Coppersmith 是干了这么一件事:今有一个 $e$ 阶的多项式 $f$, 那么可以:

  • 在模 $n$ 意义下,快速求出 $n^{1/e}$ 以内的根
  • 给定 $\beta$,快速求出模某个 $b$ 意义下较小的根,其中 $b\geq n ^ \beta$,是 $n$ 的因数。

一般采用 Sage 实现的 small_roots 方法。

Sage文档:链接
Wikipedia:链接

参考资料,一篇很全面的综述:Twenty Years of Attacks on the RSA Cryptosystem


phase 1: hash爆破

[+]proof: skr=os.urandom(8)
[+]hashlib.sha256(skr).hexdigest()=246bfcbe8c7b0be0a3ee28840a276272ba4416cb650affd846e9f7f2db2820a9
[+]skr[0:5].encode('hex')=c2183d3580
[-]skr.encode('hex')=
▲ 第一阶段。需要找出符合条件的 skr

  给出了 skr 的前 5 位,需要找到正确的 skr 使得其 sha256 为给定值。显然直接爆破后三位就行。

def phase1(pre, target):
    pre = codecs.decode(pre.encode(), 'hex')
    for x in itertools.product(range(256), repeat=3):
        skr = pre + b''.join([t.to_bytes(1, 'big') for t in x])
        if hashlib.sha256(skr).hexdigest() == target:
            print(f'find {skr}')
            return codecs.encode(skr, 'hex').decode()

phase1('c2183d3580', '246bfcbe8c7b0be0a3ee28840a276272ba4416cb650affd846e9f7f2db2820a9')

# find b'\xc2\x18=5\x80\x14Q9'
# 'c2183d3580145139'

phase 2: 已知明文高位,求低位

[+]n=13112061820685643239663831166928327119579425830632458568801544406506769461279590962772340249183569437559394200635526183698604582385769381159563710823689417274479549627596095398621182995891454516953722025068926293512505383125227579169778946631369961753587856344582257683672313230378603324005337788913902434023431887061454368566100747618582590270385918204656156089053519709536001906964008635708510672550219546894006091483520355436091053866312718431318498783637712773878423777467316605865516248176248780637132615807886272029843770186833425792049108187487338237850806203728217374848799250419859646871057096297020670904211

[+]e=3

[+]m=random.getrandbits(512)

[+]c=pow(m,e,n)=15987554724003100295326076036413163634398600947695096857803937998969441763014731720375196104010794555868069024393647966040593258267888463732184495020709457560043050577198988363754703741636088089472488971050324654162166657678376557110492703712286306868843728466224887550827162442026262163340935333721705267432790268517

[+]((m>>72)<<72)=2519188594271759205757864486097605540135407501571078627238849443561219057751843170540261842677239681908736

[-]long_to_bytes(m).encode('hex')=

  这里给出了 $m$ 的高440位,我们只需要推断剩余的低 72 位。记真实的 $m$ 为$highM + x$,则 $$m ^ 3 - c= (highM + x) ^ 3 - c = 0$$这个方程的根很小,可以直接求解。

def phase2(high_m, n, c):
    R.<x> = PolynomialRing(Zmod(n), implementation='NTL')
    m = high_m + x
    M = m((m^3 - c).small_roots()[0])
    print(hex(int(M))[2:])

n = 13112061820685643239663831166928327119579425830632458568801544406506769461279590962772340249183569437559394200635526183698604582385769381159563710823689417274479549627596095398621182995891454516953722025068926293512505383125227579169778946631369961753587856344582257683672313230378603324005337788913902434023431887061454368566100747618582590270385918204656156089053519709536001906964008635708510672550219546894006091483520355436091053866312718431318498783637712773878423777467316605865516248176248780637132615807886272029843770186833425792049108187487338237850806203728217374848799250419859646871057096297020670904211
c = 15987554724003100295326076036413163634398600947695096857803937998969441763014731720375196104010794555868069024393647966040593258267888463732184495020709457560043050577198988363754703741636088089472488971050324654162166657678376557110492703712286306868843728466224887550827162442026262163340935333721705267432790268517
high_m = 2519188594271759205757864486097605540135407501571078627238849443561219057751843170540261842677239681908736

