RSA原理

安恒网络空间安全讲武堂

发布于 2018-12-24 15:19:15

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发布于 2018-12-24 15:19:15

文章被收录于专栏：安恒网络空间安全讲武堂

知识准备

素数：素数又称质数，指在一个大于1的自然数中，除了1和此整数自身外，不能被其他自然数整除的数。

互质数：公因数只有1的两个数，叫做互质数

模运算：两个整数a，b，若它们除以正整数m所得的余数相等，则称a，b对于模m同余，记作: a ≡ b (mod m)；读作：a同余于b模m，或者，a与b关于模m同余。例如：26 ≡ 14 (mod 12)。

欧拉函数：在数论，对正整数n，欧拉函数是小于n的正整数中与n互质的数的数目（φ(1)=1）。

模反元素：如果两个正整数a和n互质，那么一定可以找到整数b，使得 ab-1 被n整除，或者说ab被n除的余数是1。这时，b就叫做a的“模反元素”。

公钥与私钥

1.找到两个的不同大素数p&q,N=pq。 2.根据欧拉函数得到 r=(p-1)(q-1) 3.选择一个小于r的整数e，求e关于模r的模反元素d。 4.销毁p，q。

这样就得到了公钥(N,e),私钥(N,d)

加密解密过程

加密只需要公钥(N,e)，对于明文x进行如下运算 x^e ≡ c (mod N) 得到密文c。

解密只需要知道私钥(N,d),对于密文c进行如下运算 c^d ≡ x （mod N）还原明文x。

CTF中的RSA例题

0x01 基础RSA加密

用公钥和密文解密出明文，这建立在N可分解的基础上，我们可以通过pq得到秘钥。

#  coding: utf-8
from Crypto.PublicKey import RSA
import gmpy2
import codecs

pub=RSA.importKey(open("/Users/a1tm4nz/Downloads/RSA/public.pem").read())
n=pub.n #n
e=pub.e #e
p=258631601377848992211685134376492365269 # 通过http://factordb.com/分解N
q=286924040788547268861394901519826758027
#d=gmpy2.invert(e,(p-1)*(q-1))
d=23071769375111040425287244625328797615295772814180109366784249976498215494337
c=int(codecs.encode(open('/Users/a1tm4nz/Downloads/RSA/flag.enc','rb').read(),'hex_codec'),16)
m=hex(pow(c,d,n))[2:].replace("L","")
if(len(m)%2==1):#16进制解密要求密文不能为奇数，在头部填0即可
    m='0'+m
print m.decode('hex')

0x02 wiener attack

当N或e都很大时，我们可以使用wiener攻击 github上有利用脚本: https://github.com/pablocelayes/rsa-wiener-attack

南邮平台上的一道题

#coding:utf-8
from Crypto.PublicKey import RSA
from Crypto.Cipher import PKCS1_v1_5 as Cipher_pkcs1_v1_5
import base64

flag=raw_input('flag:')
key=RSA.construct((1063045321283844468344531168992778520651192162100948533991539097447031440090068191835838938460807260866872379834796862916118785271062209281267667069640000501698142693389209275376843382863579650119977059768375028586326490055087394631528241983631462471709913758728591459476799115050977493979613545056736162868049L, 837165022918376318972691589160491375229372195625940137121740685432530132860541010174727630660292946071507342455170833392895060048564125597915757582027572284342507277083636059558106672685400173531425920294781499112027917632497954958437660357575400222692979844873372105801998210845285775146263117399191185379347L))
cipher = Cipher_pkcs1_v1_5.new(key)
cipher_text = base64.b64encode(cipher.encrypt(flag))
print cipher_text
#cipher_text = 'AGgt1h6dudnkeoCr7SFclkYYsYa65KZ8V29bbgbf+BDyjnyx5stCYjcyktat73aHs2EOaMgwGUwj3HwPTvT+T5LHIxM4uTnAgWOui4dnb7vF7QizN0ShY2O1h26CgLnf5I0vQWbY7WCC7kA/orNW7F5yxZiKRAawacS2M5ghP4/Q'

