Crypto：

SimpleRSA

wiener attack
利用自定义的m检验生成的d‘来判断phi是否正确

e=1072295425944136507039938677101442481213519408125148233880442849206353379681989305000570387093152236263203395726974692959819315410781180094216209100069530791407495510882640781920564732214327898099944792714253622047873152630438060151644601786843683746256407925709702163565141004356238879406385566586704226148537863811717298966607314747737551724379516675376634771455883976069007134218982435170160647848549412289128982070647832774446345062489374092673169618836701679
n=1827221992692849179244069834273816565714276505305246103435962887461520381709739927223055239953965182451252194768935702628056587034173800605827424043281673183606478736189927377745575379908876456485016832416806029254972769617393560238494326078940842295153029285394491783712384990125100774596477064482280829407856014835231711788990066676534414414741067759564102331614666713797073811245099512130528600464099492734671689084990036077860042238454908960841595107122933173
c=1079929174110820494059355415059104229905268763089157771374657932646711017488701536460687319648362549563313125268069722412148023885626962640915852317297916421725818077814237292807218952574111141918158391190621362508862842932945783059181952614317289116405878741758913351697905289993651105968169193211242144991434715552952340791545323270065763529865010326192824334684413212357708275259096202509042838081150055727650443887438253964607414944245877904002580997866300452from gmpy2 import invert
from Crypto.Util.number import *
#wiener_attack
def possible(e,alist,N): for x in alist:if x==0:continuephi = floor(e*(1/x))if phi%8==0:try :d = invert(e,phi)except:continuec = pow(3,e,n)if pow(c,d,n) == 3:return delse:continuedef wiener_attack(e,N):c=continued_fraction(e/N)alist=c.convergents()return possible(e,alist,N)d=wiener_attack(e,n)
print d
print  long_to_bytes(pow(c,d,n))

more_calc

其实很简单，p看似很大，其实q更大，所以直接化简N，用m mod p

#more_calc
from Crypto.Util.number import *
p=27405107041753266489145388621858169511872996622765267064868542117269875531364939896671662734188734825462948115530667205007939029215517180761866791579330410449202307248373229224662232822180397215721163369151115019770596528704719472424551024516928606584975793350814943997731939996459959720826025110179216477709373849945411483731524831284895024319654509286305913312306154387754998813276562173335189450448233216133842189148761197948559529960144453513191372254902031168755165124218783504740834442379363311489108732216051566953498279198537794620521800773917228002402970358087033504897205021881295154046656335865303621793069
c=350559186837488832821747843236518135605207376031858002274245004287622649330215113818719954185397072838014144973032329600905419861908678328971318153205085007743269253957395282420325663132161022100365481003745940818974280988045034204540385744572806102552420428326265541925346702843693366991753468220300070888651732502520797002707248604275755144713421649971492440442052470723153111156457558558362147002004646136522011344261017461901953583462467622428810167107079281190209731251995976003352201766861887320739990258601550606005388872967825179626176714503475557883810543445555390014562686801894528311600623156984829864743222963877167099892926717479789226681810584894066635076755996423203380493776130488170859798745677727810528672150350333480506424506676127108526488370011099147698875070043925524217837379654168009179798131378352623177947753192948012574831777413729910050668759007704596447625484384743880766558428224371417726480372362810572395522725083798926133468409600491925317437998458582723897120786458219630275616949619564099733542766297770682044561605344090394777570973725211713076201846942438883897078408067779325471589907041186423781580046903588316958615443196819133852367565049467076710376395085898875495653237178198379421129086523
e = 0x10001
c1=c%p
d1= inverse_mod(e,p-1)
print long_to_bytes(pow(c1,d1,p))

RSAssss

N=p*q*next_prime(p)*next_prime(q)

所以N存在两个十分相近的数

n1= p * next_prime( q )
n2= q * next_prime( p )

利用yafu分解可以得到n1和n2

又因为素数间隔很近，所以可以变成
n1=p∗(q+y)n2=q∗(p+x)n1 = p * (q+y) \\ n2 = q*(p+x) n1=p∗(q+y)n2=q∗(p+x)此时x,y<1500x,y<1500x,y<1500