phase2(high_m, n, c)
# 464c41477b325e3872736137353839363933666336383963373763356635323632643635343237323432377d

phase 3: 已知 p 高位,求低位

[+]n=12784625729032789592766625203074018101354917751492952685083808825504221816847310910447532133616954262271205877651255598995305639194329607493047941212754523879402744065076183778452640602625242851184095546100200565113016690161053808950384458996881574266573992526357954507491397978278604102524731393059303476350167738237822647246425836482533150025923051544431330502522043833872580483142594571802189321599016725741260254170793393777293145010525686561904427613648184843619301241414264343057368192416551134404100386155751297424616254697041043851852081071306219462991969849123668248321130382231769250865190227630009181759219

[+]e=65537

[+]m=random.getrandbits(512)

[+]c=pow(m,e,n)=627824086157119245056478875800598959553774250161670787506083253960788230737588761787385686125828765665617567887904228030839535317987589608761534500003128247164233774794784231518212804270056404565710426613938264302998015421153393879729263551292024543756422702956470022959537221269172084619081368498693930550456153543628170306324206266216348386707008661128717431426237486511309767286175518238620230507201952867261283880986868752676549613958785288914989429224582849218395471672295410036858881836363364885164276983237312235831591858044908369376855484127614933545955544787160352042318378588039587911741028067576722790778

[+]((p>>128)<<128)=97522826022187678545924975588711975512906538181361325096919121233043973599759518562689050415761485716705615149641768982838255403594331293651224395590747133152128042950062103156564440155088882592644046069208405360324372057140890317518802130081198060093576841538008960560391380395697098964411821716664506908672

[-]long_to_bytes(m).encode('hex')=

  Coppersmith 可以解决多项式在模 $n$ 的某个因数下的根。我们设 $p = pHigh + x$,然后拿去求解方程 $$p = 0 \pmod {\text{sth divides n}}$$得到 $p$ 之后即可推出私钥。

def phase3(high_p, n, c):
    R.<x> = PolynomialRing(Zmod(n), implementation='NTL')
    p = high_p + x
    x0 = p.small_roots(X = 2^128, beta = 0.1)[0]
    
    P = int(p(x0))
    Q = n // P
    
    assert n == P*Q
    
    d = inverse_mod(65537, (P-1)*(Q-1))
    print(hex(power_mod(c, d, n)))
    
n = 12784625729032789592766625203074018101354917751492952685083808825504221816847310910447532133616954262271205877651255598995305639194329607493047941212754523879402744065076183778452640602625242851184095546100200565113016690161053808950384458996881574266573992526357954507491397978278604102524731393059303476350167738237822647246425836482533150025923051544431330502522043833872580483142594571802189321599016725741260254170793393777293145010525686561904427613648184843619301241414264343057368192416551134404100386155751297424616254697041043851852081071306219462991969849123668248321130382231769250865190227630009181759219
c = 627824086157119245056478875800598959553774250161670787506083253960788230737588761787385686125828765665617567887904228030839535317987589608761534500003128247164233774794784231518212804270056404565710426613938264302998015421153393879729263551292024543756422702956470022959537221269172084619081368498693930550456153543628170306324206266216348386707008661128717431426237486511309767286175518238620230507201952867261283880986868752676549613958785288914989429224582849218395471672295410036858881836363364885164276983237312235831591858044908369376855484127614933545955544787160352042318378588039587911741028067576722790778
high_p = 97522826022187678545924975588711975512906538181361325096919121233043973599759518562689050415761485716705615149641768982838255403594331293651224395590747133152128042950062103156564440155088882592644046069208405360324372057140890317518802130081198060093576841538008960560391380395697098964411821716664506908672

phase3(high_p, n, c)

phase 4: 已知 d 低位,求 p, q

[+]n=92896523979616431783569762645945918751162321185159790302085768095763248357146198882641160678623069857011832929179987623492267852304178894461486295864091871341339490870689110279720283415976342208476126414933914026436666789270209690168581379143120688241413470569887426810705898518783625903350928784794371176183