N，e都很大。我们用wiener attack 得到d

python rsa-wiener-attack/RSAwienerHacker.py n e

得到私钥d

d=57899763801722261062891290503559835904571946557258761154422546104824094670843

接下来就是常规的RSA解密

import base64
n=1063045321283844468344531168992778520651192162100948533991539097447031440090068191835838938460807260866872379834796862916118785271062209281267667069640000501698142693389209275376843382863579650119977059768375028586326490055087394631528241983631462471709913758728591459476799115050977493979613545056736162868049

e=837165022918376318972691589160491375229372195625940137121740685432530132860541010174727630660292946071507342455170833392895060048564125597915757582027572284342507277083636059558106672685400173531425920294781499112027917632497954958437660357575400222692979844873372105801998210845285775146263117399191185379347

d=57899763801722261062891290503559835904571946557258761154422546104824094670843

m='AGgt1h6dudnkeoCr7SFclkYYsYa65KZ8V29bbgbf+BDyjnyx5stCYjcyktat73aHs2EOaMgwGUwj3HwPTvT+T5LHIxM4uTnAgWOui4dnb7vF7QizN0ShY2O1h26CgLnf5I0vQWbY7WCC7kA/orNW7F5yxZiKRAawacS2M5ghP4/Q'
c=int(base64.b64decode(m).encode('hex'),16)
flag=hex(pow(c,d,n))[2:].replace("L","")
if(len(flag)%2==1):
    flag='0'+flag
print flag.decode('hex')

0x03 有公因数的两个N

给了两对公钥，N太大无法分解，但是N1,N2有公因数。那么此时公因数就是p，可以分别求出q1，q2 题目

n1=18263905851567773440446838695766097054252159817375942220432646590577605535001102705343902666589196712209131000424743250389209817386462242094905266578654348699073317748484503797678183012090375022172700739930717847219593096973008967105897376613550069563133191469825170677181620033104899474861544205137427444083416158205978241738189319430709815369614381957092634679663073529915011800029514945250518582469896694087993939399022631417819581576165949892810231692555896017395242464371112868608767990194529216988324463096379599680586615395063392235579858007086701467453321499203151052012397135583838714605379937464734426058203
n2=16950818485762084795193828768953323876388698051219062552262211712110062204954209462306530235388240321343855913666709750794055992220667151032536667937762799073479211925880106492191394846770654371623007051501782616639485222511300384032213459590408774089539345780246233268007572472533774114330568959631749390932599046733958624832563792588926026242133422467392689761450865841250657088270966077177543599222351800102728976845282712937106806976091210265560260177661816495238213887970556095475226646345568545415814035277834069152282458515989066082948101449829801979628039212597995349260855092279108102204886522855975419755219
c1=16274856857661787783089247952446020386301296490309822420733326939579521159181274564159881569720941773424141684911497028248685883897404191432880449283023146073930043226457053587418510143359803678057561120305169670182063356905346792409675959838228170818653485027257264058185367161472527834396804757004371950225319647551718070122431050642186905590213972232201966833949845104276760241004644118590467546314025479853604227295841523010158969804175921406672115195772809154058842429049437301440993794765038365224477229612151404063782303298937771968709567577283974551173044172598459482531433545960749147311254443274915272200560
c2=9946468920119252596998213656931348575944985856629754429330209121534145245119561878513995066589817036899299533093751237144960328759208855732474853794711347203865156360078772132790431594811682581926722057546683437873159107885652842304739962490836998123152090675606004046425633751397173768982047965656687448847259753864171018963561303276197312504508548802813909914926514763930195218396740593919987596462341469781868335025782329081775818968846955110510048099746584203570892950955431181639182647914240604278151551608856971433512600491550082244566145491335738112881861092354219766862656988674738232228115996349755982641605
e1=1804229351
e2=17249876309

