Lifted ElGamal 门限加密算法

本文详细学习Lifted ElGamal 门限加密算法

门限加密体制

（1）门限加密是可以抗合谋的
（2）表现在私钥分为\(n\)份，至少需要\(t\)份才能解密成功，叫做（t-n）门限。类似于“秘密分享”。

ElGamal算法

（1）源自【A public key cryptosystem and a signature scheme based on discrete logarithms】给出了加法和乘法同态性的定义，其中加法同态只能用于小的明文域。
（2）\(G\)是阶为\(p\)的群，\(g\)是群\(G\)的生成元，基于\(DDH\)问题，公钥是\(PK=(G,p,g,h)\)，私钥是\(SK=s\)，其中\(g^s=h\)。
（3）加密：选择一个随机数\(r\in Z_p\)，计算\(Enc_{PK}(m,r)=<g^r,h^r*g^m>\)；解密：密文\(c=<\alpha,\beta>\)，计算\(g^m=\beta*\alpha^{-s}\)，最后得到\(m\)【\(m\)只能是小数据，如果太大则根据离散对数问题\(m\)是难解的】

原论文中给出的公钥加密方案是：

参考：ElGamal算法

Lifted ElGamal 门限加密算法

（1）源自【A public key cryptosystem and a signature scheme based on discrete logarithms】
（2）这里将明文放在了指数上，恢复明文，就需要计算离散对数，所以\(\rho\)选取要很小，不然很难恢复明文。
（3）通过查表（离散对数表）来获取结果，这里对应上面的exhaustive search
（4）密钥生成，加密，解密和原ElGamal类似。

开源库

github：https://github.com/aistcrypt/Lifted-ElGamal

该库实现了具有加法同态性的Lifted-ElGamal算法【A Public Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms】和实现了对受限明文空间的零知识证明【Methods for Restricting Message Space in Public-Key Encryption】。

基于Lifted-ElGamal算法给出了一个投票系统。

安装

环境：MacOS

（1）依赖库
OpenSSL：安装参考
GMP(libgmp-dev)
（2）下载

mkdir work
cd work
git clone git://github.com/aistcrypt/Lifted-ElGamal.git
git clone git://github.com/herumi/xbyak.git
git clone git://github.com/herumi/mie.git
git clone git://github.com/herumi/cybozulib.git
git clone git://github.com/herumi/cybozulib_ext.git
#如果卡的话，换成  https://github.com/***.git

其中：

• cybozulib_ext是VC（Visual C++）中使用OpenSSL和GMP库所需的
• Xbyak在Intel系CPU中提升运算速度
• Linux通过apt-get等获取OpenSSL和libgmp-dev

测试

（1）有限域\(F_p\)上进行测试

CYBOZU_TEST_AUTO(testFp)
{
	typedef mie::FpT<mie::Gmp, TagFp> Zn;
	typedef mie::ElgamalT<Fp, Zn> ElgamalFp;
	/*
		Zn = (Z/mZ) - {0}
	*/
	const int m = 65537;
	{
		std::ostringstream os;
		os << m;
		Fp::setModulo(os.str());
	}
	{
		std::ostringstream os;
		os << m - 1;
		Zn::setModulo(os.str());
	}
	ElgamalFp::PrivateKey prv;

	/*
		3^(m-1) = 1
	*/
	const int f = 3;
	{
		Fp x(f);
		Fp::power(x, x, m - 1);
		CYBOZU_TEST_EQUAL(x, 1);
	}
	prv.init(f, 17, rg);
	const ElgamalFp::PublicKey& pub = prv.getPublicKey();