所以可以通过爆破x,yx,yx,y得到两条方程组，解方程即可。

#RSAssss
from gmpy2 import *
from Crypto.Util.number import *
c=3304124639719334349997663632110579306673932777705840648575774671427424134287680988314129312593361087606243819528298610131797078262351307396831985397555390640151391138633431951746748156610463582479645561779194981806129898009876517899450840875569675976765155608446799203699927448835004756707151281044859676695533373755798273892503194753948997947653100690841880925445059175494314198605475023939567750409907217654291430615102258523998394231436796902635077995829477347316754739938980814293304289318417443493019704073164585505217658570214989150175123757038125380996050761572021986573934155470641091678664451080065719261207
e=0x10001
n=8030860507195481656424331455231443135773524476536419534745106637165762909478292141556846892146553555609301914884176422322286739546193682236355823149096731058044933046552926707682168435727800175783373045726692093694148718521610590523718813096895883533245331244650675812406540694948121258394822022998773233400623162137949381772195351339548977422564546054188918542382088471666795842185019002025083543162991739309935972705871943787733784491735500905013651061284020447578230135075211268405413254368439549259917312445348808412659422810647972872286215701325216318641985498202349281374905892279894612835009186944143298761257
n1=89615068527538836315602124154008300286636934599617334867509053076622715365809371740037316558871796433906844464070995869293654082577887578197182408045175035339285085728002838220314068474670975228778464240088084331807420720121364486765011169669747553393661650912114228227308579940164269877101973728452252879383
n2=n//n1
def get_q():for a in range(1500):for b in range(1500):c=(pow(n1-n2+a*b,2)+4*a*n2*b)if iroot(c,2)[1] is True:try:q = (-(n1+a*b-n2)+iroot(c,2)[0])/(2*a)if q>0 and  n%q==0 :return a,b,qexcept:continuea,b,q1 = get_q()
p1=n1//(q1+b)
p2=p1+a
q2=q1+b
phi=(p1-1)*(p2-1)*(q1-1)*(q2-1)
d=invert(e,phi)
print long_to_bytes(pow(c,d,n))

Exposure

已知dp高位，可以通过化简构造f(x)利用coppersmith 解决。

已知dp∗e=k∗(p−1)+1dp*e = k*(p-1)+1dp∗e=k∗(p−1)+1，因为dp<(p−1)dp<(p-1)dp<(p−1)，所以e>ke>ke>k，所以k可以通过e遍历出来

dp∗e+k−1=kpdp*e +k-1=kpdp∗e+k−1=kp ，所以 dp∗e+k−1=0modpdp*e +k-1= 0 \mod pdp∗e+k−1=0modp，设已知高位为dp1dp1dp1，低位为xxx，所以得到
e∗(dp1+x)+k−1=0modpe*(dp1+x)+k-1=0 mod pe∗(dp1+x)+k−1=0modp
为了让x的首项为1（因为只有这样才能用Coppersmith），此时两边乘上
d1 = gmpy2.invert(e,n)，变成 dp1+x+(k−1)∗d1=0modpdp1 + x +(k-1)*d1 = 0 mod pdp1+x+(k−1)∗d1=0modp
可以利用Coppersmith method ，把x解出来，然后再求p。

from Crypto.Util.number import long_to_bytes
dp1=1153696846823715458342658568392537778171840014923745253759529432977932183322553944430236879985
c=46735962204857190520476434898881001530665718155698898882603422023484998388668858692912250418134186095459060506275961050676051693220280588047233628259880712415593039977585805890920089318643002597837000049626154900908543384761210358835843974072960080857150727010985827690190496793207012355214605393036388807616
e=7621
n=140376049134934822153964243403031201922239588054133319056483413311963385321279682186354948441840374124640187894619689719746347334298621083485494086361152915457458004998419817456902929318697902819798254427945343361548635794308362823239150919240307072688623000747781103375481834571274423004856276841225675241863def factorize(e, dp):for i in range(2, e):p = (e * dp - 1 + i) // iif n % p == 0:return preturn -1def recover_p(dp1):F.<x> = PolynomialRing(Zmod(n))d1 = inverse_mod(e, n)b = 200for k in range(1235,1524):## 爆出来的k在这个范围，所以直接这样写了print kf = (dp1 << b) + x + (k - 1) * d1x0 = f.small_roots(X=2 ** (b + 1), beta=0.44, epsilon=1/32)if len(x0) != 0:dp = x0[0] + (dp1 << b)p = factorize(e, Integer(dp))if p < 0:continueelse:return pif __name__ == "__main__":p = recover_p(dp1)q = n // pphi = (p - 1) * (q - 1)d = inverse_mod(e, phi)print  long_to_bytes(pow(c, d, n))

