Table 2.

Notation and abbreviations.

Notations	Description
SC	Smart Card.
$U_i$	User.
$u_{id}$	Identity of the User $U_i$
$u_{pw}$	The password of the user $U_i$.
$u_{bi}$	Biometrics imprint of the user $U_i$.
$p,q$	Two prime numbers.
$r_i$	Random number.
$H(.)$	One-way hash function.
$E\{.\}/D\{.\}$	A pair of symmetric encryption/decryption.
$Z_q^{*}$	The non-zero integers modulus p.
$PK$	The public key of the server.
S	The secret key of the server.
$Aut{h}_u^{{}^{\prime }}$	The authenticator of the user $U_i$.
$TS$	Timestamp.
$k{s}_{u\to tgs}$	A key session between User and TGS.
$tg{s}_{sk}$	Secret key of the TGS.
$tg{s}_{id}$	Ticket granting server identity.
$f{n}_{id}$	Identity of the Fog node.
$tg{s}_{tkt}$	Ticket granting server ticket.
$k{s}_{u\to {fn}_S}$	Key session between User and FN.
$f{n}_{tkt}$	Fog Node ticket.
‖	Concatenation operation.
⊕	XOR operation.