	const int m1 = 12345;
	const int m2 = 17655;
	ElgamalFp::CipherText c1, c2;
	pub.enc(c1, m1, rg);
	pub.enc(c2, m2, rg);
	// BitVector
	{
		cybozu::BitVector bv;
		c1.appendToBitVec(bv);
		ElgamalFp::CipherText c3;
		c3.fromBitVec(bv);//c3复制c1
		CYBOZU_TEST_EQUAL(c1.c1, c3.c1);
		CYBOZU_TEST_EQUAL(c1.c2, c3.c2);
	}
	Zn dec1, dec2;
	prv.dec(dec1, c1);
	prv.dec(dec2, c2);
	// dec(enc) = id，判断是否解密成功
	CYBOZU_TEST_EQUAL(dec1, m1);
	CYBOZU_TEST_EQUAL(dec2, m2);
	// iostream
	{
		ElgamalFp::PublicKey pub2;
		ElgamalFp::PrivateKey prv2;
		ElgamalFp::CipherText cc1, cc2;
		{
			std::stringstream ss;
			ss << prv;
			ss >> prv2;
		}
		Zn d;
		prv2.dec(d, c1);
		CYBOZU_TEST_EQUAL(d, m1);
		{
			std::stringstream ss;
			ss << c1;
			ss >> cc1;
		}
		d = 0;
		prv2.dec(d, cc1);
		CYBOZU_TEST_EQUAL(d, m1);
		{
			std::stringstream ss;
			ss << pub;
			ss >> pub2;
		}
		pub2.enc(cc2, m2, rg);
		prv.dec(d, cc2);
		CYBOZU_TEST_EQUAL(d, m2);
	}
	// enc(m1) enc(m2) = enc(m1 + m2)
	c1.add(c2);
	prv.dec(dec1, c1);
	CYBOZU_TEST_EQUAL(dec1, m1 + m2);
	// enc(m1) x = enc(m1 + x)
	const int x = 555;
	pub.add(c1, x);
	prv.dec(dec1, c1);
	CYBOZU_TEST_EQUAL(dec1, m1 + m2 + x);
	// rerandomize
	c1 = c2;
	pub.rerandomize(c1, rg);
	// verify c1 != c2
	CYBOZU_TEST_ASSERT(c1.c1 != c2.c1);
	CYBOZU_TEST_ASSERT(c1.c2 != c2.c2);
	prv.dec(dec1, c1);
	// dec(c1) = dec(c2)
	CYBOZU_TEST_EQUAL(dec1, m2);

	// check neg
	{
		ElgamalFp::CipherText c;
		Zn m = 1234;
		pub.enc(c, m, rg);
		c.neg();
		Zn dec;
		prv.dec(dec, c);
		CYBOZU_TEST_EQUAL(dec, -m);
	}
	// check mul
	{
		ElgamalFp::CipherText c;
		Zn m = 1234;
		int x = 111;
		pub.enc(c, m, rg);
		c.mul(x);
		Zn dec;
		prv.dec(dec, c);
		m *= x;
		CYBOZU_TEST_EQUAL(dec, m);
	}
	// check negative value
	for (int i = -10; i < 10; i++) {
		ElgamalFp::CipherText c;
		const Zn mm = i;
		pub.enc(c, mm, rg);
		Zn dec;
		prv.dec(dec, c, 1000);
		CYBOZU_TEST_EQUAL(dec, mm);
	}

	// isZeroMessage
	for (int m = 0; m < 10; m++) {
		ElgamalFp::CipherText c0;
		pub.enc(c0, m, rg);
		if (m == 0) {
			CYBOZU_TEST_ASSERT(prv.isZeroMessage(c0));
		} else {
			CYBOZU_TEST_ASSERT(!prv.isZeroMessage(c0));
		}
	}
	// zkp
	{
		ElgamalFp::Zkp zkp;
		ElgamalFp::CipherText c;
		cybozu::crypto::Hash hash(cybozu::crypto::Hash::N_SHA256);
		pub.encWithZkp(c, zkp, 0, hash, rg);
		CYBOZU_TEST_ASSERT(pub.verify(c, zkp, hash));
		zkp.s0 += 1;
		CYBOZU_TEST_ASSERT(!pub.verify(c, zkp, hash));
		pub.encWithZkp(c, zkp, 1, hash, rg);
		CYBOZU_TEST_ASSERT(pub.verify(c, zkp, hash));
		zkp.s0 += 1;
		CYBOZU_TEST_ASSERT(!pub.verify(c, zkp, hash));
		CYBOZU_TEST_EXCEPTION_MESSAGE(pub.encWithZkp(c, zkp, 2, hash, rg), cybozu::Exception, "encWithZkp");
	}
}

