Big Data

330 

Big Data: Storage, Sharing, and Security

Then, the encryption scheme has the additively homomorphic property by ensuring

Dec(K(m),Enc(K(m), v) ⊕Enc(K( m) , v



)) = v + v



,

where ⊕ indicates vector addition, that is,

Enc(K(m),v) ⊕Enc(K(m), v



)=(e

1

+ e



1

,...,e

m

+ e



m

).

Let b be a real number. As an extension of the additively homomorphic property, we have

Dec(K(m),b Enc(K(m), v)) = b ∗v,

where b Enc(K(m),v)=(b ∗e

1

,...,b ∗e

m

). Combined with the ⊕ operation, we can get

Dec(K(m),Enc(K(m), v) ⊕(b Enc(K(m), v



))) = v + b ∗v



.

When b = −1, we can calculate v −v



from the two ciphertexts.

The multiplicative homomorphism is about calculating v ∗v



from the