解密脚本

#  coding: utf-8
from Crypto.PublicKey import RSA
import gmpy2
import codecs

n1=18263905851567773440446838695766097054252159817375942220432646590577605535001102705343902666589196712209131000424743250389209817386462242094905266578654348699073317748484503797678183012090375022172700739930717847219593096973008967105897376613550069563133191469825170677181620033104899474861544205137427444083416158205978241738189319430709815369614381957092634679663073529915011800029514945250518582469896694087993939399022631417819581576165949892810231692555896017395242464371112868608767990194529216988324463096379599680586615395063392235579858007086701467453321499203151052012397135583838714605379937464734426058203
n2=16950818485762084795193828768953323876388698051219062552262211712110062204954209462306530235388240321343855913666709750794055992220667151032536667937762799073479211925880106492191394846770654371623007051501782616639485222511300384032213459590408774089539345780246233268007572472533774114330568959631749390932599046733958624832563792588926026242133422467392689761450865841250657088270966077177543599222351800102728976845282712937106806976091210265560260177661816495238213887970556095475226646345568545415814035277834069152282458515989066082948101449829801979628039212597995349260855092279108102204886522855975419755219
c1=16274856857661787783089247952446020386301296490309822420733326939579521159181274564159881569720941773424141684911497028248685883897404191432880449283023146073930043226457053587418510143359803678057561120305169670182063356905346792409675959838228170818653485027257264058185367161472527834396804757004371950225319647551718070122431050642186905590213972232201966833949845104276760241004644118590467546314025479853604227295841523010158969804175921406672115195772809154058842429049437301440993794765038365224477229612151404063782303298937771968709567577283974551173044172598459482531433545960749147311254443274915272200560
c2=9946468920119252596998213656931348575944985856629754429330209121534145245119561878513995066589817036899299533093751237144960328759208855732474853794711347203865156360078772132790431594811682581926722057546683437873159107885652842304739962490836998123152090675606004046425633751397173768982047965656687448847259753864171018963561303276197312504508548802813909914926514763930195218396740593919987596462341469781868335025782329081775818968846955110510048099746584203570892950955431181639182647914240604278151551608856971433512600491550082244566145491335738112881861092354219766862656988674738232228115996349755982641605
e1=1804229351
e2=17249876309

q=gmpy2.gcd(n1,n2)
p1=n1//q
p2=n2//q

d1=gmpy2.invert(e1,(p1-1)*(q-1))
d2=gmpy2.invert(e2,(p2-1)*(q-1))

while d1<0:
    d1+=(p1-1)*(q-1)
while d2<0:
    d2+=(p2-1)*(q-1)
m2=hex(pow(c2,d2,n2))[2:].replace("L","")
m1=hex(pow(c1,d1,n1))[2:].replace("L","")

if(len(m2)%2==1):
    m2='0'+m2
if(len(m1)%2==1):
    m1 ='0'+ m1
print m1.decode('hex')
print m2.decode('hex')

