基于JPBC的无证书聚合签名方案实现

摘要

一开始签名方案是基于PKI的，无证书签名起源于 基于身份密码体制， 2009 年第一篇无证书签名方案1被提出，随后出现了一些列方案2,3;包括无配对的无证书聚合签名方案4,更多内容参考文献5.

暂时没有看见无证书聚合签名方案实现相关的代码，本文基于JPBC库实现，使用方法可以参考B站视频。也可以使用C++和PBC库实现。

方案概述

本文方案是基于论文3描述的。

Setup：
给定安全参数 $\kappa \in Z$，KGC选择两个循环群 $G_1$ 、$G_2$，其阶均为素数 $q$ ， $G_1$ 的生成元$P$，计算可接受的配对 $e:G_1\times G_1\to G_2$ 。 KGC 随机选择主密钥 $s\in Z_q^{\ast }$ ，设置 $P_{pub}=sP$ ，选择加密哈希函数 H 1 : { 0 , 1 } ∗ → G 1 H_1:\{0,1\}^\ast→G_1 H1​:{0,1}∗→G1​ , H 2 : { 0 , 1 } ∗ → G 1 H_2:\{0,1\}^\ast→G_1 H2​:{0,1}∗→G1​, H 3 : { 0 , 1 } ∗ → Z q ∗ H_3:\{0,1\}^\ast→Z_q^\ast H3​:{0,1}∗→Zq∗​. 系统参数为 { q , G 1 , G 2 , e , P , P p u b , H 1 , H 2 , H 3 } \{q,G_1,G_2,e,P,P_{pub},H_1,H_2,H_3 \} {q,G1​,G2​,e,P,Ppub​,H1​,H2​,H3​}, 主密钥是s。

PartialKeyGen:
给定用户身份 I D i ∈ { 0 , 1 } ∗ ID_i\in \{0,1\}^\ast IDi​∈{0,1}∗，KGC首先计算$Q_{I{D}_i}=H_1(I{D}_i)$。 然后，它设置该用户的部分密钥$ps{k}_{I{D}_i}=s{Q}_{(}I{D}_i)$并将其通过安全通道传输给对应用户。 用户可以通过检查是否正确来检查其正确性$e(ps{k}_{I{D}_i},P)=e(Q_{I{D}_i},P_{pub})$。

UserKeyGen:
用户$I{D}_i$随机选择值$x_{I{D}_i}\in Z_q^{\ast }$作为他的用户私钥$us{k}_{I{D}_i}$，并且计算用户公钥$up{k}_{I{D}_i}=x_{I{D}_i}P$。

Sign:
对于消息$m_i\in {0,1}^{\ast }$，选择状态信息$∆$(选择公开参数作为状态信息)，具有$I{D}_i$身份的签名者执行以下步骤：

1. 选择随机数$r_i\in Z_q^{\ast }$并且计算$U_i=r_{i}P\in G_1$.
2. 计算$Q=H_2(∆),h_i=H_3(m_i,I{D}_i,up{k}_{I{D}_i},U_i)$
3. 计算$V_i=ps{k}_{I{D}_i}+r_i\cdot Q+h_i\cdot x_i\cdot P_{pub}$.

输出$(U_i,V_i)$作为$m_i$的签名。

Verify ：
给定一个消息$m_i$签名$(U_i,V_i)$，其对应的身份为$I{D}_i$和公钥$up{k}_{I{D}_i}$，计算$Q_{(}I{D}_i)=H_1(I{D}_i),Q=H_2(∆)$ ,
$h_i=H_3(m_i,I{D}_i,up{k}_{I{D}_i},U_i)$
如果等式 $e(V_i,P)=e(h_i\cdot up{k}_{I{D}_i}+Q_{I{D}_i},P_{pub})e(U_i,Q)$ 成立，接收签名；否则拒绝。