[+]e=3

[+]m=random.getrandbits(512)

[+]c=pow(m,e,n)=56164378185049402404287763972280630295410174183649054805947329504892979921131852321281317326306506444145699012788547718091371389698969718830761120076359634262880912417797038049510647237337251037070369278596191506725812511682495575589039521646062521091457438869068866365907962691742604895495670783101319608530

[+]d&((1<<512)-1)=787673996295376297668171075170955852109814939442242049800811601753001897317556022653997651874897208487913321031340711138331360350633965420642045383644955

[-]long_to_bytes(m).encode('hex')=

  既然已知 $d$ 的低位,也就是已知 $d$ 在模 $2^{512}$ 意义下的值,又有 $e=3$,我们考虑等式$$\begin{aligned}ed & \equiv 1 \pmod {(p-1)(q-1)} \\ 3d &= 1 + k\cdot (p-1)(q-1)\quad \text{where }k<3 \end{aligned}$$两边对 $2^{512}$ 取模,有$$3\cdot dLow \equiv 1 + k\cdot (n - p - q + 1) \pmod{2 ^ {512}}$$以 $\frac{n}{p}$ 代替 $q$,使上面的方程成为单变量的:$$3\cdot dLow \cdot p \equiv p + k\cdot (np - p^2 - n + p) \pmod {2^{512}}$$

  这个方程是模意义下的一元二次方程,是可解的。解出来之后得到了 $p$ 的低位,通过与 phase 3 类似的方式可以得到 $p,q$.

def getFullP(low_p, n):
    R.<x> = PolynomialRing(Zmod(n), implementation='NTL')
    p = x*2^512 + low_p
    root = (p-n).monic().small_roots(X = 2^128, beta = 0.4)
    if root:
        return p(root[0])
    return None
    
def phase4(low_d, n, c):
    maybe_p = []
    for k in range(1, 4):
        p = var('p')
        p0 = solve_mod([3*p*low_d  == p + k*(n*p - p^2 - n + p)], 2^512)
        maybe_p += [int(x[0]) for x in p0]
    print(maybe_p)
    
    for x in maybe_p:
        P = getFullP(x, n)
        if P: break
    
    P = int(P)
    Q = n // P
    
    assert P*Q == n
    
    d = inverse_mod(3, (P-1)*(Q-1))
    print(hex(power_mod(c, d, n))[2:])
    


n = 92896523979616431783569762645945918751162321185159790302085768095763248357146198882641160678623069857011832929179987623492267852304178894461486295864091871341339490870689110279720283415976342208476126414933914026436666789270209690168581379143120688241413470569887426810705898518783625903350928784794371176183
c = 56164378185049402404287763972280630295410174183649054805947329504892979921131852321281317326306506444145699012788547718091371389698969718830761120076359634262880912417797038049510647237337251037070369278596191506725812511682495575589039521646062521091457438869068866365907962691742604895495670783101319608530
low_d = 787673996295376297668171075170955852109814939442242049800811601753001897317556022653997651874897208487913321031340711138331360350633965420642045383644955

phase4(low_d, n, c)
# 464c41477b325e3872736135616230383637343566366563373435363139613862363566653465633536307d

phase 5: 广播攻击

[+]e=3
[+]m=random.getrandbits(512)

