From 7975d02c6c1ee679a236087e86955c086f1a9a8e Mon Sep 17 00:00:00 2001 From: Ummon Date: Mon, 15 Dec 2014 13:35:31 +0100 Subject: [PATCH] First commit. --- .gitignore | 27 ++++++++++++++++++++ icr14_lab_03.pdf | Bin 0 -> 154135 bytes rapport/main.tex | 40 +++++++++++++++++++++++++++++ src/Rand.cpp | 65 +++++++++++++++++++++++++++++++++++++++++++++++ src/Rand.h | 36 ++++++++++++++++++++++++++ src/Rsa.cpp | 47 ++++++++++++++++++++++++++++++++++ src/Rsa.h | 27 ++++++++++++++++++++ src/RsaCrt.cpp | 1 + src/RsaCrt.h | 20 +++++++++++++++ src/Utils.cpp | 24 +++++++++++++++++ src/Utils.h | 29 +++++++++++++++++++++ src/lab3.qbs | 27 ++++++++++++++++++++ src/main.cpp | 40 +++++++++++++++++++++++++++++ 13 files changed, 383 insertions(+) create mode 100644 .gitignore create mode 100644 icr14_lab_03.pdf create mode 100644 rapport/main.tex create mode 100644 src/Rand.cpp create mode 100644 src/Rand.h create mode 100644 src/Rsa.cpp create mode 100644 src/Rsa.h create mode 100644 src/RsaCrt.cpp create mode 100644 src/RsaCrt.h create mode 100644 src/Utils.cpp create mode 100644 src/Utils.h create mode 100644 src/lab3.qbs create mode 100644 src/main.cpp diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..bc14d9e --- /dev/null +++ b/.gitignore @@ -0,0 +1,27 @@ +lab3-build/ +*.aux +*.log +*.[oa] +moc_*.cpp +*.moc +*.pb.cc +*.pb.h +*.pb.cpp +*.user +*.user.* +*.session +*.Release +*.Debug +*.exe +*.qm +*.dll +Makefile* +output/ +Installations/ +.tmp/ +/doc/doxygen/html +qrc_*.cpp +TODO +*.session +*.DS_STORE +Thumbs.db diff --git a/icr14_lab_03.pdf b/icr14_lab_03.pdf new file mode 100644 index 0000000000000000000000000000000000000000..5dd5a2a76eb70858d367a4339aa757a663907ed6 GIT binary patch literal 154135 zcma&N1C%B~(kS}1Ic?jvd)l^b+t##g+qP}n_Oy*@8+Uf^{_oy*cHem$c}|_Ih^)-0 zsL05uj7U;>VNqH}I#y`XxuxMXXeM?71_C=nO9F0gXnJWATQg^K0!Bu5R)T*5G`*;W zwX=yM0llcTfwPIQiIJVL2{bP+w3D-=iGdBY`$n>=hTQ=hg3oiU{5(38IwQABmrE?k zm?2D2xj*m~8jMJRjf=@VwF1X9@e_(4bbSD4-J9nx|HqT{4-lrQDGlQ*NJfT%ofNfA-=@vh4& z`Q?`C${DMV6ci%1LABmDg*>7*8abU3OwyziDK51#b(w51nY3;hcSJ3%Wj(R2IEUuy z4IkK2x85EZLeVwd!ocMt&tU=qtokt21p8s)a*}ajyyf@hR#ZyIVkUA2CKoa7WZlN3 z`{Uv191;AfbM%*K)5UdLPs6o-N`S0H&=kjU%DwnuLJ||d2fGUv^Vr>*wf&S~=c3O+ z$Hs*p{90a-v%Yno-UMpeSXcOC4H!&9>g`@wGpIWra*kJ(gNj;rYKNSnx@E7|O{9TQ zNVo0BZ=Q1=T)X4b^u;Lg7=Dh~O+p0?>k^Q+S*cW@m?uVqdQNmt*>I#^E50E65kg^R z-VQ_!hG;hLCEn(lP);DZ+l!nR6(ro~jnEPc+Vc7llWz{@)yv(p_H(>;%<8k+`OpQD zc{tk1S^nF^qhyCD^iB|3F?D+d^E&Tw9EZ*z7LZhXzp2PGhGT73J=!-OV#@ktd>?|O zn16h=osWvynLLcHynu`XZN0Fa&9{``2PRHv&6q+t?*Us4FmXWo!;XY}&MGUQ6CO(m z6`Y$C2>yN<%$V=@`{yCAGdWfxCXI~j|?Jx+^?13=s~4BJ-) zr;ljFSq+|UOy#{}EpBw*dain|snI$5Pfy0CezsVRY7CtOhE(A{s|({Tnqce)abxr` zh#|o$8)0RSHEi#S;+i(s44T+o6r9$b-2l8TELS5vWY;2Tod00;)mS2S~Hxu2A(I)l^E& z#bZOle0$ry!$Us!