easy_matrix

RING-LWE问题，当时构造不出来是因为只会embedding——attack这一种方法计算CVP，在modq\mod qmodq 下就不会用了。

赛后看了Nu1L的wp才发现，其实这里需要利用 The Nearest Plane Algorithm计算CVP再构造格解决。

from sage.modules.free_module_integer import IntegerLattice
row = 128
prime = 2129
column=42#这里的ma我先转置了
ma=[[133, 210, 141, 293, 30, 290, 445, 377, 292, 201, 92, 68, 374, 169, 69, 70, 195, 495, 32, 361, 202, 158, 198, 187, 95, 449, 447, 507, 315, 475, 97, 271, 339, 123, 121, 279, 41, 341, 280, 325, 24, 293, 467, 19, 59, 284, 488, 349, 188, 373, 240, 115, 50, 173, 270, 82, 157, 398, 393, 376, 365, 228, 425, 502, 375, 85, 387, 133, 450, 501, 176, 477, 340, 221, 115, 233, 451, 202, 409, 484, 418, 268, 340, 345, 134, 363, 105, 168, 385, 135, 219, 415, 488, 454, 85, 120, 215, 455, 365, 145, 249, 158, 399, 161, 344, 372, 293, 122, 275, 253, 128, 432, 326, 72, 510, 310, 137, 293, 88, 462, 471, 479, 411, 17, 141, 375, 345, 322], [22, 367, 369, 83, 326, 195, 405, 351, 235, 430, 198, 455, 495, 509, 151, 208, 171, 202, 103, 328, 449, 95, 20, 178, 288, 179, 9, 333, 60, 489, 298, 367, 326, 441, 433, 195, 197, 330, 10, 63, 358, 336, 197, 238, 424, 167, 154, 157, 63, 53, 72, 67, 363, 349, 94, 197, 184, 463, 223, 34, 295, 392, 213, 11, 303, 295, 207, 399, 370, 429, 111, 93, 257, 245, 45, 56, 27, 68, 69, 32, 24, 478, 289, 449, 82, 61, 397, 431, 103, 356, 8, 238, 316, 320, 169, 485, 368, 158, 142, 430, 22, 439, 100, 455, 374, 384, 449, 307, 38, 293, 215, 70, 335, 88, 94, 360, 364, 93, 178, 374, 51, 433, 447, 484, 450, 123, 313, 237], [100, 199, 503, 511, 131, 396, 425, 44, 473, 277, 42, 303, 355, 169, 61, 172, 103, 324, 57, 467, 195, 506, 186, 429, 230, 498, 242, 143, 86, 288, 149, 268, 500, 210, 261, 135, 57, 210, 469, 281, 72, 308, 502, 168, 488, 372, 49, 314, 368, 489, 475, 156, 224, 310, 219, 190, 189, 36, 388, 497, 197, 119, 19, 444, 292, 94, 9, 21, 24, 228, 413, 3, 97, 439, 52, 123, 317, 61, 371, 266, 263, 44, 32, 13, 361, 324, 490, 235, 99, 236, 408, 340, 417, 324, 299, 152, 159, 277, 124, 27, 508, 12, 268, 48, 15, 67, 47, 370, 201, 237, 81, 65, 475, 11, 291, 227, 425, 247, 365, 205, 364, 468, 511, 300, 491, 154, 404, 189], [73, 457, 149, 367, 38, 476, 160, 112, 400, 330, 370, 18, 239, 363, 213, 378, 138, 21, 510, 240, 18, 378, 485, 204, 422, 287, 241, 95, 340, 138, 349, 150, 338, 15, 248, 205, 97, 459, 92, 440, 66, 107, 124, 285, 36, 391, 12, 19, 38, 468, 374, 395, 453, 59, 136, 163, 336, 86, 386, 457, 107, 491, 70, 459, 113, 59, 432, 180, 164, 277, 456, 474, 16, 35, 272, 265, 115, 234, 418, 249, 78, 252, 135, 323, 31, 2, 486, 323, 174, 70, 443, 55, 90, 260, 231, 256, 486, 240, 284, 59, 388, 351, 430, 71, 317, 276, 93, 504, 378, 226, 507, 258, 167, 386, 85, 478, 177, 103, 42, 176, 285, 181, 