Aggregate：
任何人都可以充当聚合签名生成器, 对于n个用户的聚合集 { U 1 , … , U n } \{U_1,…,U_n\} {U1​,…,Un​}其对应身份为 { I D 1 , … , I D n } \{ID_1,…,ID_n \} {ID1​,…,IDn​}以及对应公钥 { u p k 1 , … , u p k n } \{upk_1,…,upk_n\} {upk1​,…,upkn​},对应消息签名对为$\{(m_1,{\sigma }_1=(U_1,V_1)),\dots ,(m_n,{\sigma }_n=(U_n,V_n))\}$,
聚合签名计算：$V={\sum }_{i=1}^n{V}_i$, 将$\sigma =(U_1,\dots ,U_n,V)$作为聚合签名。

Aggregate-Verify：
验证聚合签名$\sigma =(U_1,\dots ,U_n,V)$由n个用户 { U 1 , … , U n } \{U_1,…,U_n\} {U1​,…,Un​}身份为 { I D 1 , … , I D n } \{ID_1,…,ID_n\} {ID1​,…,IDn​}和对应的公钥 { u p k 1 , … , u p k n } \{upk_1,…,upk_n\} {upk1​,…,upkn​}；关于消息 { m 1 , … , m n } \{m_1,…,m_n \} {m1​,…,mn​} 验证者执行以下操作:
计算 $Q_{I{D}_i}=H_1(I{D}_i),Q=H_2(∆),h_i=H_3(m_i,I{D}_i,up{k}_{I{D}_i},U_i),i=1,\dots ,n$.
验证：$e(V,P)=e({\sum }_{i=1}^n[h_i\cdot up{k}_{I{D}_i}+Q_{I{D}_i}],P_{pub})e({\sum }_{i=1}^n{U}_i,Q)$.如果满足则通过；否则验证失败。

代码实现

整体思路， 将相关密钥信息作为文件保存（序列化后可以通过网络传输），根据方案不同的功能设计不同的函数。

关于文件a_181_603.properties的内容

type=a
q=98826429041171753291515535532523512299028170537954154869719707264887274916552228805607584116490046284509883309001532457986879277885241872021906840932513241346999389365188296460009947
h=32243626948934860887488490158437299489453513352745889246437755713701521031193083418924110592954582395114812811896992400310730276
r=3064991081731777546575510593831386635550174528483098623
exp2=181
exp1=127
sign1=-1
sign0=-1