[+]n1=78642188663937191491235684351005990853149481644703243255021321296087539054265733392095095639539412823093600710316645130404423641473150336492175402885270861906530337207734106926328737198871118125840680572148601743121884788919989184318198417654263598170932154428514561079675550090698019678767738203477097731989
[+]c1=pow(m,e,n1)=23419685303892339080979695469481275906709035609088426118328601771163101123641599051556995351678670765521269546319724616458499631461037359417701720430452076029312714313804716888119910334476982840024696320503747736428099717113471541651211596481005191146454458591558743268791485623924245960696651150688621664860

[+]n2=98174485544103863705821086588292917749386955237408645745685476234349659452606822650329076955303471252833860010724515777826660887118742978051231030080666542833950748806944312437614585352818344599399156268450521239843157288915059003487783576003027303399985723834248634230998110618288843582573006048070816520647
[+]c2=pow(m,e,n2)=72080679612442543693944655041130370753964497034378634203383617624269927191363529233872659451561571441107920350406295389613006330637565645758727103723546610079332161151567096389071050158035757745766399510575237344950873632114050632573903701015749830874081198250578516967517980592506626547273178363503100507676

[+]n3=91638855323231795590642755267985988356764327384001022396221901964430032527111968159623063760057482761918901490239790230176524505469897183382928646349163030620342744192731246392941227433195249399795012672172947919435254998997253131826888070173526892674308708289629739522194864912899817994807268945141349669311
[+]c3=pow(m,e,n3)=22149989692509889061584875630258740744292355239822482581889060656197919681655781672277545701325284646570773490123892626601106871432216449814891757715588851851459306683123591338089745675044763551335899599807235257516935037356212345033087798267959242561085752109746935300735969972249665700075907145744305255616

[-]long_to_bytes(m).encode('hex')=

  相同的消息用三个不同的公钥加密,且 $e=3$,直接通过中国剩余定理得到 $m^3$ 的确切值,开根号即可。

def phase5(n1, c1, n2, c2, n3, c3):
    r = CRT([c1, c2, c3], [n1, n2, n3])
    m = int(r)^(1/3)
    print(hex(m)[2:])
    
n1 = 78642188663937191491235684351005990853149481644703243255021321296087539054265733392095095639539412823093600710316645130404423641473150336492175402885270861906530337207734106926328737198871118125840680572148601743121884788919989184318198417654263598170932154428514561079675550090698019678767738203477097731989
c1 = 23419685303892339080979695469481275906709035609088426118328601771163101123641599051556995351678670765521269546319724616458499631461037359417701720430452076029312714313804716888119910334476982840024696320503747736428099717113471541651211596481005191146454458591558743268791485623924245960696651150688621664860
n2 = 98174485544103863705821086588292917749386955237408645745685476234349659452606822650329076955303471252833860010724515777826660887118742978051231030080666542833950748806944312437614585352818344599399156268450521239843157288915059003487783576003027303399985723834248634230998110618288843582573006048070816520647
c2 = 72080679612442543693944655041130370753964497034378634203383617624269927191363529233872659451561571441107920350406295389613006330637565645758727103723546610079332161151567096389071050158035757745766399510575237344950873632114050632573903701015749830874081198250578516967517980592506626547273178363503100507676
n3 = 91638855323231795590642755267985988356764327384001022396221901964430032527111968159623063760057482761918901490239790230176524505469897183382928646349163030620342744192731246392941227433195249399795012672172947919435254998997253131826888070173526892674308708289629739522194864912899817994807268945141349669311
c3 = 22149989692509889061584875630258740744292355239822482581889060656197919681655781672277545701325284646570773490123892626601106871432216449814891757715588851851459306683123591338089745675044763551335899599807235257516935037356212345033087798267959242561085752109746935300735969972249665700075907145744305255616

phase5(n1,c1,n2,c2,n3,c3)
# 464c41477b325e3872736138633566336366663462633039353334396665633635666332323633653837387d

phase 6: Franklin-Reiter 相关消息攻击

[+]n= 113604829563460357756722229849309932731534576966155520277171862442445354404910882358287832757024693652075211204635679309777620586814014894544893424988818766425089667672311645586528776360047956843961901352792631908859388801090108188344342619580661377758180391734771694803991493164412644148805229529911069578061