0x04 低指数攻击

e=3时进行低指数攻击

# coding:utf-8
import gmpy2

c=545666236924510340010249577709750283325731706774285241719627277546281629429734726717293022303311450772262647904537263500252284243393598944613964442974546950954108203106726282255676706429218187217515454665602130999856741523362906632677988245886500953095201122016935004088287862399317170828388632964668574391252399791901016522260191839164586088073933168096433230663402492577707149742261018318811473591856287943664733276898405984282679026758294364432874973387827086342720762945025346962005339728347282927842299962927871005260338747371451546554777112213044710533502191671159066680035742327279159127279685106716107705888068319962657817786581813767331740609788885735155741039564703781141646102609725965697004923161084032164730408824475517786576979990372940555488021025837456038491436690372760376483602299268887032528766383572923258228355911069631275397149328319966792315903921085816103476508992023873616148326626245855060470294978538631677232260545724075728912626994884533001056079733734460116442499311813113038763837974777469202302071221647473459505245546281400799833123812072606012604323510933244028733287443734697557314202167934768160824072400916728008549350662843995750077421616789178835625661267955774815287104291379928002318796086248
n=721059527572145959497866070657244746540818298735241721382435892767279354577831824618770455583435147844630635953460258329387406192598509097375098935299515255208445013180388186216473913754107215551156731413550416051385656895153798495423962750773689964815342291306243827028882267935999927349370340823239030087548468521168519725061290069094595524921012137038227208900579645041589141405674545883465785472925889948455146449614776287566375730215127615312001651111977914327170496695481547965108836595145998046638495232893568434202438172004892803105333017726958632541897741726563336871452837359564555756166187509015523771005760534037559648199915268764998183410394036820824721644946933656264441126738697663216138624571035323231711566263476403936148535644088575960271071967700560360448191493328793704136810376879662623765917690163480410089565377528947433177653458111431603202302962218312038109342064899388130688144810901340648989107010954279327738671710906115976561154622625847780945535284376248111949506936128229494332806622251145622565895781480383025403043645862516504771643210000415216199272423542871886181906457361118669629044165861299560814450960273479900717138570739601887771447529543568822851100841225147694940195217298482866496536787241
e=3
i = 0
while True:
    if gmpy2.iroot(c + i * n, 3)[1] == True:
        print "Success!"
        m=gmpy2.iroot(c + i * n, 3)
        print m
        break
    i += 1
# m=440721643740967258786371951429849843897639673893942371730874939742481383302887786063966117819631425015196093856646526738786745933078032806737504580146717737115929461581126895844008044713461807791172016433647699394456368658396746134702627548155069403689581548233891848149612485605022294307233116137509171389596747894529765156771462793389236431942344003532140158865426896855377113878133478689191912682550117563858186L
m=hex(m)[2:].replace("L","")
if (len(m) % 2 == 1):
    m= '0' + m
print m.decode('hex')

0x05 共模攻击

当有两个e两个c时，可以进行公模攻击

# coding:utf8
import gmpy2
import libnum
import codecs

n=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
c1=int(codecs.encode(open("/Users/a1tm4nz/crypto/veryhardRSA/flag.enc1",'rb').read(),'hex_codec'),16)
c2=int(codecs.encode(open("/Users/a1tm4nz/crypto/veryhardRSA/flag.enc2",'rb').read(),'hex_codec'),16)
e1=17
e2=65537
s1=libnum.xgcd(e1,e2)[0]
s2=libnum.xgcd(e1,e2)[1]
#此处判断s1和s2是否小于0，因为pow()函数里s1和s2不能为负，
if(s1<0):
    s1=-s1
    c1=gmpy2.invert(c1,n)#若s1为负,s1取正，c1取逆
if(s2<0):
    s2=-s2
    c2=gmpy2.invert(c2,n)
m=libnum.n2s((pow(c1,s1,n)*pow(c2,s2,n)%n))
print(m)

0x06 Lattice based attacks on RSA

题目

# n=0x79982a272b9f50b2c2bc8b862ccc617bb39720a6dc1a22dc909bbfd1243cc0a03dd406ec0b1a78fa75ce5234e8c57e0aab492050906364353b06ccd45f90b7818b04be4734eeb8e859ef92a306be105d32108a3165f96664ac1e00bba770f04627da05c3d7513f5882b2807746090cebbf74cd50c0128559a2cc9fa7d88f7b2d
# e=0x3
# c=0x381db081852c92d268b49a1b9486d724e4ecf49fc97dc5f20d1fad902b5cdfb49c8cc1e968e36f65ae9af7e8186f15ccdca798786669a3d2c9fe8767a7ae938a4f9115ae8fed4928d95ad550fddd3a9c1497785c9e2279edf43f04601980aa28b3b52afb55e2b34e5b175af25d5b3bd71db88b3b31e48a177a469116d957592c
# b=0xfedcba98765432100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
# m=b+x (x:64bit)

n无法分解，低指数爆破失败，但我们已用明文的高位。参考:https://github.com/mimoo/RSA-and-LLL-attacks 使用sage可以解出明文