代码

import it.unisa.dia.gas.jpbc.Element;
import it.unisa.dia.gas.jpbc.Pairing;
import it.unisa.dia.gas.plaf.jpbc.pairing.PairingFactory;import java.io.FileInputStream;
import java.io.FileOutputStream;
import java.io.IOException;
import java.security.MessageDigest;
import java.security.NoSuchAlgorithmException;
import java.util.*;public class CLAS {//生成公私钥对,并保存到文件中public static void Setup(String pairingParametersFileName, String pubParamFileName, String KGC_SK_FileName) {Pairing bp = PairingFactory.getPairing(pairingParametersFileName);//G1的生成元PElement P = bp.getG1().newRandomElement().getImmutable();//计算主私钥和公钥Element s = bp.getZr().newRandomElement().getImmutable();Element P_Pub = P.mulZn(s);Properties pubParamProp = new Properties();//后面对写的元素统一采用如下方法：首先将元素转为字节数组，然后进行Base64编码为可读字符串pubParamProp.setProperty("P", Base64.getEncoder().encodeToString(P.toBytes()));pubParamProp.setProperty("P_Pub", Base64.getEncoder().encodeToString(P_Pub.toBytes()));storePropToFile(pubParamProp, pubParamFileName);Properties param_s = new Properties();//私钥不存入公开参数中,由KGC自己保存param_s.setProperty("s", Base64.getEncoder().encodeToString(s.toBytes()));storePropToFile(param_s, KGC_SK_FileName);}//根据用户id生成私钥public static void PartialPrivateKeyGen(String pairingParametersFileName, String id, String KGC_SK_FileName) throws NoSuchAlgorithmException {Pairing bp = PairingFactory.getPairing(pairingParametersFileName);//使用HASH 将 id 转为QIDbyte[] idHash = HASH(id);Element QID = bp.getG1().newElementFromHash(idHash, 0, idHash.length).getImmutable();//从文件中读取 主私钥Properties mskProp = loadPropFromFile(KGC_SK_FileName);String sString = mskProp.getProperty("s");Element s = bp.getZr().newElementFromBytes(Base64.getDecoder().decode(sString)).getImmutable();  //Base64编码后对应的恢复元素的方法//计算用户私钥, 这里应该将私钥安全的传输给用户//方便模拟，统一存入一个文件中Element psk_ID = QID.powZn(s).getImmutable();Properties pskProp = new Properties();pskProp.setProperty("psk", Base64.getEncoder().encodeToString(psk_ID.toBytes()));storePropToFile(pskProp, id + ".properties");}//生成用户私钥public static void UserKeyGen(String pairingParametersFileName, String pubParamFileName, String id) {Pairing bp = PairingFactory.getPairing(pairingParametersFileName);//从文件中读取公钥Properties pkProp = loadPropFromFile(pubParamFileName);String PString = pkProp.getProperty("P");Element P = bp.getG1().newElementFromBytes(Base64.getDecoder().decode(PString)).getImmutable();//生成随机数x，作为用户的私钥Element x = bp.getZr().newRandomElement().getImmutable();//计算用户公钥Element upk = P.mulZn(x);Properties userProp = new Properties();userProp.setProperty("usk", Base64.getEncoder().encodeToString(x.toBytes()));userProp.setProperty("upk", Base64.getEncoder().encodeToString(upk.toBytes()));storePropToFile(userProp, id + ".properties");}//签名public static Element[] Sign(String pairingParametersFileName, String pubParamFileName, String id, byte[] message) throws NoSuchAlgorithmException {Pairing bp = PairingFactory.getPairing(pairingParametersFileName);//获取公开参数Properties pubProp = loadPropFromFile(pubParamFileName);Element P = bp.getG1().newElementFromBytes(Base64.getDecoder().decode(pubProp.getProperty("P"))).getImmutable();Element P_Pub = bp.getG1().newElementFromBytes(Base64.getDecoder().decode(pubProp.getProperty("P_Pub"))).getImmutable();//获取用户自己的信息Properties userProp = loadPropFromFile(id + ".properties");Element upk = bp.getG1().newElementFromBytes(Base64.getDecoder().decode(userProp.getProperty("upk"))).getImmutable();Element x = bp.getZr().newElementFromBytes(Base64.getDecoder().decode(userProp.getProperty("usk"))).getImmutable();Element psk = bp.getG1().newElementFromBytes(Base64.getDecoder().decode(userProp.getProperty("psk"))).getImmutable();//选择随机数Element r = bp.getZr().newRandomElement();Element U = P.mulZn(r).getImmutable();//获取状态信息，将公开参数作为状态信息String