（2）测试加解密和同态计算

CYBOZU_TEST_AUTO(testEc)
{
	typedef mie::FpT<mie::Gmp, TagEc> Zn;
	typedef mie::ElgamalT<Ec, Zn> ElgamalEc;
	Fp::setModulo(para.p);
	Zn::setModulo(para.n);
	Ec::setParam(para.a, para.b);
	const Fp x0(para.gx);
	const Fp y0(para.gy);
	const size_t bitLen = Zn(-1).getBitLen();
	const Ec P(x0, y0);
	/*
		Zn = <P>
	*/
	ElgamalEc::PrivateKey prv;
	prv.init(P, bitLen, rg);
	prv.setCache(0, 60000);
	const ElgamalEc::PublicKey& pub = prv.getPublicKey();

	const int m1 = 12345;
	const int m2 = 17655;
	ElgamalEc::CipherText c1, c2;
	pub.enc(c1, m1, rg);
	pub.enc(c2, m2, rg);
	// BitVector
	{
		cybozu::BitVector bv;
		c1.appendToBitVec(bv);
		ElgamalEc::CipherText c3;
		c3.fromBitVec(bv);
		CYBOZU_TEST_EQUAL(c1.c1, c3.c1);
		CYBOZU_TEST_EQUAL(c1.c2, c3.c2);
	}
	Zn dec1, dec2;
	prv.dec(dec1, c1);
	prv.dec(dec2, c2);
	// dec(enc) = id
	CYBOZU_TEST_EQUAL(dec1, m1);
	CYBOZU_TEST_EQUAL(dec2, m2);
	// iostream
	{
		ElgamalEc::PublicKey pub2;
		ElgamalEc::PrivateKey prv2;
		ElgamalEc::CipherText cc1, cc2;
		{
			std::stringstream ss;
			ss << prv;
			ss >> prv2;
		}
		prv.setCache(-200, 60000);
		Zn d;
		prv2.dec(d, c1);
		CYBOZU_TEST_EQUAL(d, m1);
		{
			std::stringstream ss;
			ss << c1;
			ss >> cc1;
		}
		d = 0;
		prv2.dec(d, cc1);
		CYBOZU_TEST_EQUAL(d, m1);
		{
			std::stringstream ss;
			ss << pub;
			ss >> pub2;
		}
		pub2.enc(cc2, m2, rg);
		prv.dec(d, cc2);
		CYBOZU_TEST_EQUAL(d, m2);
	}
	// enc(m1) enc(m2) = enc(m1 + m2)
	c1.add(c2);
	prv.dec(dec1, c1);
	CYBOZU_TEST_EQUAL(dec1, m1 + m2);
	// enc(m1) x = enc(m1 + x)
	const int x = 555;
	pub.add(c1, x);
	prv.dec(dec1, c1);
	CYBOZU_TEST_EQUAL(dec1, m1 + m2 + x);
	// rerandomize
	c1 = c2;
	pub.rerandomize(c1, rg);
	// verify c1 != c2
	CYBOZU_TEST_ASSERT(c1.c1 != c2.c1);
	CYBOZU_TEST_ASSERT(c1.c2 != c2.c2);
	prv.dec(dec1, c1);
	// dec(c1) = dec(c2)
	CYBOZU_TEST_EQUAL(dec1, m2);

	// check neg
	{
		ElgamalEc::CipherText c;
		Zn m = 1234;
		pub.enc(c, m, rg);
		c.neg();
		Zn dec;
		prv.dec(dec, c);
		CYBOZU_TEST_EQUAL(dec, -m);
	}
	// check mul
	{
		ElgamalEc::CipherText c;
		Zn m = 123;
		int x = 111;
		pub.enc(c, m, rg);
		Zn dec;
		prv.dec(dec, c);
		c.mul(x);
		prv.dec(dec, c);
		m *= x;
		CYBOZU_TEST_EQUAL(dec, m);
	}