%qb+H73e@7&6i5%^>_(!z6bXIJlUhcq$9dq$^*fau@YH)L8GBH zI+5DR-edH_Ooxd;dK>TtB$C$_ap?$sKHA`hsWoTw)gSlz1JBy@rvTwsAm$+6tuAGP zGrn4sZrZ%f#d*5HMK~J}*U=2Y(?|Ny!nK42IwSz)MDqKvc`K0ybXvop`34PyEC)po z*f7}Y{1`yt6COxH<&yk-_&mzUiha%Mb?dll%^OqU#{CH6(XfZ2m45#LImQ!b+gfS8 z_N0Sv*6iR0*?3|b6=?>tnQuApu$$3p@p)zVz#YjRxZW!-koz?uN2J+sM#0*H7bo)( zDzLY6lIrag1M80RW@Ub3bn3LRvKB$gQX;EnRMtmU*Us*rXMd(;lWTGM&TkcH@f@}{ zf6MBO6RL5gxz}brk&n8v%NWMnx;dD4;R*+m;anSvk7gbC2U1oNd<*4<5C+BGqahgf z7v2|0pkntSw27_pzbw?>+F$Di&BXkFnK@+-dlLeBbvZ*z6C-D6dLpdT|pAGjnGGMot!JdO-_kCwUV`Av+s;J6jW5X9AA@$6&H?{-c4A zowc2#lD&b^-_C_iTrG@D6vYJpj|CMLQvNsl`MvhN3qX(*l@J900RaF&e+%Gy6A%P| z0tZJzLO?}9LPA49K}AJH`3rw9@F;(A|0pOZ=-6a<=$OQq=qUK~1jOW&G&D4*c+9K} zRIFswG?YM4(9m$Oa9Hs0STr~|B!7vNlr;Y+e=GmI{uA*J>Z@C@2UdFfhnJ zUkDfx1e6eo5lleA036xDpNJ_wAEo~Wf>=<|uzr5e@s>HjDWQSn@57it0FeJz&wqyk z1_1^8OQJvo0RIIbP+%aie~kz;{G0C$0PNpE|56YEe1H+g zqC%Uharuqz?$No}aUJz-ffsvDeYs_s#%-y>{iz4NY;;{# z`3AuHWZ0GCBTEU%30^6RxnrC*IzY@_R2;SsB}m_I!#*xkTacF#7|GZxh(@{)(D!70D(Nt=njq2d$5YjpIov^D>T{@asP-KhZ zB3iXduuV)Y$gsqXlS{T)HY?=_$u^7il{(hp-d%oOICEXj8&cp42&hJMRhtG>T4%>Z zgepnfi!K-Bu}lJO zxs`|s;^ul_zEeZ9O%ZAFUNDR9KB}p*r(xo*Bzan-sy>|`=PwFoiq52Rb#Cm>L0uB# zFICW5RX#WU%x2^bUz{U8iPpQWN29;ZB&ljyO(c!<|vy2{o&LbfVC(^rY;C_M}CHAw#j)iwaUm zYAj3hOhOS2SlPB;30874M>1bvR-~X_ps=IWRxJwjCMEUbj9`|)I0g*_#sFbUNYcs) zD^Y^nG>x$YRa2_ZUZqm^Ykp!=&R(>bFgfbxl@H`eo)4`wBey2Bvfpi5b-isU>o7Pk znKsX>9UQ2s<^K`T*$xK$+?l7GPKxSDcj{k;mQm>f$(5ZfVx?Uvoe~k+%>QQj=jI0& zhn%4PtgU3veila9<3PSVkYV ?qzR0c-EB}sq;rV6%=E*z?oBw6IDWq1=4v&KM; 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znccmHJ<@6$)25(SLg`)#*Ig9{uF{@+g;CSL1 z`Y@VG-bf*niSC>>$5?1rQB@ooe03mPx?a;Ab|w~s)Nu-#O~1VM5$P2FpC~a=^BqKF zcOWGt0Q4VLXJd}>06_Jy?wD|pJq!egBX0qq7T%GVKSco48e|U#NrT|TM)QCGJhAbQ z$h`TV5gq}=+8>Vnr-lOADazKC|Px)+&l~^udS_xM8P#xK>xQ1 zah-qBNras2KWy^h51d6g zoY#0*-KY#X_pcsCt*f4(r#I44E))c@oJlSesA8N(t(80}CH4GgPBNy17L={emA;+zk)@q>7SYk& zcFE-OiPG}PbhWR*4t<@9RnXr7XtI6Auhih;|9$wuco!T#ocL1%0GE-Lg-HXzU@a4E Gz<&W$eB(9% literal 0 HcmV?d00001 diff --git a/rapport/main.tex b/rapport/main.tex new file mode 100644 index 0000000..19a579f --- /dev/null +++ b/rapport/main.tex @@ -0,0 +1,40 @@ +\documentclass[a4paper,10pt]{article} + +\usepackage[francais]{babel} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{lmodern} + +\usepackage{graphicx} +\usepackage{listings} +\usepackage{url} +\usepackage{upquote} +\usepackage{color} +\usepackage[usenames,dvipsnames]{xcolor} + +\title{ICR - Labo \#3 : \textit{Attaque par faute contre RSA-CRT}} +\author{G.Burri} + + +\begin{document} + +\nocite{*} + +\maketitle + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Introduction} + + + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Conclusion} + + +%\bibliographystyle{plain} +%\bibliography{main} + +\end{document} diff --git a/src/Rand.cpp b/src/Rand.cpp new file mode 100644 index 0000000..c82b83f --- /dev/null +++ b/src/Rand.cpp @@ -0,0 +1,65 @@ +#include "Rand.h" + +#include +#include +#include +using namespace std; + +mpz_class Rand::randSize(int sizeBits, bool forceSize) +{ + const int nbBitsFirstByte = ((sizeBits - 1) % 8) + 1; // >= 1 and <= 8 + const int nbBytes = sizeBits / 8 + (nbBitsFirstByte != 8 ? 