p = pubProp.getProperty("P");String pPub = pubProp.getProperty("P_Pub");byte[] hash = HASH(p + pPub);Element Q = hashToG(bp, hash);// 计算m, id, upk,U组合的hash值byte[] res = hashCombination(message, id.getBytes(), upk.toBytes(), U.toBytes());Element h = hashToZ(bp, res);Element V = P_Pub.mulZn(x).mulZn(h).add(Q.mulZn(r)).add(psk).getImmutable();Element[] sigma = new Element[2];sigma[0] = U;sigma[1] = V;return sigma;}//验证public static boolean Verify(String pairingParametersFileName, String pubParamFileName, String id, byte[] message, Element[] sigma) throws NoSuchAlgorithmException {Pairing bp = PairingFactory.getPairing(pairingParametersFileName);//使用sha1 将 id 转为QIDbyte[] idHash = HASH(id);Element QID = bp.getG1().newElementFromHash(idHash, 0, idHash.length).getImmutable();//获取公开参数Properties pubProp = loadPropFromFile(pubParamFileName);Element P = bp.getG1().newElementFromBytes(Base64.getDecoder().decode(pubProp.getProperty("P"))).getImmutable();Element P_Pub = bp.getG1().newElementFromBytes(Base64.getDecoder().decode(pubProp.getProperty("P_Pub"))).getImmutable();//获取状态信息，将公开参数作为状态信息String p = pubProp.getProperty("P");String pPub = pubProp.getProperty("P_Pub");byte[] hash = HASH(p + pPub);Element Q = hashToG(bp, hash);//获取用户的公钥，Properties userProp = loadPropFromFile(id + ".properties");Element upk = bp.getG1().newElementFromBytes(Base64.getDecoder().decode(userProp.getProperty("upk"))).getImmutable();// 计算m, id, upk,U组合的hash值byte[] res = hashCombination(message, id.getBytes(), upk.toBytes(), sigma[0].toBytes());Element h = hashToZ(bp, res);Element left = bp.pairing(sigma[1], P);Element right = bp.pairing(QID.add(upk.mulZn(h)), P_Pub).mul(bp.pairing(sigma[0], Q));return left.isEqual(right);}//聚合签名, U_1,U_2,...,U_n,Vpublic static Element[] Aggregate(List<Element[]> sigmaList) {Element[] res = new Element[sigmaList.size() + 1];Element[] init = sigmaList.get(0);res[0] = init[0];Element V = init[1];for (int i = 1; i < sigmaList.size(); i++) {res[i] = sigmaList.get(i)[0];V = V.add(sigmaList.get(1)[1]);}res[sigmaList.size()] = V;return res;}// 聚合签名验证public static boolean AggregateVerify(String pairingParametersFileName, String pubParamFileName, String[] idx, byte[][] message, Element[] sigma) throws NoSuchAlgorithmException {Pairing bp = PairingFactory.getPairing(pairingParametersFileName);Element[] QIDX = new Element[idx.length];for (int i = 0; i < idx.length; i++) {byte[] idHash = HASH(idx[i]);Element QID = bp.getG1().newElementFromHash(idHash, 0, idHash.length).getImmutable();QIDX[i] = QID;}//获取公开参数信息Properties pubProp = loadPropFromFile(pubParamFileName);Element P = bp.getG1().newElementFromBytes(Base64.getDecoder().decode(pubProp.getProperty("P"))).getImmutable();Element P_Pub = bp.getG1().newElementFromBytes(Base64.getDecoder().decode(pubProp.getProperty("P_Pub"))).getImmutable();String p = pubProp.getProperty("P");String pPub = pubProp.getProperty("P_Pub");byte[] hash = HASH(p + pPub);Element Q = hashToG(bp, hash);Element[] PKX = new Element[idx.length];for (int i = 0; i < idx.length; i++) {Properties userProp = loadPropFromFile(idx[i] + ".properties");Element upk = bp.getG1().newElementFromBytes(Base64.getDecoder().decode(userProp.getProperty("upk"))).getImmutable();PKX[i] = upk;}Element[] hx = new Element[idx.length];for (int i = 0; i < idx.length; i++) {byte[] res = hashCombination(message[i], idx[i].getBytes(), PKX[i].toBytes(), sigma[i].toBytes());Element h = hashToZ(bp, res);hx[i] = h;}Element left = bp.pairing(sigma[idx.length], P);Element U = sigma[0];for (int i = 1; i < idx.length; i++) {U = U.add(sigma[i]);}Element part = QIDX[0].add(PKX[0].mulZn(hx[0]));for (int i = 1; i < idx.length; i++) {part = part.add(QIDX[i].add(PKX[i].mulZn(hx[i])));}Element right = bp.pairing(part, P_Pub).mul(bp.pairing(U, Q));return left.isEqual(right);}//一种可行的组合方法public static byte[] hashCombination(byte[] message, byte[] id, byte[] upk, byte[] U) throws NoSuchAlgorithmException {int m_len = message.length, id_len = id.length, upk_len = upk.length, u_len = U.length;int total_len = m_len + id_len + upk_len + u_len;byte[] res = new byte[total_len];for (int i = 0; i < m_len; i++) {res[i] = message[i];}for (int i = 0; i < id_len; i++) {res[i + m_len] = id[i];}for (int i = 0; i < upk_len; i++) {res[i + m_len + id_len] = upk[i];}for (int i = 0; i < u_len; i++) {res[i + m_len + id_len + upk_len] = U[i];}MessageDigest instance = MessageDigest.getInstance("SHA-256");instance.update(res);return instance.digest();}public static void storePropToFile(Properties prop, String fileName) {try (FileOutputStream out = new FileOutputStream(fileName, true)) {prop.store(out, null);} catch (IOException e) {e.printStackTrace();System.out.println(fileName + " save failed!");System.exit(-1);}}public static Properties loadPropFromFile(String fileName) {Properties prop = new Properties();try (FileInputStream in = new FileInputStream(fileName)) {prop.load(in);} catch (IOException e) {e.printStackTrace();System.out.println(fileName + " load failed!");System.exit(-1);}return prop;}public static byte[] HASH(String content) throws NoSuchAlgorithmException {MessageDigest instance = MessageDigest.getInstance("SHA-1");instance.update(content.getBytes());return instance.digest();}public static Element hashToG(Pairing pb, byte[] code) {return pb.getG1().newElementFromHash(code, 0, code.length).getImmutable();}public static Element hashToZ(Pairing pb, byte[] code) {return pb.getZr().newElementFromHash(code, 0, code.length).getImmutable();}public static void main(String[] args) throws Exception {String pairingParametersFileName = "a_181_603.properties";String idAlice = "alice@example.com";String idBob = "bob@example.com";String pubParamFileName = "data/pub.properties";String KGCFileName = "data/kgc.properties";Setup(pairingParametersFileName, pubParamFileName, KGCFileName);PartialPrivateKeyGen(pairingParametersFileName, idAlice, KGCFileName);PartialPrivateKeyGen(pairingParametersFileName, idBob, KGCFileName);UserKeyGen(pairingParametersFileName, pubParamFileName, idAlice);UserKeyGen(pairingParametersFileName, pubParamFileName, idBob);String message_a = "Alice,This is a message from Alice!";String message_b = "Bob,This is a message from Bob!";Element[] sigma1 = Sign(pairingParametersFileName, pubParamFileName, idAlice, message_a.getBytes());Element[] sigma2 = Sign(pairingParametersFileName, pubParamFileName, idBob, message_b.getBytes());boolean result = Verify(pairingParametersFileName, pubParamFileName, idAlice, message_a.getBytes(), sigma1);System.out.println("Alice 验证签名通过？ " + result);result = Verify(pairingParametersFileName, pubParamFileName, idBob, message_b.getBytes(), sigma2);System.out.println("Bob 验证签名通过？ " + result);List<Element[]> sigmaList = new ArrayList<>();sigmaList.add(sigma1);sigmaList.add(sigma2);Element[] SIGMA = Aggregate(sigmaList);String[] idx = {idAlice, idBob};byte[][] message = {message_a.getBytes(), message_b.getBytes()};
//        message[0][1] = 1;//假如消息被篡改result = AggregateVerify(pairingParametersFileName, pubParamFileName, idx, message, SIGMA);System.out.println("聚合签名验证通过？ " + result);}}

参考文献

1. A new certificateless aggregate signature scheme

2. Cryptanalysis and improvement of a certificateless aggregate signature scheme

3. A certificateless aggregate signature scheme for healthcare wireless sensor network

4. Efficient certificateless aggregate signcryption scheme without bilinear pairings

5. Security issues in IoT applications using certificateless aggregate signcryption schemes: An overview