Crypto
Swedish RSA
polynomial_rsa.sage:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 flag = bytearray(raw_input()) flag = list(flag) length = len(flag) bits = 16 ## Prime for Finite Field. p = random_prime(2^bits-1, False, 2^(bits-1)) file_out = open("downloads/polynomial_rsa.txt", "w") file_out.write("Prime: " + str(p) + "\n") ## Univariate Polynomial Ring in y over Finite Field of size p R.<y> = PolynomialRing(GF(p)) ## Analogous to the primes in Z def gen_irreducable_poly(deg): while True: out = R.random_element(degree=deg) if out.is_irreducible(): return out ## Polynomial "primes" P = gen_irreducable_poly(ZZ.random_element(length, 2*length)) Q = gen_irreducable_poly(ZZ.random_element(length, 2*length)) ## Public exponent key e = 65537 ## Modulus N = P*Q file_out.write("Modulus: " + str(N) + "\n") ## Univariate Quotient Polynomial Ring in x over Finite Field of size 659 with modulus N(x) S.<x> = R.quotient(N) ## Encrypt m = S(flag) print m c = m^e file_out.write("Ciphertext: " + str(c)) file_out.close()
polynomial_rsa.txt：
1 2 3 Prime: 43753 Modulus: 34036*y^177 + 23068*y^176 + 13147*y^175 + 36344*y^174 + 10045*y^173 + 41049*y^172 + 17786*y^171 + 16601*y^170 + 7929*y^169 + 37570*y^168 + 990*y^167 + 9622*y^166 + 39273*y^165 + 35284*y^164 + 15632*y^163 + 18850*y^162 + 8800*y^161 + 33148*y^160 + 12147*y^159 + 40487*y^158 + 6407*y^157 + 34111*y^156 + 8446*y^155 + 21908*y^154 + 16812*y^153 + 40624*y^152 + 43506*y^151 + 39116*y^150 + 33011*y^149 + 23914*y^148 + 2210*y^147 + 23196*y^146 + 43359*y^145 + 34455*y^144 + 17684*y^143 + 25262*y^142 + 982*y^141 + 24015*y^140 + 27968*y^139 + 37463*y^138 + 10667*y^137 + 39519*y^136 + 31176*y^135 + 27520*y^134 + 32118*y^133 + 8333*y^132 + 38945*y^131 + 34713*y^130 + 1107*y^129 + 43604*y^128 + 4433*y^127 + 18110*y^126 + 17658*y^125 + 32354*y^124 + 3219*y^123 + 40238*y^122 + 10439*y^121 + 3669*y^120 + 8713*y^119 + 21027*y^118 + 29480*y^117 + 5477*y^116 + 24332*y^115 + 43480*y^114 + 33406*y^113 + 43121*y^112 + 1114*y^111 + 17198*y^110 + 22829*y^109 + 24424*y^108 + 16523*y^107 + 20424*y^106 + 36206*y^105 + 41849*y^104 + 3584*y^103 + 26500*y^102 + 31897*y^101 + 34640*y^100 + 27449*y^99 + 30962*y^98 + 41434*y^97 + 22125*y^96 + 24314*y^95 + 3944*y^94 + 18400*y^93 + 38476*y^92 + 28904*y^91 + 27936*y^90 + 41867*y^89 + 25573*y^88 + 25659*y^87 + 33443*y^86 + 18435*y^85 + 5934*y^84 + 38030*y^83 + 17563*y^82 + 24086*y^81 + 36782*y^80 + 20922*y^79 + 38933*y^78 + 23448*y^77 + 10599*y^76 + 7156*y^75 + 29044*y^74 + 23605*y^73 + 7657*y^72 + 28200*y^71 + 2431*y^70 + 3860*y^69 + 23259*y^68 + 14590*y^67 + 33631*y^66 + 15673*y^65 + 36049*y^64 + 29728*y^63 + 22413*y^62 + 18602*y^61 + 18557*y^60 + 23505*y^59 + 17642*y^58 + 12595*y^57 + 17255*y^56 + 15316*y^55 + 8948*y^54 + 38*y^53 + 40329*y^52 + 