	// check negative value
	for (int i = -10; i < 10; i++) {
		ElgamalEc::CipherText c;
		const Zn mm = i;
		pub.enc(c, mm, rg);
		Zn dec;
		prv.dec(dec, c, 1000);
		CYBOZU_TEST_EQUAL(dec, mm);
	}

	// isZeroMessage
	for (int m = 0; m < 10; m++) {
		ElgamalEc::CipherText c0;
		pub.enc(c0, m, rg);
		if (m == 0) {
			CYBOZU_TEST_ASSERT(prv.isZeroMessage(c0));
		} else {
			CYBOZU_TEST_ASSERT(!prv.isZeroMessage(c0));
		}
	}
	// zkp
	{
		ElgamalEc::Zkp zkp;
		ElgamalEc::CipherText c;
//		cybozu::Sha1 hash;
		cybozu::crypto::Hash hash(cybozu::crypto::Hash::N_SHA256);
		pub.encWithZkp(c, zkp, 0, hash, rg);
		CYBOZU_TEST_ASSERT(pub.verify(c, zkp, hash));
		zkp.s0 += 1;
		CYBOZU_TEST_ASSERT(!pub.verify(c, zkp, hash));
		pub.encWithZkp(c, zkp, 1, hash, rg);
		CYBOZU_TEST_ASSERT(pub.verify(c, zkp, hash));
		zkp.s0 += 1;
		CYBOZU_TEST_ASSERT(!pub.verify(c, zkp, hash));
		CYBOZU_TEST_EXCEPTION_MESSAGE(pub.encWithZkp(c, zkp, 2, hash, rg), cybozu::Exception, "encWithZkp");
	}
	// cache
	{
		const int m1 = 9876;
		const int m2 = -3142;
		ElgamalEc::CipherText c1, c2;
		pub.enc(c1, m1, rg);
		pub.enc(c2, m2, rg);
		prv.setCache(-10000, 10000);
		int dec1 = prv.dec(c1);
		int dec2 = prv.dec(c2);
		CYBOZU_TEST_EQUAL(m1, dec1);
		CYBOZU_TEST_EQUAL(m2, dec2);
		c1.add(c2);
		bool b;
		int dec = prv.dec(c1, &b);
		CYBOZU_TEST_EQUAL(m1 + m2, dec);
		CYBOZU_TEST_ASSERT(b);
		prv.clearCache();
		prv.dec(c1, &b);
		CYBOZU_TEST_ASSERT(!b);
	}
	// benchmark
	{
		int m = 12345;
		ElgamalEc::CipherText c;
		CYBOZU_BENCH("enc", pub.enc, c, m, rg);
		prv.setCache(0, 20000);
		CYBOZU_BENCH("dec", prv.dec, c);
		CYBOZU_BENCH("rand", pub.rerandomize, c, rg);
	}
}

（3）功能测试

#include <mcl/fp.hpp>
#include <mcl/ecparam.hpp>
#include <mcl/elgamal.hpp>

struct TagZn;
typedef mcl::FpT<> Fp;
typedef mcl::FpT<TagZn> Zn;
typedef mcl::EcT<Fp> Ec;
typedef mcl::ElgamalT<Ec, Zn> ElgamalEc;

const mcl::EcParam& para = mcl::ecparam::secp192k1;
cybozu::RandomGenerator g_rg;

int main()
{
	Ec P;
	mcl::initCurve<Ec, Zn>(para.curveType, &P);
	const size_t bitSize = Zn::getBitSize();
	/*
		Zn = <P>
	*/
	ElgamalEc::PrivateKey prv;
	prv.init(P, bitSize, g_rg);
	prv.setCache(0, 60000);
	const ElgamalEc::PublicKey& pub = prv.getPublicKey();