1 : 0); + + // A fixed rand value used during benchmark tests. + // char randBytes[] {(char)0x3d, (char)0x3f, (char)0x74, (char)0xa3, (char)0xcb, (char)0xba, (char)0x2e, (char)0x26, (char)0xf4, (char)0x5a, (char)0x05, (char)0xb0, (char)0x91, (char)0x2d, (char)0x13, (char)0x4c, (char)0x81, (char)0xa4, (char)0x4e, (char)0x8d, (char)0x50, (char)0xc6, (char)0xa5, (char)0x6e, (char)0x69, (char)0x3b, (char)0x1d, (char)0xa1, (char)0x68, (char)0x79, (char)0x87, (char)0xbe, (char)0xbf, (char)0xc0, (char)0x3d, (char)0x06, (char)0x4a, (char)0x74, (char)0xc8, (char)0x36, (char)0xa0, (char)0x92, (char)0x0c, (char)0xb4, (char)0x99, (char)0x6e, (char)0x64, (char)0x27, (char)0x61, (char)0x18, (char)0xe5, (char)0x07, (char)0xf1, (char)0xcb, (char)0x45, (char)0x57, (char)0x84, (char)0xa8, (char)0xbc, (char)0x25, (char)0x00, (char)0x1c, (char)0x7f, (char)0x1c, (char)0xf7, (char)0x40, (char)0xc5, (char)0x09, (char)0x73, (char)0xbf, (char)0xde, (char)0xb2, (char)0x65, (char)0x15, (char)0x64, (char)0xb4, (char)0x94, (char)0x8f, (char)0x1c, (char)0x6b, (char)0x84, (char)0x4b, (char)0x69, (char)0x6f, (char)0xc7, (char)0x71, (char)0x26, (char)0x3c, (char)0x58, (char)0x5f, (char)0x08, (char)0x34, (char)0x29, (char)0x9a, (char)0xfb, (char)0x95, (char)0x9d, (char)0x4f, (char)0x27, (char)0x49, (char)0x94, (char)0xc4, (char)0x26, (char)0x76, (char)0xc1, (char)0x8e, (char)0x30, (char)0x7f, (char)0x93, (char)0x63, (char)0x68, (char)0x3c, (char)0x0b, (char)0x8f, (char)0x5d, (char)0xff, (char)0x09, (char)0x68, (char)0x1b, (char)0x41, (char)0xea, (char)0xfd, (char)0xe1, (char)0xd5, (char)0x77, (char)0x1d, (char)0x3b, (char)0xda, (char)0x6f, (char)0xb8, (char)0xbe, (char)0x86, (char)0x25, (char)0x8a, (char)0xc1, (char)0x2c, (char)0x87, (char)0xd0, (char)0x24, (char)0xe9, (char)0x0d, (char)0xb3, (char)0x23, (char)0xcb, (char)0x40, (char)0xa3, (char)0x40, (char)0xc7, (char)0x2c, (char)0x3e, (char)0x08, (char)0xdf, (char)0xdc, (char)0xaa, (char)0x7a, (char)0x4a, (char)0x02, (char)0x23, (char)0x73, (char)0xb3, (char)0xc9, (char)0xe5, (char)0x2a, (char)0x36, (char)0xf3, (char)0xd1, (char)0x59, (char)0xfd, (char)0x74, (char)0x63, (char)0x46, (char)0x92, (char)0x47, (char)0x48, (char)0xf0, (char)0x05, (char)0x4c, (char)0xf9, (char)0x92, (char)0xe2, (char)0xe8, (char)0x7e, (char)0x52, (char)0x1a, (char)0x84, (char)0x47, (char)0x9a, (char)0xb5, (char)0x25, (char)0xdc, (char)0xb2, (char)0xed, (char