9823*y^51 + 5798*y^50 + 6379*y^49 + 8662*y^48 + 34640*y^47 + 38321*y^46 + 18760*y^45 + 13135*y^44 + 15926*y^43 + 34952*y^42 + 28940*y^41 + 13558*y^40 + 42579*y^39 + 38015*y^38 + 33788*y^37 + 12381*y^36 + 195*y^35 + 13709*y^34 + 31500*y^33 + 32994*y^32 + 30486*y^31 + 40414*y^30 + 2578*y^29 + 30525*y^28 + 43067*y^27 + 6195*y^26 + 36288*y^25 + 23236*y^24 + 21493*y^23 + 15808*y^22 + 34500*y^21 + 6390*y^20 + 42994*y^19 + 42151*y^18 + 19248*y^17 + 19291*y^16 + 8124*y^15 + 40161*y^14 + 24726*y^13 + 31874*y^12 + 30272*y^11 + 30761*y^10 + 2296*y^9 + 11017*y^8 + 16559*y^7 + 28949*y^6 + 40499*y^5 + 22377*y^4 + 33628*y^3 + 30598*y^2 + 4386*y + 23814 Ciphertext: 5209*x^176 + 10881*x^175 + 31096*x^174 + 23354*x^173 + 28337*x^172 + 15982*x^171 + 13515*x^170 + 21641*x^169 + 10254*x^168 + 34588*x^167 + 27434*x^166 + 29552*x^165 + 7105*x^164 + 22604*x^163 + 41253*x^162 + 42675*x^161 + 21153*x^160 + 32838*x^159 + 34391*x^158 + 832*x^157 + 720*x^156 + 22883*x^155 + 19236*x^154 + 33772*x^153 + 5020*x^152 + 17943*x^151 + 26967*x^150 + 30847*x^149 + 10306*x^148 + 33966*x^147 + 43255*x^146 + 20342*x^145 + 4474*x^144 + 3490*x^143 + 38033*x^142 + 11224*x^141 + 30565*x^140 + 31967*x^139 + 32382*x^138 + 9759*x^137 + 1030*x^136 + 32122*x^135 + 42614*x^134 + 14280*x^133 + 16533*x^132 + 32676*x^131 + 43070*x^130 + 36009*x^129 + 28497*x^128 + 2940*x^127 + 9747*x^126 + 22758*x^125 + 16615*x^124 + 14086*x^123 + 13038*x^122 + 39603*x^121 + 36260*x^120 + 32502*x^119 + 17619*x^118 + 17700*x^117 + 15083*x^116 + 11311*x^115 + 36496*x^114 + 1300*x^113 + 13601*x^112 + 43425*x^111 + 10376*x^110 + 11551*x^109 + 13684*x^108 + 14955*x^107 + 6661*x^106 + 12674*x^105 + 21534*x^104 + 32132*x^103 + 34135*x^102 + 43684*x^101 + 837*x^100 + 29311*x^99 + 4849*x^98 + 26632*x^97 + 26662*x^96 + 10159*x^95 + 32657*x^94 + 12149*x^93 + 17858*x^92 + 35805*x^91 + 19391*x^90 + 30884*x^89 + 42039*x^88 + 17292*x^87 + 4694*x^86 + 1497*x^85 + 1744*x^84 + 31071*x^83 + 26246*x^82 + 24402*x^81 + 22068*x^80 + 39263*x^79 + 23703*x^78 + 21484*x^77 + 12241*x^76 + 28821*x^75 + 32886*x^74 + 43075*x^73 + 35741*x^72 + 19936*x^71 + 37219*x^70 + 33411*x^69 + 8301*x^68 + 12949*x^67 + 28611*x^66 + 42654*x^65 + 6910*x^64 + 18523*x^63 + 31144*x^62 + 21398*x^61 + 36298*x^60 + 27158*x^59 + 918*x^58 + 38601*x^57 + 4269*x^56 + 5699*x^55 + 36444*x^54 + 34791*x^53 + 37978*x^52 + 32481*x^51 + 8039*x^50 + 11012*x^49 + 11454*x^48 + 30450*x^47 + 1381*x^46 + 32403*x^45 + 8202*x^44 + 8404*x^43 + 37648*x^42 + 43696*x^41 + 34237*x^40 + 36490*x^39 + 41423*x^38 + 35792*x^37 + 36950*x^36 + 31086*x^35 + 38970*x^34 + 12439*x^33 + 7963*x^32 + 16150*x^31 + 11382*x^30 + 3038*x^29 + 20157*x^28 + 23531*x^27 + 32866*x^26 + 5428*x^25 + 21132*x^24 + 13443*x^23 + 28909*x^22 + 42716*x^21 + 6567*x^20 + 24744*x^19 + 8727*x^18 + 14895*x^17 + 28172*x^16 + 30903*x^15 + 26608*x^14 + 27314*x^13 + 42224*x^12 + 42551*x^11 + 37726*x^10 + 11203*x^9 + 36816*x^8 + 5537*x^7 + 20301*x^6 + 17591*x^5 + 41279*x^4 + 7999*x^3 + 33753*x^2 + 34551*x + 9659