98, 120, 239, 400, 71, 215], [192, 277, 52, 44, 432, 302, 58, 172, 292, 104, 482, 328, 143, 14, 1, 246, 272, 58, 331, 131, 226, 413, 168, 303, 429, 419, 453, 429, 457, 166, 248, 490, 470, 420, 97, 502, 387, 133, 58, 464, 272, 149, 438, 57, 70, 121, 86, 371, 304, 253, 160, 192, 297, 311, 254, 244, 104, 168, 81, 237, 365, 108, 58, 130, 341, 170, 243, 129, 492, 394, 17, 354, 124, 460, 124, 381, 212, 205, 141, 369, 144, 395, 32, 446, 451, 137, 458, 85, 230, 428, 364, 210, 409, 229, 41, 194, 393, 508, 497, 302, 433, 80, 412, 5, 141, 380, 190, 191, 7, 290, 492, 382, 447, 25, 158, 431, 41, 264, 307, 488, 236, 411, 133, 160, 342, 83, 162, 436], [287, 413, 405, 325, 112, 378, 41, 43, 292, 195, 263, 70, 427, 243, 306, 288, 391, 352, 92, 78, 171, 12, 119, 252, 17, 394, 486, 411, 99, 422, 92, 408, 176, 476, 420, 286, 258, 399, 439, 374, 312, 430, 118, 469, 264, 414, 341, 169, 84, 28, 10, 491, 197, 80, 111, 196, 263, 211, 198, 264, 112, 70, 34, 429, 54, 74, 316, 147, 336, 123, 81, 321, 136, 486, 277, 103, 104, 256, 92, 379, 400, 486, 295, 359, 65, 457, 51, 242, 441, 69, 448, 158, 66, 422, 354, 208, 373, 135, 78, 402, 120, 7, 461, 508, 281, 496, 313, 195, 459, 504, 466, 289, 405, 1, 23, 341, 48, 175, 411, 58, 339, 209, 509, 240, 36, 199, 379, 248], [340, 187, 50, 85, 473, 355, 19, 133, 266, 243, 1, 290, 352, 141, 30, 303, 324, 19, 380, 360, 75, 83, 426, 435, 1, 60, 129, 408, 171, 52, 439, 160, 263, 65, 236, 294, 250, 371, 270, 61, 133, 407, 212, 14, 330, 467, 59, 258, 414, 163, 40, 122, 10, 370, 424, 487, 331, 261, 76, 375, 178, 91, 424, 273, 489, 20, 382, 306, 470, 317, 395, 287, 252, 232, 430, 274, 143, 146, 242, 96, 289, 53, 312, 353, 250, 268, 180, 446, 228, 199, 508, 280, 491, 238, 124, 230, 234, 286, 180, 153, 336, 26, 316, 356, 96, 40, 286, 136, 408, 440, 266, 181, 223, 473, 341, 4, 494, 38, 172, 258, 154, 306, 34, 94, 368, 236, 452, 270], [483, 13, 49, 109, 226, 479, 40, 333, 275, 290, 110, 23, 406, 302, 178, 389, 126, 324, 221, 89, 476, 246, 98, 22, 456, 235, 8, 22, 11, 509, 8, 493, 181, 210, 196, 128, 432, 245, 255, 85, 317, 147, 91, 17, 54, 464, 498, 383, 445, 191, 461, 307, 311, 456, 508, 211, 491, 362, 88, 283, 24, 328, 143, 238, 11, 485, 108, 335, 231, 98, 391, 166, 80, 1, 174, 26, 492, 141, 60, 323, 197, 204, 262, 16, 8, 474, 501, 33, 374, 211, 50, 61, 384, 146, 372, 161, 228, 314, 181, 171, 16, 478, 357, 145, 118, 467, 363, 410, 82, 498, 215, 410, 91, 481, 402, 215, 142, 283, 374, 405, 104, 262, 171, 92, 384, 479, 22, 328], [298, 281, 454, 291, 338, 412, 372, 461, 49, 43, 497, 428, 499, 148, 244, 207, 79, 26, 216, 470, 197, 66, 286, 211, 356, 77, 276, 110, 287, 66, 86, 380, 504, 84, 372, 475, 340, 26, 339, 437, 89, 182, 491, 