[+]e=7   # 题目出错了!这里应该为 3,但 BUUCTF 的靶机返回了 7,我暂且蒙在鼓里


[+]m=random.getrandbits(512)


[+]c=pow(m,e,n)=112992730284209629010217336632593897028023711212853788739137950706145189880318698604512926758021533447981943498594790549326550460216939216988828130624120379925895123186121819609415184887470233938291227816332249857236198616538782622327476603338806349004620909717360739157545735826670038169284252348037995399308


[+]x=pow(m+1,e,n)=112992730284209629010217336632593897028023711212853788739137950706145189880318698604512926758021552486915464025361447529153776277710423467951041523831865232164370127602772602643378592695459331174613894578701940837730590029577336924367384969935652616989527416027725713616493815764725131271563545176286794438175


[-]long_to_bytes(m).encode('hex')=

  注意到 $m$ 是下面方程组的解:$$\begin{cases}x ^ e - c _1 = 0 \\ (x+1) ^ e - c_2 = 0\end{cases}$$

  于是 $(x-m)$ 是这两个多项式的公因式。把两个多项式求 gcd,即可得到 $m$.

n = 113604829563460357756722229849309932731534576966155520277171862442445354404910882358287832757024693652075211204635679309777620586814014894544893424988818766425089667672311645586528776360047956843961901352792631908859388801090108188344342619580661377758180391734771694803991493164412644148805229529911069578061
c1 = 112992730284209629010217336632593897028023711212853788739137950706145189880318698604512926758021533447981943498594790549326550460216939216988828130624120379925895123186121819609415184887470233938291227816332249857236198616538782622327476603338806349004620909717360739157545735826670038169284252348037995399308
c2 = 112992730284209629010217336632593897028023711212853788739137950706145189880318698604512926758021552486915464025361447529153776277710423467951041523831865232164370127602772602643378592695459331174613894578701940837730590029577336924367384969935652616989527416027725713616493815764725131271563545176286794438175
e = 3

# c1 = m^e
# c2 = (m+1)^e

R.<x> = PolynomialRing(Zmod(n))
g1 = x^e - c1
g2 = (x+1)^e - c2

def myGcd(x, y):
    if y == 0:
        return x.monic()
    return myGcd(y, x%y)

v = myGcd(g2, g1)
M = n - v.coefficients()[0]

assert g1(M) == 0
print(hex(M))
# 0x464c41477b325e3872736133393863663864663763323636363162623763623635623262396661653235657d

phase 7: Boneh Durfee 攻击

[+]n=0xbadd260d14ea665b62e7d2e634f20a6382ac369cd44017305b69cf3a2694667ee651acded7085e0757d169b090f29f3f86fec255746674ffa8a6a3e1c9e1861003eb39f82cf74d84cc18e345f60865f998b33fc182a1a4ffa71f5ae48a1b5cb4c5f154b0997dc9b001e441815ce59c6c825f064fdca678858758dc2cebbc4d27L

[+]d=random.getrandbits(1024*0.270)

[+]e=invmod(d,phin)

[+]hex(e)=0x11722b54dd6f3ad9ce81da6f6ecb0acaf2cbc3885841d08b32abc0672d1a7293f9856db8f9407dc05f6f373a2d9246752a7cc7b1b6923f1827adfaeefc811e6e5989cce9f00897cfc1fc57987cce4862b5343bc8e91ddf2bd9e23aea9316a69f28f407cfe324d546a7dde13eb0bd052f694aefe8ec0f5298800277dbab4a33bbL

[+]m=random.getrandbits(512)

[+]c=pow(m,e,n)=0xe3505f41ec936cf6bd8ae344bfec85746dc7d87a5943b3a7136482dd7b980f68f52c887585d1c7ca099310c4da2f70d4d5345d3641428797030177da6cc0d41e7b28d0abce694157c611697df8d0add3d900c00f778ac3428f341f47ecc4d868c6c5de0724b0c3403296d84f26736aa66f7905d498fa1862ca59e97f8f866cL

[-]long_to_bytes(m).encode('hex')=

  知道 $e$,另外 $d \leq n ^ {0.27}$. 查阅资料知使用 Boneh and Durfee attack. 现成脚本:

mimoo/RSA-and-LLL-attacks
attacking RSA via lattice reductions (LLL). Contribute to mimoo/RSA-and-LLL-attacks development by creating an account on GitHub.

