基于JPBC的无证书聚合签名方案实现

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 κ ∈ Z \kappa \in Z κ∈Z，KGC选择两个循环群 G 1 G_1 G1 、 G 2 G_2 G2，其阶均为素数 q q q ， G 1 G_1 G1 的生成元 P P P，计算可接受的配对 e : G 1 × G 1 → G 2 e:G_1×G_1→G_2 e:G1×G1→G2 。 KGC 随机选择主密钥 s ∈ Z q ∗ s\in Z_q^\ast s∈Zq∗ ，设置 P p u b = s P P_{pub}=sP Ppub=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 ) Q_{ID_i }=H_1 (ID_i) QIDi=H1(IDi)。 然后，它设置该用户的部分密钥 p s k I D i = s Q ( I D i ) psk_{ID_i }=sQ_(ID_i ) pskIDi=sQ(IDi)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 ( p s k I D i , P ) = e ( Q I D i , P p u b ) e(psk_{ID_i },P)=e(Q_{ID_i },P_{pub}) e(pskIDi,P)=e(QIDi,Ppub)。

UserKeyGen :

用户 I D i ID_i IDi随机选择值 x I D i ∈ Z q ∗ x_{ID_i }\in Z_q^\ast xIDi∈Zq∗作为他的用户私钥 u s k I D i usk_{ID_i } uskIDi，并且计算用户公钥 u p k I D i = x I D i P upk_{ID_i}=x_{ID_i } P upkIDi=xIDiP。

Sign :

对于消息 m i ∈ 0 , 1 ∗ m_i∈{0,1}^\ast mi∈0,1∗，选择状态信息 ∆ ∆ ∆(选择公开参数作为状态信息)，具有 I D i ID_i IDi身份的签名者执行以下步骤：

1. 选择随机数 r i ∈ Z q ∗ r_i∈Z_q^\ast ri∈Zq∗并且计算 U i = r i P ∈ G 1 U_i=r_i P∈G_1 Ui=riP∈G1.
2. 计算 Q = H 2 ( ∆ ) , h i = H 3 ( m i , I D i , u p k I D i , U i ) Q=H_2(∆),h_i=H_3(m_i,ID_i,upk_{ID_i},U_i) Q=H2(∆),hi=H3(mi,IDi,upkIDi,Ui)
3. 计算 V i = p s k I D i + r i ⋅ Q + h i ⋅ x i ⋅ P p u b V_i=psk_{ID_i }+r_i⋅Q+h_i⋅x_i⋅P_{pub} Vi=pskIDi+ri⋅Q+hi⋅xi⋅Ppub.

输出 ( U i , V i ) (U_i,V_i ) (Ui,Vi)作为 m i m_i mi的签名。

Verify ：

给定一个消息 m i m_i mi签名 ( U i , V i ) (U_i,V_i ) (Ui,Vi)，其对应的身份为 I D i ID_i IDi和公钥 u p k I D i upk_{ID_i} upkIDi，计算 Q ( I D i ) = H 1 ( I D i ) , Q = H 2 ( ∆ ) Q_(ID_i )=H_1 (ID_i ),Q=H_2 (∆) Q(IDi)=H1(IDi),Q=H2(∆) ,
h i = H 3 ( m i , I D i , u p k I D i , U i ) h_i=H_3 (m_i,ID_i,upk_{ID_i },U_i ) hi=H3(mi,IDi,upkIDi,Ui)

如果等式 e ( V i , P ) = e ( h i ⋅ u p k I D i + Q I D i , P p u b ) e ( U i , Q ) e(V_i,P)=e(h_i⋅upk_{ID_i }+Q_{ID_i },P_{pub} )e(U_i,Q) e(Vi,P)=e(hi⋅upkIDi+QIDi,Ppub)e(Ui,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 , σ 1 = ( U 1 , V 1 ) ) , ... , ( m n , σ n = ( U n , V n ) ) } \{(m_1,σ_1=(U_1,V_1 )),...,(m_n,σ_n=(U_n,V_n))\} {(m1,σ1=(U1,V1)),...,(mn,σn=(Un,Vn))},

聚合签名计算： V = ∑ i = 1 n V i V=∑_{i=1}^n V_i V=∑i=1nVi, 将 σ = ( U 1 , ... , U n , V ) σ=(U_1,...,U_n,V) σ=(U1,...,Un,V)作为聚合签名。

Aggregate-Verify ：

验证聚合签名 σ = ( U 1 , ... , U n , V ) σ=(U_1,...,U_n,V) σ=(U1,...,Un,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 , u p k I D i , U i ) , i = 1 , ... , n Q_{ID_i }=H_1 (ID_i ),Q=H_2 (∆),h_i=H_3 (m_i,ID_i,upk_{ID_i },U_i ),i=1,...,n QIDi=H1(IDi),Q=H2(∆),hi=H3(mi,IDi,upkIDi,Ui),i=1,...,n.

验证： e ( V , P ) = e ( ∑ i = 1 n [ h i ⋅ u p k I D i + Q I D i ] , P p u b ) e ( ∑ i = 1 n U i , Q ) e(V,P)=e(∑{i=1}^n [h_i⋅upk{ID_i}+Q_{ID_i} ] ,P_{pub} )e(∑_{i=1}^n U_i ,Q) e(V,P)=e(∑i=1n[hi⋅upkIDi+QIDi],Ppub)e(∑i=1nUi,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P
        Element 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QID
        byte[] 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QID
        byte[] 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,V
    public 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

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