可见这是多项式的RSA，我之前的WriteUp里有提到过：https://ce-automne.github.io/2019/04/11/0CTF-2019-babyrsa-WriteUp/
但是这次还是没做出来。。。你说菜不菜
其实思路差不多，主要有一点搞错了，就是求解phi的时候，没有考虑到GF(43753)对应的phi:
1 phi = (43753**p.degree()-1)*(43753**q.degree()-1)
用sage来做：
1 2 3 4 5 6 7 8 9 10 11 R.<y> = PolynomialRing(GF(43753 )) n = R('34036*y^177 + 23068*y^176 + 13147*y^175 + 36344*y^174 + 10045*y^173 + 41049*y^172 + 17786*y^171 + 16601*y^170 + 7929*y^169 + 37570*y^168 + 990*y^167 + 9622*y^166 + 39273*y^165 + 35284*y^164 + 15632*y^163 + 18850*y^162 + 8800*y^161 + 33148*y^160 + 12147*y^159 + 40487*y^158 + 6407*y^157 + 34111*y^156 + 8446*y^155 + 21908*y^154 + 16812*y^153 + 40624*y^152 + 43506*y^151 + 39116*y^150 + 33011*y^149 + 23914*y^148 + 2210*y^147 + 23196*y^146 + 43359*y^145 + 34455*y^144 + 17684*y^143 + 25262*y^142 + 982*y^141 + 24015*y^140 + 27968*y^139 + 37463*y^138 + 10667*y^137 + 39519*y^136 + 31176*y^135 + 27520*y^134 + 32118*y^133 + 8333*y^132 + 38945*y^131 + 34713*y^130 + 1107*y^129 + 43604*y^128 + 4433*y^127 + 18110*y^126 + 17658*y^125 + 32354*y^124 + 3219*y^123 + 40238*y^122 + 10439*y^121 + 3669*y^120 + 8713*y^119 + 21027*y^118 + 29480*y^117 + 5477*y^116 + 24332*y^115 + 43480*y^114 + 33406*y^113 + 43121*y^112 + 1114*y^111 + 17198*y^110 + 22829*y^109 + 24424*y^108 + 16523*y^107 + 20424*y^106 + 36206*y^105 + 41849*y^104 + 3584*y^103 + 26500*y^102 + 31897*y^101 + 34640*y^100 + 27449*y^99 + 30962*y^98 + 41434*y^97 + 22125*y^96 + 24314*y^95 + 3944*y^94 + 18400*y^93 + 38476*y^92 + 28904*y^91 + 27936*y^90 + 41867*y^89 + 25573*y^88 + 25659*y^87 + 33443*y^86 + 18435*y^85 + 5934*y^84 + 38030*y^83 + 17563*y^82 + 24086*y^81 + 36782*y^80 + 20922*y^79 + 38933*y^78 + 23448*y^77 + 10599*y^76 + 7156*y^75 + 29044*y^74 + 23605*y^73 + 7657*y^72 + 28200*y^71 + 2431*y^70 + 3860*y^69 + 23259*y^68 + 14590*y^67 + 33631*y^66 + 15673*y^65 + 36049*y^64 + 29728*y^63 + 22413*y^62 + 18602*y^61 + 18557*y^60 + 23505*y^59 + 17642*y^58 + 12595*y^57 + 17255*y^56 + 15316*y^55 + 8948*y^54 + 38*y^53 + 40329*y^52 + 9823*y^51 + 5798*y^50 + 6379*y^49 + 8662*y^48 + 34640*y^47 + 38321*y^46 + 18760*y^45 + 13135*y^44 + 15926*y^43 + 34952*y^42 + 28940*y^41 + 13558*y^40 + 42579*y^39 + 38015*y^38 + 33788*y^37 + 12381*y^36 + 195*y^35 + 13709*y^34 + 31500*y^33 + 32994*y^32 + 30486*y^31 + 40414*y^30 + 2578*y^29 + 30525*y^28 + 43067*y^27 + 6195*y^26 + 36288*y^25 + 23236*y^24 + 21493*y^23 + 15808*y^22 + 34500*y^21 + 6390*y^20 + 42994*y^19 + 42151*y^18 + 19248*y^17 + 19291*y^16 + 8124*y^15 + 40161*y^14 + 24726*y^13 + 31874*y^12 + 30272*y^11 + 30761*y^10 + 2296*y^9 + 11017*y^8 + 16559*y^7 + 28949*y^6 + 40499*y^5 + 22377*y^4 + 33628*y^3 + 30598*y^2 + 4386*y + 23814' ) p,q = n.factor() p,q = p[0 ],q[0 ] phi = (43753 **p.degree()-1 )*(43753 **q.degree()-1 ) e = 65537 d = inverse_mod(e,phi) c = R('5209*y^176 + 10881*y^175 + 31096*y^174 + 23354*y^173 + 28337*y^172 + 15982*y^171 + 13515*y^170 + 21641*y^169 + 10254*y^168 + 34588*y^167 + 27434*y^166 + 29552*y^165 + 7105*y^164 + 22604*y^163 + 41253*y^162 + 42675*y^161 + 21153*y^160 + 32838*y^159 + 34391*y^158 + 832*y^157 + 720*y^156 + 22883*y^155 + 19236*y^154 + 33772*y^153 + 5020*y^152 + 17943*y^151 + 26967*y^150 + 30847*y^149 + 10306*y^148 + 33966*y^147 + 43255*y^146 + 20342*y^145 + 4474*y^144 + 3490*y^143 + 38033*y^142 + 11224*y^141 + 30565*y^140 + 31967*y^139 + 32382*y^138 + 9759*y^137 + 1030*y^136 + 32122*y^135 + 42614*y^134 + 14280*y^133 + 16533*y^132 + 32676*y^131 + 43070*y^130 + 36009*y^129 + 28497*y^128 + 2940*y^127 + 9747*y^126 + 22758*y^125 + 16615*y^124 + 14086*y^123 + 13038*y^122 + 39603*y^121 + 36260*y^120 + 32502*y^119 + 17619*y^118 + 17700*y^117 + 15083*y^116 + 11311*y^115 + 36496*y^114 + 1300*y^113 + 13601*y^112 + 43425*y^111 + 10376*y^110 + 11551*y^109 + 13684*y^108 + 14955*y^107 + 6661*y^106 + 12674*y^105 + 21534*y^104 + 32132*y^103 + 34135*y^102 + 43684*y^101 + 837*y^100 + 29311*y^99 + 4849*y^98 + 26632*y^97 + 26662*y^96 + 10159*y^95 + 32657*y^94 + 12149*y^93 + 17858*y^92 + 35805*y^91 + 19391*y^90 + 30884*y^89 + 42039*y^88 + 17292*y^87 + 4694*y^86 + 1497*y^85 + 1744*y^84 + 31071*y^83 + 26246*y^82 + 24402*y^81 + 22068*y^80 + 39263*y^79 + 23703*y^78 + 21484*y^77 + 12241*y^76 + 28821*y^75 + 32886*y^74 + 43075*y^73 + 35741*y^72 + 19936*y^71 + 37219*y^70 + 33411*y^69 + 8301*y^68 + 12949*y^67 + 28611*y^66 + 42654*y^65 + 6910*y^64 + 18523*y^63 + 31144*y^62 + 21398*y^61 + 36298*y^60 + 27158*y^59 + 918*y^58 + 38601*y^57 + 4269*y^56 + 5699*y^55 + 36444*y^54 + 34791*y^53 + 37978*y^52 + 32481*y^51 + 8039*y^50 + 11012*y^49 + 11454*y^48 + 30450*y^47 + 1381*y^46 + 32403*y^45 + 8202*y^44 + 8404*y^43 + 37648*y^42 + 43696*y^41 + 34237*y^40 + 36490*y^39 + 41423*y^38 + 35792*y^37 + 36950*y^36 + 31086*y^35 + 38970*y^34 + 12439*y^33 + 7963*y^32 + 16150*y^31 + 11382*y^30 + 3038*y^29 + 20157*y^28 + 23531*y^27 + 32866*y^26 + 5428*y^25 + 21132*y^24 + 13443*y^23 + 28909*y^22 + 42716*y^21 + 6567*y^20 + 24744*y^19 + 8727*y^18 + 14895*y^17 + 28172*y^16 + 30903*y^15 + 26608*y^14 + 27314*y^13 + 42224*y^12 + 42551*y^11 + 37726*y^10 + 11203*y^9 + 36816*y^8 + 5537*y^7 + 20301*y^6 + 17591*y^5 + 41279*y^4 + 7999*y^3 + 33753*y^2 + 34551*y + 9659' ) m = pow(c,d,n) print(m) print("" .join([chr(c) for c in m.list()]))
运行，即可得到flag
ECC-RSA ecc-rsa.py
