154168674029601999161233768448, 2356485115966396582702905060308337348059203998322467536896, 4712970231932793165405810120616674696118407996644935073792, 9425940463865586330811620241233349392236815993289870147584, 18851880927731172661623240482466698784473631986579740295168, 37703761855462345323246480964933397568947263973159480590336, 75407523710924690646492961929866795137894527946318961180672, 150815047421849381292985923859733590275789055892637922361344, 301630094843698762585971847719467180551578111785275844722688, 603260189687397525171943695438934361103156223570551689445376, 1206520379374795050343887390877868722206312447141103378890752, 2413040758749590100687774781755737444412624894282206757781504, 4826081517499180201375549563511474888825249788564413515563008, 9652163034998360402751099127022949777650499577128827031126016, 19304326069996720805502198254045899555300999154257654062252032, 38608652139993441611004396508091799110601998308515308124504064, 77217304279986883222008793016183598221203996617030616249008128, 154434608559973766444017586032367196442407993234061232498016256, 308869217119947532888035172064734392884815986468122464996032512, 617738434239895065776070344129468785769631972936244929992065024, 1235476868479790131552140688258937571539263945872489859984130048, 2470953736959580263104281376517875143078527891744979719968260096, 2271772865869042562343933210469277026878003414201539042867658625, 1873411123687967160823236878372080794476954459114657688666455683, 1076687639325816357781844214177688329674856548940894980264049799, 2153375278651632715563688428355376659349713097881789960528099598, 1636615949253147467262747314144280059420373826475159523987337629, 603097290456176970660865085722086859561695283661898650905813691, 1206194580912353941321730171444173719123390567323797301811627382, 2412389161824707882643460342888347438246781134647594603623254764, 2154643715599297801422291143210221617214509900006768810177647961]
c = 9300864384064097558471271281235466391616213967975393508307623247
x = 2412389161824707882643460342888347438246781134647594603623254764
y = 2154643715599297801422291143210221617214509900006768810177647961

# 计算a和b
b = 2 * x - y
n = public_key[0]
factors = factorint(n)
g = list(factors.keys())[0]
a = list(factors.keys())[1]

# 得到private_key
private_key = []
private_key.append(g)
for i in range(len(public_key) - 1):
    g = g * 2
    private_key.append(g)
print(private_key)

# 计算c*a^(-1)的结果
c = c * pow(a, -1, b) % b

# 倒循环计算出flag的高位-->低位
flag = ""
for i in private_key[::-1]:
    if c >= i:
        c -= i
        flag = "1" + flag
    else:
        flag = "0" + flag

print(bytes.fromhex(hex(int(flag, 2))[2:]))