  然而这是 Python2 的。手工改成 Python3 版本:

import time

"""
Setting debug to true will display more informations
about the lattice, the bounds, the vectors...
"""
debug = True

"""
Setting strict to true will stop the algorithm (and
return (-1, -1)) if we don't have a correct 
upperbound on the determinant. Note that this 
doesn't necesseraly mean that no solutions 
will be found since the theoretical upperbound is
usualy far away from actual results. That is why
you should probably use `strict = False`
"""
strict = False

"""
This is experimental, but has provided remarkable results
so far. It tries to reduce the lattice as much as it can
while keeping its efficiency. I see no reason not to use
this option, but if things don't work, you should try
disabling it
"""
helpful_only = True
dimension_min = 7 # stop removing if lattice reaches that dimension

############################################
# Functions
##########################################

# display stats on helpful vectors
def helpful_vectors(BB, modulus):
    nothelpful = 0
    for ii in range(BB.dimensions()[0]):
        if BB[ii,ii] >= modulus:
            nothelpful += 1

    print (nothelpful, "/", BB.dimensions()[0], " vectors are not helpful")

# display matrix picture with 0 and X
def matrix_overview(BB, bound):
    for ii in range(BB.dimensions()[0]):
        a = ('%02d ' % ii)
        for jj in range(BB.dimensions()[1]):
            a += '0' if BB[ii,jj] == 0 else 'X'
            if BB.dimensions()[0] < 60:
                a += ' '
        if BB[ii, ii] >= bound:
            a += '~'
        print (a)

# tries to remove unhelpful vectors
# we start at current = n-1 (last vector)
def remove_unhelpful(BB, monomials, bound, current):
    # end of our recursive function
    if current == -1 or BB.dimensions()[0] <= dimension_min:
        return BB

    # we start by checking from the end
    for ii in range(current, -1, -1):
        # if it is unhelpful:
        if BB[ii, ii] >= bound:
            affected_vectors = 0
            affected_vector_index = 0
            # let's check if it affects other vectors
            for jj in range(ii + 1, BB.dimensions()[0]):
                # if another vector is affected:
                # we increase the count
                if BB[jj, ii] != 0:
                    affected_vectors += 1
                    affected_vector_index = jj

            # level:0
            # if no other vectors end up affected
            # we remove it
            if affected_vectors == 0:
                print ("* removing unhelpful vector", ii)
                BB = BB.delete_columns([ii])
                BB = BB.delete_rows([ii])
                monomials.pop(ii)
                BB = remove_unhelpful(BB, monomials, bound, ii-1)
                return BB

            # level:1
            # if just one was affected we check
            # if it is affecting someone else
            elif affected_vectors == 1:
                affected_deeper = True
                for kk in range(affected_vector_index + 1, BB.dimensions()[0]):
                    # if it is affecting even one vector
                    # we give up on this one
                    if BB[kk, affected_vector_index] != 0:
                        affected_deeper = False
                # remove both it if no other vector was affected and
                # this helpful vector is not helpful enough
                # compared to our unhelpful one
                if affected_deeper and abs(bound - BB[affected_vector_index, affected_vector_index]) < abs(bound - BB[ii, ii]):
                    print ("* removing unhelpful vectors", ii, "and", affected_vector_index)
                    BB = BB.delete_columns([affected_vector_index, ii])
                    BB = BB.delete_rows([affected_vector_index, ii])
                    monomials.pop(affected_vector_index)
                    monomials.pop(ii)
                    BB = remove_unhelpful(BB, monomials, bound, ii-1)
                    return BB
    # nothing happened
    return BB