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 from fastecdsa.curve import P521 as Curvefrom fastecdsa.point import Pointfrom Crypto.Util.number import bytes_to_long, isPrimefrom os import urandomfrom random import getrandbitsdef gen_rsa_primes (G) : urand = bytes_to_long(urandom(521 //8 )) while True : s = getrandbits(521 ) ^ urand Q = s*G if isPrime(Q.x) and isPrime(Q.y): print("ECC Private key:" , hex(s)) print("RSA primes:" , hex(Q.x), hex(Q.y)) print("Modulo:" , hex(Q.x * Q.y)) return (Q.x, Q.y) flag = int.from_bytes(input(), byteorder="big" ) ecc_p = Curve.p a = Curve.a b = Curve.b Gx = Curve.gx Gy = Curve.gy G = Point(Gx, Gy, curve=Curve) e = 0x10001 p, q = gen_rsa_primes(G) n = p*q file_out = open("downloads/ecc-rsa.txt" , "w" ) file_out.write("ECC Curve Prime: " + hex(ecc_p) + "\n" ) file_out.write("Curve a: " + hex(a) + "\n" ) file_out.write("Curve b: " + hex(b) + "\n" ) file_out.write("Gx: " + hex(Gx) + "\n" ) file_out.write("Gy: " + hex(Gy) + "\n" ) file_out.write("e: " + hex(e) + "\n" ) file_out.write("p * q: " + hex(n) + "\n" ) c = pow(flag, e, n) file_out.write("ciphertext: " + hex(c) + "\n" )
ecc-rsa.txt
1 2 3 4 5 6 7 8 ECC Curve Prime: 0x1ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff Curve a: -0x3 Curve b: 0x51953eb9618e1c9a1f929a21a0b68540eea2da725b99b315f3b8b489918ef109e156193951ec7e937b1652c0bd3bb1bf073573df883d2c34f1ef451fd46b503f00 Gx: 0xc6858e06b70404e9cd9e3ecb662395b4429c648139053fb521f828af606b4d3dbaa14b5e77efe75928fe1dc127a2ffa8de3348b3c1856a429bf97e7e31c2e5bd66 Gy: 0x11839296a789a3bc0045c8a5fb42c7d1bd998f54449579b446817afbd17273e662c97ee72995ef42640c550b9013fad0761353c7086a272c24088be94769fd16650 e: 0x10001 p * q: 0x118aaa1add80bdd0a1788b375e6b04426c50bb3f9cae0b173b382e3723fc858ce7932fb499cd92f5f675d4a2b05d2c575fc685f6cf08a490d6c6a8a6741e8be4572adfcba233da791ccc0aee033677b72788d57004a776909f6d699a0164af514728431b5aed704b289719f09d591f5c1f9d2ed36a58448a9d57567bd232702e9b28f ciphertext: 0x3862c872480bdd067c0c68cfee4527a063166620c97cca4c99baff6eb0cf5d42421b8f8d8300df5f8c7663adb5d21b47c8cb4ca5aab892006d7d44a1c5b5f5242d88c6e325064adf9b969c7dfc52a034495fe67b5424e1678ca4332d59225855b7a9cb42db2b1db95a90ab6834395397e305078c5baff78c4b7252d7966365afed9e
阅读代码发现，flag是通过RSA加密的，要想解密，需要分解RSA的模数，很明显通过常见的方法是分解不了的。
p和q是通过下面的代码生成的
1 p, q = gen_rsa_primes(G)
对应的函数
1 2 3 4 5 6 7 8 9 10 11 def gen_rsa_primes (G) : urand = bytes_to_long(urandom(521 //8 )) while True : s = getrandbits(521 ) ^ urand Q = s*G if isPrime(Q.x) and isPrime(Q.y): print("ECC Private key:" , hex(s)) print("RSA primes:" , hex(Q.x), hex(Q.y)) print("Modulo:" , hex(Q.x * Q.y)) return (Q.x, Q.y)
可见ECC的私钥是随机产生，p和q即Q.x和Q.y，是ECC里的公钥，G是ECC的一组解，从ecc-rsa.txt可以发现这组解是已知的。
现在来梳理一下，目前知道ECC方程：y^2 = x^3 + a*x + b mod P 里的(a,b,P)的值，注意这里P的值是通过
1 from fastecdsa.curve import P521 as Curve
确定的。
然后将公钥的这组解(p,q)代入到ECC方程里:
1 q^2 = p^3 + a*p + b mod P (1)
接着因为
将式(1)等号两边都乘以p^2
