Logical Analysis and Verification of Cryptographic Protocols - Loria

1.2. CRYPTOGRAPHIC PRIMITIVES 5
1.2.1 Concatenation
The concatenation, denoted by “·”, is commonly used in cryptographic protocols
to construct messages. The concatenation algorithm “·” is represented by a
function that takes a couple of messages as input and outputs their concatenation,
for example the concatenation of “a” and “b” is “a· b”.
We give below the algebraic properties of the concatenation:
1.2.2 Exclusive or
Associativity : (x · y) · z = x · (y · z)
Unary : x · ɛ = ɛ · x = x
The exclusive or, denoted by “⊕” or “XOR”, is commonly used in cryptographic
protocols to construct messages, it has the following algebraic properties:
1.2.3 Pairing
Associativity : (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z)
Commutativity : x ⊕ y = y ⊕ x
Unary : x ⊕ ɛ = x
Nilpotence : x ⊕ x = ɛ
The pairing, denoted by “〈−, −〉”, is commonly used in cryptographic protocols
to construct messages, it has the following algebraic properties:
1 st projection : π1〈x, y〉 = x
2 nd projection : π2〈x, y〉 = y
where π1 and π2 denote the projection functions. The main difference between
the pairing and the concatenation is that the pairing is not associative, i.e.
〈x, 〈y, z〉〉 �= 〈〈x, y〉, z〉.
1.2.4 Encryption schemes
We have two types of encryption schemes: the public (or asymmetric) encryption
schemes and the symmetric encryption schemes.
Public encryption schemes
The public encryption schemes, also called asymmetric encryption schemes, have
been initially introduced by W. Diffie and M. Hellman [101] and R. Merkle [154].
A “public encryption scheme” is defined by three algorithms: the encryption
algorithm “enc p ”, the decryption algorithm “dec p ”, and the key generation algorithm