IEEE Computational Intelligence Magazine - August 2022 - 16

m
Raw Message
Public key (kp)
Private key (ks)
HE Deep
Learning User
y
Raw Result
Decoding
Processed
Plaintext
ΓΘ(y)
-1
y
DΘ(y, ks)
Decryption
y
Processed
Ciphertext
FIGURE 1 Machine and deep learning processing chain based on the BFV homomorphic encryption scheme.
computation of a set of operations directly on encrypted data.
This goal is achieved thanks to the HE scheme's ability to maintain
the algebraic structure of the data during the encrypted processing.
In other words, HE allows a party to compute operations
between ciphertexts, guaranteeing that the obtained result, when
decrypted, will be equal (under certain assumptions, which are
detailed as follows) to that obtained by computing the same
operations between the corresponding plaintexts.
HE schemes can generally be classified into four categories,
which are characterized by increasing complexity in terms of
the number and type of operations. First, partially HE schemes
are HE schemes that can support the homomorphic
computation of only one class of operations. An example of a
partially HE scheme is Rivest-Shamir-Adleman (RSA) [12].
TABLE II Summary of the notations used in this paper.
NOTATION
m
n
p
q
MEANING
Raw message
Polynomial modulus degree
Plaintext coefficient modulus
H
U n
ppZ []/( )
Rx xUn
m
mu
Rx xUn
qqZ []/( )
=
=
()
= CH
-
()
mmC= H
mm1
ks, kp
mE (, )mkp
mD (, )mks
6@ g
6 hg
u = H
u
= H
6@g Un ,p
Ciphertext coefficient modulus
Encryption parameters (n, p, q)
+
Second, somewhat HE schemes support the homomorphic computation
of an unbounded number of additions and a single
multiplication. A notable example in this category is the
Boneh-Goh-Nissim (BGN) scheme [13]. Third, leveled HE
schemes are schemes that support the homomorphic computation
of a predetermined number of additions and multiplications.
The BFV scheme [14] (which is the reference HE
scheme considered in this paper) and the Cheon-Kim-Kim-
Song (CKKS) scheme [15] belong to this category. Finally,
fully HE schemes support the homomorphic computation of an
unbounded number of operations, often binary ones (AND,
NOT...). While the HE schemes in this category provide the
greatest flexibility in terms of processing abilities, configuring
and managing them is very difficult. An example of a fully HE
scheme is Torus-Fully-HE (TFHE) [16]. As previously mentioned,
this paper focuses on the BFV scheme [14], which is a
popular leveled HE scheme based on the ring learning with
errors (RLWE) problem [17].

IEEE Computational Intelligence Magazine - August 2022

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