22 2. PRELIMINARIES
is called a partially homomorphic encryption algorithm. If it supports both operations it is called
a fully homomorphic encryption algorithm. Encryption schemes, such as Paillier and Elgamal,
are partially homomorphic encryption algorithms while Gentry’s FHE [66] is an example of a
fully homomorphic encryption algorithm. Homomorphic encryption offers us the ability to per-
form a restricted set of mathematical operations on encrypted data, thus eliminating the need to
decrypt data. We will use homomorphic encryption in Chapter 5 when we present secure data
aggregation algorithms that perform addition over the ciphertext on wireless sensors.
2.2.2 PAILLIER ENCRYPTION
e Paillier cryptosystem [106] is a public key encryption scheme based on the Decisional Com-
posite Residuosity Assumption (DCRA). e DCRA states that given a composite number
n D n
1
n
2
, for primes n
1
and n
2
, and an integer z, it is hard to find out whether there exists
a number y such that z y
n
mod n. A message M 2 Z
n
in Paillier encryption is encrypted as
C D g
M
:r
n
mod n
2
;
where g 2 Z
n
2
is a public element, while r 2 Z
n
is chosen randomly by the encryptor. e
ciphertext C can be decrypted as
M D L.C
mod n
2
/: mod n;
where D lcm.n
1
1; n
2
1/, D .L.g
mod n
2
//
1
mod n and L.x/ D .x 1/=n.
Paillier encryption exhibits the property of additive homomorphism which will be used in
Chapter 6. Given C
m
1
and C
m
2
, paillier ciphertexts for m
1
and m
2
, the ciphertext for m
1
C m
2
can be generated by multiplying C
m
1
and C
m
2
:
C
m
1
Cm
2
D C
m
1
:C
m
2
:
C
m
1
Cm
2
can then be decrypted by the private key to retrieve m
1
C m
2
.
2.2.3 ELLIPTIC CURVE CRYPTOGRAPHY
Public key cryptography, also known as asymmetric cryptography, is widely used for various
purposes such as key distribution, secure communication, message signing, etc. However, tra-
ditional public key algorithms are generally energy-hungry and, therefore, not suitable for wire-
less sensors. Elliptic curve cryptography (ECC) is a form of public key cryptography that has
emerged as an attractive and viable public key system for constrained environments.
ECC offers considerably greater security for a given key size, compared to the traditional
public key systems. is means that smaller keys can be used for equivalent security which makes
possible more compact implementations that are faster (Table 2.1). is also leads to less power
consumption and in turn less heat production. Heat production is an often overlooked issue on
wireless sensors that can affect the WSN operations in two different ways. First, heat is just
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