)0x02, (char)0x65, (char)0x76, (char)0x3e, (char)0xad, (char)0xc0, (char)0x3c, (char)0xa0, (char)0x30, (char)0x83, (char)0x0c, (char)0xbe, (char)0x09, (char)0x63, (char)0x36, (char)0x3c, (char)0xc4, (char)0x94, (char)0x79, (char)0x74, (char)0x30, (char)0x71, (char)0x1b, (char)0xac, (char)0x0c, (char)0x2f, (char)0xcf, (char)0xf4, (char)0x68, (char)0x26, (char)0xd9, (char)0xb4, (char)0xb5, (char)0x3c, (char)0x07, (char)0x41, (char)0xa4, (char)0xde, (char)0x80, (char)0x00, (char)0x70, (char)0x06, (char)0x8c, (char)0xa7, (char)0x1b, (char)0x81, (char)0xcb, (char)0x6b, (char)0xc7, (char)0x20, (char)0x45, (char)0x0e, (char)0x23, (char)0x19, (char)0xb8, (char)0xa3, (char)0x29, (char)0x29, (char)0xf4, (char)0x99, (char)0xde, (char)0xfb, (char)0xad, (char)0xa5 }; + + char randBytes[nbBytes]; + readRand(randBytes, nbBytes); + + // We discard the n most significant bits where n >= 0 && n <= 7. + if (nbBitsFirstByte != 8) + randBytes[0] &= 0xFF >> (8 - nbBitsFirstByte); + + // If asked we set the most significant bit to 1. + if (forceSize) + randBytes[0] |= 0x01 << (nbBitsFirstByte - 1); + + mpz_class n; + mpz_import(n.get_mpz_t(), nbBytes, 1, sizeof(char), 1, 0, randBytes); // Big-endian -> most significant byte first. + return n; +} + +mpz_class Rand::randLimits(const mpz_class& from, const mpz_class& to) +{ + const mpz_class nbValues = to - from + 1; + const size_t nbBits = mpz_sizeinbase(mpz_class(nbValues - 1).get_mpz_t(), 2); + + // A 'nbBytes' size random number is generated, if it matches the given limits we return it. + for (;;) + { + mpz_class n = randSize(nbBits, false) + from; + if (n >= from && n <= to) + return n; + } +} + +mpz_class Rand::randPrime(const int size_bits) +{ + mpz_class r = randSize(size_bits); + mpz_nextprime(r.get_mpz_t(), r.get_mpz_t()); + return r; +} + +void Rand::readRand(char* to, int size) +{ + const string RANDOM_SOURCE("/dev/urandom"); + + static fstream ifs(RANDOM_SOURCE, ios_base::in); + if (!ifs) + { + cerr << "Unable to read the random source: " << RANDOM_SOURCE << endl; + throw 1; + } + + ifs.read(to, size); +} diff --git a/src/Rand.h b/src/Rand.h new file mode 100644 index 0000000..2ee12a1 --- /dev/null +++ b/src/Rand.h @@ -0,0 +1,36 @@ +#ifndef RAND_H +#define RAND_H + +#include + +class Rand +{ +public: + /** + * Returns a random number with a maximum size of 'sizeBits'. + * If 'forceSize' is true the size is exactly 'sizeBits', so the highest bit will always be 1. + * + * For example, for 'sizeBits' = 12 and 'forceSize' = true, the returned number will be in the following range: + * [2^11, 2^12-1] + * + * For 'sizeBits' = 12 and 'forceSize' = false, the returned number will be in the following range: + * [0, 2^12-1] + */ + static mpz_class randSize(int sizeBits, bool forceSize = true); + + /** + * Return a random number n such as: + * 'from' <= n <= 'to' + */ + static mpz_class randLimits(const mpz_class& from, const mpz_class& to); + + static mpz_class randPrime(const int size_bits); + +private: + /** + * Read a rand from "/dev/urandom" of size 'size' in byte to the given array 'to'. + */ + static void readRand(char* to, int size); +}; + +#endif diff --git a/src/Rsa.cpp b/src/Rsa.cpp new file mode 100644 index 0000000..be21998 --- /dev/null +++ b/src/Rsa.cpp @@ -0,0 +1,47 @@ +#include "Rsa.h" + +using namespace std; + +#include "Rand.h" +#include "Utils.h" + +pair Rsa::generateRSAKeys(uint exponent, uint keySizeBits) +{ + mpz_class p, q, phi; + KeyPub kPub; + KeyPriv kPriv; + + do + { + kPub.e = exponent; + p = Rand::randPrime(keySizeBits / 2); + q = Rand::randPrime(keySizeBits / 2); + + kPriv.n = kPub.n = p * q; + phi = (p - 1) * (q - 1); + + } while (mpz_invert(kPriv.d.get_mpz_t(), kPub.e.get_mpz_t(), phi.get_mpz_t()) == 0); + +// For debugging purpose. +// print("p", p); +// print("q", q); +// print("n", kPub.n); +// print("phi", phi); +// print("d", kPriv.d); + + return make_pair(kPub, kPriv); +} + +mpz_class Rsa::sign(const mpz_class& m, const KeyPriv& kPriv) +{ + mpz_class result; + mpz_powm_sec(result.get_mpz_t(), m.get_mpz_t(), kPriv.d.get_mpz_t(), kPriv.n.get_mpz_t()); + return result; +} + +bool Rsa::verifySignature(const mpz_class& m, const mpz_class& sig, const KeyPub& kPub) +{ + mpz_class mPrime; + mpz_powm_sec(mPrime.get_mpz_t(), sig.get_mpz_t(), kPub.e.get_mpz_t(), kPub.n.get_mpz_t()); + return m == mPrime; +} diff --git a/src/Rsa.h b/src/Rsa.h new file mode 100644 index 0000000..cbc3676 --- /dev/null +++ b/src/Rsa.h @@ -0,0 +1,27 @@ +#ifndef RSA_H +#define RSA_H + +#include +#include + +class Rsa +{ +public: + struct KeyPub { + mpz_class n; + mpz_class e; + }; + + struct KeyPriv { + mpz_class n; + mpz_class d; + }; + + static std::pair generateRSAKeys(uint exponent, uint keySizeBits); + + static mpz_class sign(const mpz_class& m, const KeyPriv& kPriv); + + static bool verifySignature(const mpz_class& m, const mpz_class& sig, const KeyPub& kPub); +}; + +#endif diff --git a/src/RsaCrt.cpp b/src/RsaCrt.cpp new file mode 100644 index 0000000..43f0b97 --- /dev/null +++ b/src/RsaCrt.cpp @@ -0,0 +1 @@ +#include "RsaCrt.h" diff --git a/src/RsaCrt.h b/src/RsaCrt.h new file mode 100644 index 0000000..29bc4b9 --- /dev/null +++ b/src/RsaCrt.h @@ -0,0 +1,20 @@ +#ifndef RSACRT_H +#define RSACRT_H + +#include + +class RsaCrt +{ +public: + RsaCrt(); + + struct KeyPrivCRT { + mpz_class p; + mpz_class q; + mpz_class dp; + mpz_class dq; + mpz_class qInv; + }; +}; + +#endif diff --git a/src/Utils.cpp b/src/Utils.cpp new file mode 100644 index 0000000..b60be50 --- /dev/null +++ b/src/Utils.cpp @@ -0,0 +1,24 @@ +#include "Utils.h" + +#include +using namespace std; + +void Utils::print(string name, mpz_class value) +{ + cout << name << ": " << value << endl; +} + +Timer::Timer() : + time(chrono::high_resolution_clock::now()) +{} + +int Timer::ms() const +{ + return chrono::duration_cast(chrono::high_resolution_clock::now() - this->time).count(); +} + +ostream& operator<<(ostream& os, const Timer& t) +{ + os << t.ms() << " ms"; + return os; +} diff --git a/src/Utils.h b/src/Utils.h new file mode 100644 index 0000000..d765da0 --- /dev/null +++ b/src/Utils.h @@ -0,0 +1,29 @@ +#ifndef UTILS_H +#define UTILS_H + +#include +#include +#include + +#include + +class Utils +{ +public: + static void print(std::string name, mpz_class value); +}; + +class Timer +{ +public: + Timer(); + int ms() const; + friend std::ostream& operator<<(std::ostream& os, const Timer& t); + +private: + std::chrono::time_point time; +}; + +std::ostream& operator<<(std::ostream& os, const Timer& t); + +#endif diff --git a/src/lab3.qbs b/src/lab3.qbs new file mode 100644 index 0000000..2117c09 --- /dev/null +++ b/src/lab3.qbs @@ -0,0 +1,27 @@ +import qbs 1.0 + +Product { + name: "lab3" + type: "application" + + files: ["main.cpp", "Rand.h", "Rand.cpp", "Rsa.h", "Rsa.cpp", "RsaCrt.h", "RsaCrt.cpp", "Utils.h", "Utils.cpp"] + + cpp.commonCompilerFlags: ["-std=c++11"] + cpp.staticLibraries: ["gmp", "gmpxx"] + + Properties { + condition: qbs.buildVariant == "release" + + //cpp.commonCompilerFlags: outer.concat("") + + // Override the default optimization (-02). + cpp.optimization: "none" + cpp.cxxFlags: "-O3" + } + Properties { + condition: qbs.buildVariant == "debug" + cpp.debugInformation: true + } + + Depends { name: "cpp" } +} diff --git a/src/main.cpp b/src/main.cpp new file mode 100644 index 0000000..ac75df1 --- /dev/null +++ b/src/main.cpp @@ -0,0 +1,40 @@ +/** + * ICR - Labo 3. + * It uses GMPxx lib and C++11. + * It needs the RNG file "/dev/urandom". + * + * Author: Grégory Burri + */ + +#include +using namespace std; + +#include + +#include "Rsa.h" +#include "RsaCrt.h" + +const uint KEY_SIZE_BITS = 1024; +const uint RSA_PUBLIC_EXPONENT = 65537; + +int main(int argc, char** argv) +{ + vector args; + for (int i = 0; i < argc; i++) + args.push_back(string(argv[i])); + + const auto& keys = Rsa::generateRSAKeys(RSA_PUBLIC_EXPONENT, KEY_SIZE_BITS); + const auto& kPub = keys.first; + const auto& kPriv = keys.second; + + mpz_class message(42); + mpz_class signature = Rsa::sign(message, kPriv); + + mpz_class message2(42); + cout << "verify: " << Rsa::verifySignature(message2, signature, kPub) << endl; + + mpz_class message3(43); + cout << "verify: " << Rsa::verifySignature(message3, signature, kPub) << endl; + + return 0; +} -- 2.45.2