# [787, 1574, 3148, 6296, 12592, 25184, 50368, 100736, 201472, 402944, 805888, 1611776, 3223552, 6447104, 12894208, 25788416, 51576832, 103153664, 206307328, 412614656, 825229312, 1650458624, 3300917248, 6601834496, 13203668992, 26407337984, 52814675968, 105629351936, 211258703872, 422517407744, 845034815488, 1690069630976, 3380139261952, 6760278523904, 13520557047808, 27041114095616, 54082228191232, 108164456382464, 216328912764928, 432657825529856, 865315651059712, 1730631302119424, 3461262604238848, 6922525208477696, 13845050416955392, 27690100833910784, 55380201667821568, 110760403335643136, 221520806671286272, 443041613342572544, 886083226685145088, 1772166453370290176, 3544332906740580352, 7088665813481160704, 14177331626962321408, 28354663253924642816, 56709326507849285632, 113418653015698571264, 226837306031397142528, 453674612062794285056, 907349224125588570112, 1814698448251177140224, 3629396896502354280448, 7258793793004708560896, 14517587586009417121792, 29035175172018834243584, 58070350344037668487168, 116140700688075336974336, 232281401376150673948672, 464562802752301347897344, 929125605504602695794688, 1858251211009205391589376, 3716502422018410783178752, 7433004844036821566357504, 14866009688073643132715008, 29732019376147286265430016, 59464038752294572530860032, 118928077504589145061720064, 237856155009178290123440128, 475712310018356580246880256, 951424620036713160493760512, 1902849240073426320987521024, 3805698480146852641975042048, 7611396960293705283950084096, 15222793920587410567900168192, 30445587841174821135800336384, 60891175682349642271600672768, 121782351364699284543201345536, 243564702729398569086402691072, 487129405458797138172805382144, 974258810917594276345610764288, 1948517621835188552691221528576, 3897035243670377105382443057152, 7794070487340754210764886114304, 15588140974681508421529772228608, 31176281949363016843059544457216, 62352563898726033686119088914432, 124705127797452067372238177828864, 249410255594904134744476355657728, 498820511189808269488952711315456, 997641022379616538977905422630912, 1995282044759233077955810845261824, 3990564089518466155911621690523648, 7981128179036932311823243381047296, 15962256358073864623646486762094592, 31924512716147729247292973524189184, 63849025432295458494585947048378368, 127698050864590916989171894096756736, 255396101729181833978343788193513472, 510792203458363667956687576387026944, 1021584406916727335913375152774053888, 2043168813833454671826750305548107776, 4086337627666909343653500611096215552, 8172675255333818687307001222192431104, 16345350510667637374614002444384862208, 32690701021335274749228004888769724416, 65381402042670549498456009777539448832, 130762804085341098996912019555078897664, 261525608170682197993824039110157795328, 523051216341364395987648078220315590656, 1046102432682728791975296156440631181312, 2092204865365457583950592312881262362624, 4184409730730915167901184625762524725248, 8368819461461830335802369251525049450496, 16737638922923660671604738503050098900992, 33475277845847321343209477006100197801984, 66950555691694642686418954012200395603968, 133901111383389285372837908024400791207936, 267802222766778570745675816048801582415872, 535604445533557141491351632097603164831744, 1071208891067114282982703264195206329663488, 2142417782134228565965406528390412659326976, 4284835564268457131930813056780825318653952, 8569671128536914263861626113561650637307904, 17139342257073828527723252227123301274615808, 34278684514147657055446504454246602549231616, 68557369028295314110893008908493205098463232, 137114738056590628221786017816986410196926464, 274229476113181256443572035633972820393852928, 548458952226362512887144071267945640787705856, 1096917904452725025774288142535891281575411712, 2193835808905450051548576285071782563150823424, 4387671617810900103097152570143565126301646848, 8775343235621800206194305140287130252603293696, 17550686471243600412388610280574260505206587392, 35101372942487200824777220561148521010413174784, 70202745884974401649554441122297042020826349568, 140405491769948803299108882244594084041652699136, 280810983539897606598217764489188168083305398272, 561621967079795213196435528978376336166610796544, 1123243934159590426392871057956752672333221593088, 2246487868319180852785742115913505344666443186176, 4492975736638361705571484231827010689332886372352, 8985951473276723411142968463654021378665772744704, 17971902946553446822285936927308042757331545489408, 35943805893106893644571873854616085514663090978816, 71887611786213787289143747709232171029326181957632, 143775223572427574578287495418464342058652363915264, 287550447144855149156574990836928684117304727830528, 575100894289710298313149981673857368234609455661056, 1150201788579420596626299963347714736469218911322112, 2300403577158841193252599926695429472938437822644224, 4600807154317682386505199853390858945876875645288448, 9201614308635364773010399706781717891753751290576896, 18403228617270729546020799413563435783507502581153792, 36806457234541459092041598827126871567015005162307584, 73612914469082918184083197654253743134030010324615168, 147225828938165836368166395308507486268060020649230336, 294451657876331672736332790617014972536120041298460672, 588903315752663345472665581234029945072240082596921344, 1177806631505326690945331162468059890144480165193842688, 2355613263010653381890662324936119780288960330387685376, 4711226526021306763781324649872239560577920660775370752, 9422453052042613527562649299744479121155841321550741504, 18844906104085227055125298599488958242311682643101483008, 37689812208170454110250597198977916484623365286202966016, 75379624416340908220501194397955832969246730572405932032, 150759248832681816441002388795911665938493461144811864064, 301518497665363632882004777591823331876986922289623728128, 603036995330727265764009555183646663753973844579247456256, 1206073990661454531528019110367293327507947689158494912512, 2412147981322909063056038220734586655015895378316989825024, 4824295962645818126112076441469173310031790756633979650048, 9648591925291636252224152882938346620063581513267959300096, 19297183850583272504448305765876693240127163026535918600192, 38594367701166545008896611531753386480254326053071837200384, 77188735402333090017793223063506772960508652106143674400768, 154377470804666180035586446127013545921017304212287348801536, 308754941609332360071172892254027091842034608424574697603072, 617509883218664720142345784508054183684069216849149395206144, 1235019766437329440284691569016108367368138433698298790412288]
# b'flag{e2zy_puBlic_crypt0}'