# partial_m.sage

n = 0x79982a272b9f50b2c2bc8b862ccc617bb39720a6dc1a22dc909bbfd1243cc0a03dd406ec0b1a78fa75ce5234e8c57e0aab492050906364353b06ccd45f90b7818b04be4734eeb8e859ef92a306be105d32108a3165f96664ac1e00bba770f04627da05c3d7513f5882b2807746090cebbf74cd50c0128559a2cc9fa7d88f7b2d
e = 3

m = randrange(n)
c = pow(m, e, n)

beta = 1
epsilon = beta^2/7

nbits = n.nbits()
kbits = floor(nbits*(beta^2/e-epsilon))
#mbar = m & (2^nbits-2^kbits)
mbar = 0xfedcba98765432100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
c = 0x381db081852c92d268b49a1b9486d724e4ecf49fc97dc5f20d1fad902b5cdfb49c8cc1e968e36f65ae9af7e8186f15ccdca798786669a3d2c9fe8767a7ae938a4f9115ae8fed4928d95ad550fddd3a9c1497785c9e2279edf43f04601980aa28b3b52afb55e2b34e5b175af25d5b3bd71db88b3b31e48a177a469116d957592c
print "upper %d bits (of %d bits) is given" % (nbits-kbits, nbits)

PR.<x> = PolynomialRing(Zmod(n))
f = (mbar + x)^e - c

print m
x0 = f.small_roots(X=2^kbits, beta=1)[0]  # find root < 2^kbits with factor = n
print mbar + x0
print x0

// 在线sage网站 http://sagecell.sagemath.org/

0x07 广播攻击

使用不同的模数n，相同的公钥指数e加密相同的信息。就会得到多个(m^e) ==ci (mod ni)，将(m^e)视为一个整体M，这就是典型的中国剩余定理适用情况。按照本文的中国剩余定理小节容易求得m^e的值，当e较小时直接开e方即可，可使用gmpy2.iroot(M,e)。例题和脚本

import gmpy2
def broadcast(n1, n2 ,n3, c1, c2, c3):
    n = [n1, n2, n3]
    C = [c1, c2, c3]
    N = 1
    for i in n:
        N *= i

    Ni = []
    for i in n:
        Ni.append(N / i)

    T = []
    for i in xrange(3):
        T.append(long(gmpy2.invert(Ni[i], n[i])))

    X = 0
    for i in xrange(3):
        X += C[i] * Ni[i] * T[i]

    m3 = X % N
    m = gmpy2.iroot(m3, 3)
    return m
n1=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
c1=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
n2=0x60d175fdb0a96eca160fb0cbf8bad1a14dd680d353a7b3bc77e620437da70fd9153f7609efde652b825c4ae7f25decf14a3c8240ea8c5892003f1430cc88b0ded9dae12ebffc6b23632ac530ac4ae23fbffb7cfe431ff3d802f5a54ab76257a86aeec1cf47d482fec970fc27c5b376fbf2cf993270bba9b78174395de3346d4e221d1eafdb8eecc8edb953d1ccaa5fc250aed83b3a458f9e9d947c4b01a6e72ce4fee37e77faaf5597d780ad5f0a7623edb08ce76264f72c3ff17afc932f5812b10692bcc941a18b6f3904ca31d038baf3fc1968d1cc0588a656d0c53cd5c89cedba8a5230956af2170554d27f524c2027adce84fd4d0e018dc88ca4d5d26867
c2=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
n3=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
c3=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

m = broadcast(n1,n2,n3,c1,c2,c3)
print hex(int(m[0])).replace("L","")

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公钥与私钥

加密解密过程

CTF中的RSA例题

0x01 基础RSA加密

0x02 wiener attack

0x03 有公因数的两个N

0x04 低指数攻击

0x05 共模攻击

0x06 Lattice based attacks on RSA

0x07 广播攻击

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