# Homomorphic multiplication by a scalar

Few homomorphic encryption schemes like Paillier , Ring-LWE support homomorphic multiplication operation by a scalar apart from additive homomorphic property.

Loosely they could be defined as below

$Additive : Dec(Enc(a)+Enc(b)) = (a+b)$ and $MultiByScalar: Dec(Enc(a)\times b) = a \times b$

Is this any class of homomorphic encryption schemes that supports only these two functions ?

Not sure if they could be called Partial Homomorphic Encryption(PHE) or Some What Homomorphic (SHE) schemes.

Side notes: I saw that few papers are referring $MultiByScalar$ as $absorb$ function too.

## 1 Answer

I'm not sure I completely understand the point of your question. First of all, the property Additive implies the property MultiByScalar. Indeed, to multiply a ciphertext $E(a)$ by a scalar $b$, you can simply use a square and multiply algorithm, adding a ciphertext to itself to multiply its plaintext by two, which allows to multiply in polynomial time $E(a)$ by $2^k$, for all the positions $k$ corresponding to a $1$ in the bit decomposition of $b$, then summing all those ciphertexts to get $E(ab)$.

Note that the additive variant of the ElGamal ciphertext does also satisfy this property, but unlike Paillier or LWE/RLWE, the decryption is not efficient.

Ciphertexts with the additive property have automatically this multiplicative property; they are sometimes called partially homomorphic. Somewhat homomorphic encryption schemes refer to schemes in which you can both add and multiply encrypted value (and not just an encrypted value and a scalar), but you can do so a limited number of times (say, lots of additions and a few multiplications).

So, every additively homomorphic encryption scheme (Okamoto-Uchiyama, Goldwasser-Micali, Benaloh, ElGamal, LWE/RLWE...) satisfies your two properties (and only these two properties, regarding homomorphism: only some very specific lattice-based encryption scheme are both additively and multiplicatively homomorphic).