Homomorphic operation

Homomorphic encryption schemes in ECLib are categorized into multiplicatively, additively and fully homomorphic encryption. Each class of encryption scheme supports different types of homomorphic operations.

Multiplicatively homomorphic encryption

Multiplicatively homomorphic encryption supports multiplication as a homomorphic operation.

>>> c1 = encrypt(params, pk, m1)
>>> c2 = encrypt(params, pk, m2)
>>> c3 = mult(params, c1, c2)

The decryption of c3 is expected to be m1 * m2. It can also compute the Hadamard product for 1D and 2D array-like ciphertexts.

>>> c_v1 = encrypt(params, pk, [m1, m2, m3])
>>> c_v2 = encrypt(params, pk, [m4, m5, m6])
>>> c_v3 = mult(params, c_v1, c_v2)

The decryption of c_v3 is expected to be the array that consists of m1 * m4, m2 * m5, and m3 * m6.

Additively homomorphic encryption

Additively homomorphic encryption supports addition as a homomorphic operation.

>>> c4 = add(params, c1, c2)

The decryption of c4 is expected to be m1 + m2. It can also compute the addition for 1D and 2D array-like ciphertexts.

>>> c_v4 = add(params, c_v1, c_v2)

The decryption of c_v4 is expected to be the array that consists of m1 + m4, m2 + m5, and m3 + m6. Furthermore, additively homomorphic encryption supports integer multiplication, a product between a plaintext and a ciphertext.

>>> c5 = int_mult(params, m1, c2)
>>> c_v5 = int_mult(params, [m1, m2, m3], c_v2)

The decryption of c3 and c_v3 are m1 * m2 and the inner product of [m1, m2, m3] and [m4, m5, m6], respectively. Note that additively homomorphic encryption schemes in ECLib implement element-wise addition elementwise_add() and element-wise integer multiplication elementwise_int_mult() for convenience (see API reference).

Fully homomorphic encryption

Fully homomorphic encryption supports both addition and multiplication as homomorphic operations.

>>> c3 = mult(params, c1, c2)
>>> c4 = add(params, c1, c2)

Moreover, it supports integer multiplication.

>>> c5 = int_mult(params, c1, c2)

The decryptions of c3, c4, and c5 are expected to be coincide to ones in multiplicatively and additively homomorphic encryption. The functions allows 1D and 2D array-like ciphertexts.

>>> c_v3 = mult(params, c_v1, c_v2)
>>> c_v4 = add(params, c_v1, c_v2)
>>> c_v5 = int_mult(params, [m1, m2, m3], c_v2)

Similar to additively homomorphic encryption, fully homomorphic encryption schemes in ECLib implement element-wise addition elementwise_add(), element-wise integer multiplication elementwise_int_mult(), and element-wise multiplication elementwise_mult() for convenience (see API reference).