""" 
Returns:
* 0,0   if it fails
* -1,-1 if `strict=true`, and determinant doesn't bound
* x0,y0 the solutions of `pol`
"""
def boneh_durfee(pol, modulus, mm, tt, XX, YY):
    """
    Boneh and Durfee revisited by Herrmann and May
    
    finds a solution if:
    * d < N^delta
    * |x| < e^delta
    * |y| < e^0.5
    whenever delta < 1 - sqrt(2)/2 ~ 0.292
    """

    # substitution (Herrman and May)
    PR.<u, x, y> = PolynomialRing(ZZ)
    Q = PR.quotient(x*y + 1 - u) # u = xy + 1
    polZ = Q(pol).lift()

    UU = XX*YY + 1

    # x-shifts
    gg = []
    for kk in range(mm + 1):
        for ii in range(mm - kk + 1):
            xshift = x^ii * modulus^(mm - kk) * polZ(u, x, y)^kk
            gg.append(xshift)
    gg.sort()

    # x-shifts list of monomials
    monomials = []
    for polynomial in gg:
        for monomial in polynomial.monomials():
            if monomial not in monomials:
                monomials.append(monomial)
    monomials.sort()
    
    # y-shifts (selected by Herrman and May)
    for jj in range(1, tt + 1):
        for kk in range(floor(mm/tt) * jj, mm + 1):
            yshift = y^jj * polZ(u, x, y)^kk * modulus^(mm - kk)
            yshift = Q(yshift).lift()
            gg.append(yshift) # substitution
    
    # y-shifts list of monomials
    for jj in range(1, tt + 1):
        for kk in range(floor(mm/tt) * jj, mm + 1):
            monomials.append(u^kk * y^jj)

    # construct lattice B
    nn = len(monomials)
    BB = Matrix(ZZ, nn)
    for ii in range(nn):
        BB[ii, 0] = gg[ii](0, 0, 0)
        for jj in range(1, ii + 1):
            if monomials[jj] in gg[ii].monomials():
                BB[ii, jj] = gg[ii].monomial_coefficient(monomials[jj]) * monomials[jj](UU,XX,YY)

    # Prototype to reduce the lattice
    if helpful_only:
        # automatically remove
        BB = remove_unhelpful(BB, monomials, modulus^mm, nn-1)
        # reset dimension
        nn = BB.dimensions()[0]
        if nn == 0:
            print ("failure")
            return 0,0

    # check if vectors are helpful
    if debug:
        helpful_vectors(BB, modulus^mm)
    
    # check if determinant is correctly bounded
    det = BB.det()
    bound = modulus^(mm*nn)
    if det >= bound:
        print ("We do not have det < bound. Solutions might not be found.")
        print ("Try with highers m and t.")
        if debug:
            diff = (log(det) - log(bound)) / log(2)
            print ("size det(L) - size e^(m*n) = ", floor(diff))
        if strict:
            return -1, -1
    else:
        print ("det(L) < e^(m*n) (good! If a solution exists < N^delta, it will be found)")

    # display the lattice basis
    if debug:
        matrix_overview(BB, modulus^mm)

    # LLL
    if debug:
        print ("optimizing basis of the lattice via LLL, this can take a long time")

    BB = BB.LLL()

    if debug:
        print ("LLL is done!")

