Symmetric Homomorphic Encryption

Symmetric Homomorphic Encryption (SHE) is a Somewhat Homomorphic Encryption that supports both homomorphic addition and homomorphic multiplication.

Homomorphic Properties

Homomorphic Addition-1: Two ciphertexts E(m1) and E(m2) satisfy E(m1) + E(m2) mod N -> E(m1+m2)

Homomorphic Multiplication-1: Two ciphertexts E(m1) and E(m2) satisfy E(m1) * E(m2) mod N -> E(m1*m2)

Homomorphic Addition-2: A ciphertext E(m1) and a plaintext message m2 satisfy E(m1) + m2 mod N -> E(m1+m2)

Homomorphic Multiplication-2: A ciphertext E(m1) and a plaintext message m2 > 0 satisfy E(m1) * m2 mod N -> E(m1 * m2)

Maximum Multiplicative Depth

SHE can support unlimited rounds of homomorphic addition-1,2, and homomorphic multiplication-2, but it can only provide a limited depth of homomorphic multiplication-1. Usually, we set up the maximum depth as (k0 /k2 - 1), where k0 and k2 are security parameters of the encryption algorithm. The ciphertext will be decrypted incorrectly if it exceeds the maximum depth.

Getting Start

Temporally, this repo is not available at Maven Central. You can download package and add the dependency locally.

Maven example:

        <dependency>
            <groupId>guojuntang-github</groupId>
            <artifactId>symhomenc</artifactId>
            <version>0.4</version>
            <scope>system</scope>
            <systemPath>${project.basedir}/lib/symhomenc.jar</systemPath>
        </dependency>

Usage

Initialization

Set up the SHE security parameters(k0 >> k2 > k1, see the reference for details.) and require the private key, public key, and public parameter.

        // Using the SecureRandom class as the default random number generator
        SHEParameters param = new SHEParameters(SHEParameters.K0, SHEParameters.K1, SHEParameters.K2);
        SHEPrivateKey sk = param.getSHEPrivateKey();
        SHEPublicKey pk = param.getSHEPublicKey();
        SHEPublicParameter pb = param.getSHEPublicParameter();

Also, you can also use the other Random class as the random number generator.

        Random random = new Random();
        SHEParameters parameters = new SHEParameters(SHEParameters.K0, SHEParameters.K1, SHEParameters.K2, random);

Encryption and Decryption

        // Encryption with private key
        SHECipher a = SymHomEnc.enc(123456, sk);
        // Encryption with public key
        SHECipher b = SymHomEnc.enc(123456, pk);
        // Decryption with private key
        BigInteger c = SymHomEnc.dec(a, sk);

Also, you can also use the other Random class as the random number generator.

        Random random = new Random();
        SHECipher a = SymHomEnc.enc(123456, sk, random);

Homomorphic properties

// a0 + 3
        SHECipher a0 = SymHomEnc.enc(12, pk);
                SHECipher b0 = SymHomEnc.hm_add(a0, 3, pb);
                assertEquals(SymHomEnc.dec(b0, sk), BigInteger.valueOf(15));

                // a1 + 3
                SHECipher a1 = SymHomEnc.enc(-12, pk);
                SHECipher b1 = SymHomEnc.hm_add(a1, 3, pb);
                assertEquals(SymHomEnc.dec(b1, sk), BigInteger.valueOf(-9));

                // a1 * a1
                SHECipher a2 = SymHomEnc.hm_mul(a1, a1, pb);
                assertEquals(SymHomEnc.dec(a2, sk), BigInteger.valueOf(144));


                //-a2 + a0
                SHECipher a3 = SymHomEnc.enc(-1, sk);
                SHECipher b2 = SymHomEnc.hm_mul(a2, a3, pb);
                SHECipher b3 = SymHomEnc.hm_add(b2, a0, pb);
                assertEquals(SymHomEnc.dec(b3,sk), BigInteger.valueOf(-132));

                // a1 * 100
                SHECipher b4 = SymHomEnc.hm_mul(a2, 100, pb);
                assertEquals(SymHomEnc.dec(b4, sk), BigInteger.valueOf(14400));

Reference

H. Mahdikhani, R. Lu, Y. Zheng, J. Shao and A. A. Ghorbani, "Achieving O(log³n) Communication-Efficient Privacy-Preserving Range Query in Fog-Based IoT," in IEEE Internet of Things Journal, vol. 7, no. 6, pp. 5220-5232, June 2020, doi: 10.1109/JIOT.2020.2977253.