Lifted ElGamal 門限加密演算法

PamShao 發表於 2022-06-07
演算法 加密

本文詳細學習Lifted ElGamal 門限加密演算法

門限加密體制

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(1)門限加密是可以抗合謀
(2)表現在私鑰分為\(n\)份,至少需要\(t\)份才能解密成功,叫做(t-n)門限。類似於“祕密分享”。

ElGamal演算法

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(1)源自【A public key cryptosystem and a signature scheme based on discrete logarithms】給出了加法和乘法同態性的定義,其中加法同態只能用於小的明文域。
(2)\(G\)是階為\(p\)的群,\(g\)是群\(G\)的生成元,基於\(DDH\)問題,公鑰是\(PK=(G,p,g,h)\),私鑰是\(SK=s\),其中\(g^s=h\)
(3)加密:選擇一個隨機數\(r\in Z_p\),計算\(Enc_{PK}(m,r)=<g^r,h^r*g^m>\);解密:密文\(c=<\alpha,\beta>\),計算\(g^m=\beta*\alpha^{-s}\),最後得到\(m\)\(m\)只能是小資料,如果太大則根據離散對數問題\(m\)是難解的】

原論文中給出的公鑰加密方案是:

參考:ElGamal演算法

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Lifted ElGamal 門限加密演算法

image
image
(1)源自【A public key cryptosystem and a signature scheme based on discrete logarithms】
(2)這裡將明文放在了指數上,恢復明文,就需要計算離散對數,所以\(\rho\)選取要很小,不然很難恢復明文。
(3)通過查表(離散對數表)來獲取結果,這裡對應上面的exhaustive search
(4)金鑰生成,加密,解密和原ElGamal類似。

開源庫

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

該庫實現了具有加法同態性的Lifted-ElGamal演算法【A Public Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms】和實現了非互動式的零知識證明【Methods for Restricting Message Space in Public-Key Encryption】。

基於Lifted-ElGamal演算法給出了一個投票系統。

安裝

環境:MacOS

(1)依賴庫
OpenSSL:安裝參考
GMP(libgmp-dev)
(2)下載

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
#如果卡的話,換成  https://github.com/***.git

其中:

  • cybozulib_ext是VC(Visual C++)中使用OpenSSL和GMP庫所需的
  • Xbyak在Intel系CPU中提升運算速度
  • Linux通過apt-get等獲取OpenSSL和libgmp-dev

測試

(1)有限域\(F_p\)上進行測試

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複製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,判斷是否解密成功
	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)測試加解密和同態計算

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);
	}
}

同態計算:
(1)enc(m1) enc(m2) = enc(m1 + m2)
(2)enc(m1) x = enc(m1 + x)

投票例子

介紹

每個投票者對“0”或“1”進行加密,並單獨將其密文傳送到伺服器,伺服器計算結果,而不知道每次投票或結果本身,示例程式碼模擬了該方案。
編譯後得到檔案:vote_tool.exe

執行

執行命令:

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

初始化系統並生成公鑰(vote\u pub.txt)和金鑰(vote\u prv.txt)。secp192k1用作EC ElGamal加密的引數。
(2)投票

vote_tool.exe vote [-l a bit vector]

輸入一個長度為n 位向量v[i](1bit),表示第i個投票人的投票

使用公鑰對v[i]加密,並打亂密文序列的順序。然後將每個密文儲存在vote_0.txt,...,vote_n.txt中的任何一個檔案中。由於順序被打亂了,伺服器無法檢測哪個檔案包含誰的投票。此過程模擬每個投票者單獨傳送使用自己的公鑰加密的密文。
(3)統計

vote_tool.exe count

該程式從檔案中讀取所有密文,並在不解密的情況下檢查是否是“0”或“1”加密的。這是通過使用第三方庫提供的非互動式零知識證明來實現的。該程式在不解密的情況下聚合密文並驗證所有密文。
(4)開啟

vote_tool.exe open

該程式解密密文寫入的result.txt,並在控制檯上顯示結果。

結果:

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

原始碼

參考

1、集合交集問題的安全計算
2、Simple, Fast Malicious Multiparty Private Set Intersection

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