149, 331, 500, 371, 262, 339, 304, 453, 362, 101, 36, 270, 288, 288, 473, 107, 313, 363, 143, 475, 55, 167, 56, 33, 152, 274, 212, 309, 89, 162, 226, 274, 429, 324, 169, 371, 99, 146, 202, 177, 381, 215, 74, 120, 93, 488, 11, 114, 61, 358, 90, 221, 318, 452, 454, 35, 202, 127, 234, 129, 233, 428, 502, 417, 472, 245, 491, 396, 64, 277, 259, 114, 30, 455, 271, 183, 358, 171, 309, 366, 88, 35, 112, 94, 135], [268, 179, 137, 341, 379, 113, 338, 78, 112, 394, 6, 336, 244, 241, 425, 438, 89, 87, 493, 171, 445, 399, 269, 18, 506, 252, 310, 452, 499, 361, 252, 343, 138, 240, 246, 477, 99, 463, 57, 153, 482, 72, 22, 318, 504, 20, 374, 62, 119, 370, 23, 369, 172, 228, 91, 289, 19, 405, 183, 152, 281, 26, 151, 188, 207, 81, 9, 68, 249, 199, 234, 316, 19, 70, 157, 317, 286, 310, 93, 426, 57, 149, 34, 156, 279, 407, 131, 359, 86, 430, 272, 169, 387, 102, 489, 227, 467, 445, 382, 118, 91, 42, 120, 328, 505, 0, 181, 451, 225, 216, 303, 424, 32, 141, 180, 315, 145, 287, 499, 111, 315, 411, 416, 380, 126, 228, 346, 10], [124, 480, 127, 192, 138, 285, 126, 327, 230, 85, 171, 295, 461, 195, 140, 289, 386, 59, 261, 453, 403, 194, 12, 511, 95, 122, 193, 213, 265, 70, 55, 372, 428, 392, 409, 219, 476, 113, 264, 279, 475, 259, 297, 458, 345, 49, 161, 314, 312, 49, 130, 48, 16, 240, 502, 294, 186, 276, 267, 290, 278, 107, 18, 453, 213, 69, 209, 214, 52, 175, 142, 236, 164, 489, 334, 125, 420, 195, 16, 16, 185, 333, 478, 305, 1, 504, 240, 248, 36, 411, 479, 260, 222, 422, 116, 273, 459, 165, 59, 164, 35, 220, 446, 131, 492, 485, 326, 34, 297, 62, 257, 56, 64, 190, 302, 434, 402, 80, 224, 294, 226, 180, 243, 510, 299, 329, 78, 293], [254, 411, 419, 227, 475, 360, 107, 381, 346, 48, 67, 53, 45, 426, 322, 6, 102, 179, 403, 443, 394, 455, 102, 406, 2, 289, 228, 417, 133, 66, 211, 338, 96, 65, 343, 157, 244, 432, 71, 6, 377, 6, 10, 473, 146, 20, 367, 0, 204, 391, 46, 321, 410, 501, 298, 167, 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148, 502, 4, 508, 457, 61, 419, 344, 114, 306, 127, 254, 228, 491, 225, 4, 127, 161, 136, 290, 258, 487, 18, 254, 412, 7, 213, 456, 464, 167, 340, 14, 225, 342, 323, 251, 111, 0, 211, 344, 307, 340, 259, 320, 83, 197, 132, 229, 231, 186, 137, 394, 72, 223, 362, 197, 270, 135, 480, 93, 509, 75, 309, 210, 48, 343, 132, 230, 471, 439, 173, 497, 282, 341, 67, 310, 268, 34, 325, 321, 219, 129, 54, 13, 397, 471, 504, 419, 94, 242, 15, 65, 2, 480, 447, 20, 195, 160, 13, 173, 387, 488, 151, 115, 314, 110, 498, 105, 385, 291, 34, 500, 283, 27, 279, 89, 464, 80, 231, 390, 179, 80, 144], [197, 425, 464, 316, 93, 495, 356, 73, 369, 255, 246, 231, 35, 51, 202, 65, 266, 308, 52, 71, 410, 332, 85, 502, 100, 496, 496, 443, 6, 242, 164, 