1 n^2 = p^5 + a*p^3 + b*p^2 mod P
上面的等式只有一个未知数p，第一反应是丢到z3里去解，没有解出来。
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 from z3 import *n = "0x118aaa1add80bdd0a1788b375e6b04426c50bb3f9cae0b173b382e3723fc858ce7932fb499cd92f5f675d4a2b05d2c575fc685f6cf08a490d6c6a8a6741e8be4572adfcba233da791ccc0aee033677b72788d57004a776909f6d699a0164af514728431b5aed704b289719f09d591f5c1f9d2ed36a58448a9d57567bd232702e9b28f" print(int(n,16 )) n = 12916538708895236593179535507652696958130212902754938727546178509610568654166129500188100142747506875538155738050834024841293249875243322231997953335130773335593387031913128963790434108123650368664271877756430432011758177188901760143619102842627935254653509129413728763871069738643074735737955568513819337277682319 a = -3 b = 1093849038073734274511112390766805569936207598951683748994586394495953116150735016013708737573759623248592132296706313309438452531591012912142327488478985984 P0 = "0x1ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff" print(int(P0,16 )) P = "6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151" x = Int('x' ) y = Int('y' ) solve(n**2 +P*y==x**5 +a*x**3 +b*x**2 ,x>0 )
那就尝试分解多项式，令
1 f = p^5 + a*p^3 + b*p^2 - n^2 mod P
在其因式中找到其中度为1的多项式（即p+C这种形式，其中C为任意整数），然后判断该多项式的解和n是否存在1以外的公因子，也就是gcd(p,n) != 1的公因子，为什么有这个设定，是因为gcd(p,n) = 1就表示p和n是互质的，所以这样的p是无法满足p*q = n的设定的。也就是说如果存在1以外的公因子，那么这个公因子就是解。
另外如果计算出来的解为负数，需要将其对P取模来模正（这里的运算需要再补习一下）
最终的sage利用代码：
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 from binascii import * P = 0x1ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff e = 0x10001 a = -0x3 b = 0x51953eb9618e1c9a1f929a21a0b68540eea2da725b99b315f3b8b489918ef109e156193951ec7e937b1652c0bd3bb1bf073573df883d2c34f1ef451fd46b503f00 n = 0x118aaa1add80bdd0a1788b375e6b04426c50bb3f9cae0b173b382e3723fc858ce7932fb499cd92f5f675d4a2b05d2c575fc685f6cf08a490d6c6a8a6741e8be4572adfcba233da791ccc0aee033677b72788d57004a776909f6d699a0164af514728431b5aed704b289719f09d591f5c1f9d2ed36a58448a9d57567bd232702e9b28f c = 0x3862c872480bdd067c0c68cfee4527a063166620c97cca4c99baff6eb0cf5d42421b8f8d8300df5f8c7663adb5d21b47c8cb4ca5aab892006d7d44a1c5b5f5242d88c6e325064adf9b969c7dfc52a034495fe67b5424e1678ca4332d59225855b7a9cb42db2b1db95a90ab6834395397e305078c5baff78c4b7252d7966365afed9e R.<x> = PolynomialRing(GF(P)) f = x**5 + a*x**3 + b*x**2 - n**2 factors = f.factor() print factors for i in factors: #print i i = i[0] print i if i.degree()==1: print i[0] tmp = Integer(mod(-i[0],P)) #tmp = Integer(i[0]) print tmp print gcd(tmp,n) if gcd(tmp,n) != 1: p = tmp q = n/tmp print p,q phi = (p-1)*(q-1) d = inverse_mod(e,phi) m = pow(c,d,n) print unhexlify(hex(Integer(m)))
运算结果：
参考链接：
https://www.anquanke.com/post/id/196010#h2-2
https://github.com/wat3vr/watevrCTF-2019/blob/master/challenges/crypto/ECC-RSA/ecc-rsa_solution.sage