    # transform vector i & j -> polynomials 1 & 2
    if debug:
        print ("looking for independent vectors in the lattice")
    found_polynomials = False
    
    for pol1_idx in range(nn - 1):
        for pol2_idx in range(pol1_idx + 1, nn):
            # for i and j, create the two polynomials
            PR.<w,z> = PolynomialRing(ZZ)
            pol1 = pol2 = 0
            for jj in range(nn):
                pol1 += monomials[jj](w*z+1,w,z) * BB[pol1_idx, jj] / monomials[jj](UU,XX,YY)
                pol2 += monomials[jj](w*z+1,w,z) * BB[pol2_idx, jj] / monomials[jj](UU,XX,YY)

            # resultant
            PR.<q> = PolynomialRing(ZZ)
            rr = pol1.resultant(pol2)

            # are these good polynomials?
            if rr.is_zero() or rr.monomials() == [1]:
                continue
            else:
                print ("found them, using vectors", pol1_idx, "and", pol2_idx)
                found_polynomials = True
                break
        if found_polynomials:
            break

    if not found_polynomials:
        print ("no independant vectors could be found. This should very rarely happen...")
        return 0, 0
    
    rr = rr(q, q)

    # solutions
    soly = rr.roots()

    if len(soly) == 0:
        print ("Your prediction (delta) is too small")
        return 0, 0

    soly = soly[0][0]
    ss = pol1(q, soly)
    solx = ss.roots()[0][0]

    #
    return solx, soly

def example():
    ############################################
    # How To Use This Script
    ##########################################

    #
    # The problem to solve (edit the following values)
    #

    # the modulus
    N = 0xbadd260d14ea665b62e7d2e634f20a6382ac369cd44017305b69cf3a2694667ee651acded7085e0757d169b090f29f3f86fec255746674ffa8a6a3e1c9e1861003eb39f82cf74d84cc18e345f60865f998b33fc182a1a4ffa71f5ae48a1b5cb4c5f154b0997dc9b001e441815ce59c6c825f064fdca678858758dc2cebbc4d27
    # the public exponent
    e = 0x11722b54dd6f3ad9ce81da6f6ecb0acaf2cbc3885841d08b32abc0672d1a7293f9856db8f9407dc05f6f373a2d9246752a7cc7b1b6923f1827adfaeefc811e6e5989cce9f00897cfc1fc57987cce4862b5343bc8e91ddf2bd9e23aea9316a69f28f407cfe324d546a7dde13eb0bd052f694aefe8ec0f5298800277dbab4a33bb

    # the hypothesis on the private exponent (the theoretical maximum is 0.292)
    delta = 0.280 # this means that d < N^delta

    #
    # Lattice (tweak those values)
    #

    # you should tweak this (after a first run), (e.g. increment it until a solution is found)
    m = 4 # size of the lattice (bigger the better/slower)

    # you need to be a lattice master to tweak these
    t = int((1-2*delta) * m)  # optimization from Herrmann and May
    X = 2*floor(N^delta)  # this _might_ be too much
    Y = floor(N^(1/2))    # correct if p, q are ~ same size

    #
    # Don't touch anything below
    #

    # Problem put in equation
    P.<x,y> = PolynomialRing(ZZ)
    A = int((N+1)/2)
    pol = 1 + x * (A + y)

    #
    # Find the solutions!
    #

    # Checking bounds
    if debug:
        print ("=== checking values ===")
        print ("* delta:", delta)
        print ("* delta < 0.292", delta < 0.292)
        print ("* size of e:", int(log(e)/log(2)))
        print ("* size of N:", int(log(N)/log(2)))
        print ("* m:", m, ", t:", t)

    # boneh_durfee
    if debug:
        print ("=== running algorithm ===")
        start_time = time.time()

    solx, soly = boneh_durfee(pol, e, m, t, X, Y)

    # found a solution?
    if solx > 0:
        print ("=== solution found ===")
        if False:
            print ("x:", solx)
            print ("y:", soly)

        d = int(pol(solx, soly) / e)
        print ("private key found:", d)
    else:
        print ("=== no solution was found ===")

    if debug:
        print("=== %s seconds ===" % (time.time() - start_time))

if __name__ == "__main__":
    example()

  最终拿到 flag: flag{b12a39f1-b96b-4a46-8bc7-7c3871242b9c}