170, 414, 79, 272, 308, 249, 284, 298, 213, 127, 464, 195, 382, 377, 141, 56, 389, 382, 438, 285, 21, 478, 141, 155, 130, 396, 213, 396, 362, 160, 256, 494, 374, 371, 356, 287, 410, 438, 296, 44, 68, 223, 167, 288, 87, 41, 288, 153, 69, 66, 407, 494, 369, 451, 163, 194, 323, 137, 33, 57, 214, 265, 196, 14, 191, 264, 79, 430, 51, 7, 505, 303, 69, 351, 461, 292, 379, 334, 167, 125, 31, 279, 54, 16, 306, 166, 94, 292, 226, 171, 288, 393, 151, 120, 163, 130, 466], [426, 168, 76, 144, 450, 201, 454, 72, 339, 416, 442, 60, 428, 120, 61, 507, 72, 160, 306, 313, 288, 70, 348, 396, 418, 113, 92, 244, 84, 343, 402, 171, 360, 438, 309, 478, 84, 411, 345, 106, 204, 137, 51, 7, 227, 147, 265, 392, 395, 158, 502, 123, 480, 52, 477, 142, 252, 408, 405, 196, 137, 256, 429, 510, 511, 393, 26, 348, 495, 169, 46, 143, 353, 348, 249, 312, 103, 483, 384, 332, 431, 133, 222, 275, 161, 10, 116, 71, 383, 173, 169, 30, 26, 351, 399, 64, 242, 247, 317, 187, 81, 147, 269, 511, 58, 42, 130, 368, 255, 478, 119, 462, 457, 62, 415, 382, 64, 275, 493, 489, 486, 306, 414, 380, 384, 217, 347, 490]]
res=[2087, 1418, 498, 2090, 539, 424, 1452, 61, 1447, 334, 963, 389, 1875, 514, 644, 977, 1473, 2062, 2082, 1501, 1087, 1948, 674, 2026, 1867, 1065, 1413, 1913, 525, 1486, 793, 54, 569, 474, 239, 1237, 713, 1881, 937, 961, 1539, 1252, 1766, 614, 1786, 1675, 2008, 1713, 1126, 637, 312, 241, 239, 1144, 118, 1451, 1905, 826, 226, 457, 437, 762, 1882, 1581, 1734, 1028, 1935, 1709, 23, 310, 496, 1395, 1248, 694, 1180, 1651, 108, 1, 13, 1386, 1300, 129, 525, 2127, 1845, 2026, 1972, 1099, 600, 2080, 1428, 452, 132, 158, 146, 267, 10, 1372, 939, 881, 2110, 2077, 1519, 1926, 200, 502, 2006, 1577, 1636, 1024, 1887, 116, 43, 1085, 945, 71, 438, 936, 1235, 72, 1646, 581, 1654, 635, 1977, 897, 1902, 29]
W=matrix(ZZ,ma)cc= vector(ZZ,res)# Babai's Nearest Plane algorithm
def Babai_closest_vector(M, G, target):small = targetfor _ in xrange(5):for i in reversed(range(M.nrows())):c = ((small * G[i]) / (G[i] * G[i])).round()small -=  M[i] * creturn target - smallA1=matrix.identity(42)
Ap=matrix.identity(128)*2129
B=block_matrix([[Ap],[W]])
lattice = IntegerLattice(B, lll_reduce=True)
print("LLL done")
gram = lattice.reduced_basis.gram_schmidt()[0]
target = vector(ZZ, res)
re = Babai_closest_vector(lattice.reduced_basis, gram, target)
print("Closest Vector: {}".format(re))R = IntegerModRing(prime)
M = Matrix(R, ma)
M=M.transpose()ingredients = M.solve_right(re)
print("Ingredients: {}".format(ingredients))m=''
for i in range(len(ingredients)):m+=chr(ingredients[i])
print m

Conclusion

有待提升的地方还有很多，一些密码学分析的基本方法还不能使用自如，格的学习也学得不够多，希望下次继续努力。

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