	//加解密测试
	const int m1 = 12345;
	const int m2 = 17655;
	ElgamalEc::CipherText c1, c2;
	pub.enc(c1, m1, g_rg);
	pub.enc(c2, m2, g_rg);
	std::cout<<"c1="<<c1.getStr()<<std::endl;
	std::cout<<"c2="<<c2.getStr()<<std::endl;
	std::cout<<"dec1="<<prv.dec(c1)<<std::endl;
	std::cout<<"dec2="<<prv.dec(c2)<<std::endl;
	std::cout<<"加解密测试：ok!"<<std::endl;

	//enc(m1) enc(m2) = enc(m1 + m2)
	c1.add(c2);
	std::cout<<"m1+m2="<<prv.dec(c1)<<std::endl;
	std::cout<<"同态加法（c2c）测试：ok!"<<std::endl;

	// enc(m1) x = enc(m1 + x)
	const int x = 2;
	pub.add(c2, x);
	std::cout<<"m2+x="<<prv.dec(c2)<<std::endl;
	std::cout<<"同态加法(c2m)测试：ok!"<<std::endl;

	// check mul
	Zn dec2;	
	c2.mul(x);
	prv.dec(dec2, c2);
	std::cout<<"m2*x="<<dec2<<std::endl;
	std::cout<<"同态乘法(c2m)测试：ok!"<<std::endl;
	return 0;
}

同态计算：
（1）enc(m1) enc(m2) = enc(m1 + m2)
（2）enc(m1) x = enc(m1 + x)

投票例子

介绍

每个投票者对“0”或“1”进行加密，并单独将其密文发送到服务器，服务器计算结果，而不知道每次投票或结果本身，示例代码模拟了该方案。
编译后得到文件：vote_tool.exe

这里的Lifted-ElGamal程序只生成了一对公私钥，并不是如算法一样，生成一个公钥和多个私钥

运行

运行命令：

vote_tool.exe [opt] mode mode: select any one of init/vote/count/open -l: input a bit vector

（1）初始化

vote_tool.exe init

初始化系统并生成公钥（vote\u pub.txt）和密钥（vote\u prv.txt）。secp192k1用作EC ElGamal加密的参数。
（2）投票

vote_tool.exe vote [-l a bit vector]

输入一个长度为n 位向量v[i]（1bit），表示第i个投票人的投票

使用公钥对v[i]加密，并打乱密文序列的顺序。然后将每个密文存储在vote_0.txt,...,vote_n.txt中的任何一个文件中。由于顺序被打乱了，服务器无法检测哪个文件包含谁的投票。此过程模拟每个投票者单独发送使用自己的公钥加密的密文。
（3）统计

vote_tool.exe count

该程序从文件中读取所有密文，并在不解密的情况下检查是否是“0”或“1”加密的。这是通过使用第三方库提供的非交互式零知识证明来实现的。该程序在不解密的情况下聚合密文并验证所有密文。
（4）打开

vote_tool.exe open

该程序解密密文写入的result.txt，并在控制台上显示结果。

结果：

PamdeMacBook-Air:bin pam$ ./vote_toold.exe init
mode=init
make privateKey=vote_prv.txt, publicKey=vote_pub.txt
PamdeMacBook-Air:bin pam$ ./vote_toold.exe vote -l 101010011mode=vote
voters=101010011
shuffle
each voter votes
make vote_5.txt
make vote_1.txt
make vote_6.txt
make vote_4.txt
make vote_7.txt
make vote_8.txt
make vote_0.txt
make vote_2.txt
make vote_3.txt
PamdeMacBook-Air:bin pam$ ./vote_toold.exe count
mode=count
aggregate votes
add vote_0.txt
add vote_1.txt
add vote_2.txt
add vote_3.txt
add vote_4.txt
add vote_5.txt
add vote_6.txt
add vote_7.txt
add vote_8.txt
create result file : vote_ret.txt
PamdeMacBook-Air:bin pam$ ./vote_toold.exe open
mode=open
result of vote count 5

源码

见github

参考

1、集合交集问题的安全计算
2、Simple, Fast Malicious Multiparty Private Set Intersection