Table 6.

Local sets specification for principal U.

Principal U
POSS(U) = {pwU, bU, {IDU}}	(U16) Update(IDU, IDSn, X, TU, α, ω)
BEL(U) = {#(pwU), #(bU)}	(U17) Receive(Sn, {Y, TS, δ, τ})
BL(U)	(U18) Check-freshness(T′S)
Phase 1	(U19) Check (δ, Hash(h(·); Contat(IDU, X, X′,TU, IDSn, Y, TS)))
(U1) ${\overline{\mathit{\text{pw}}}}_U\leftarrow \text{Hash}(h(\cdot );\text{XOR}(p{w}_U,b_U))$
(U2) Send (GW,{IDU, ${\overline{pw}}_U$})	(U20) KUS ← Scalar-multiplication(rU, Y)
(U3) Update(IDU, ${\overline{pw}}_U$)	(U21) Check(τ, Hash(h(·); Contat(Y, T′S, δ, KUS)))
(U4) Receive(GW, {AU, BU, WU, h(·)})	(U22) sk ← Hash(h(·); Contat(X, Y, KUS))
Phase 2	Phase 3
(U5) ${\overline{\mathit{\text{pw}}}}_U\leftarrow \text{Hash}(h(\cdot );\text{XOR}(p{w}_U,b_U))$	(U23) ${\overline{\mathit{\text{pw}}}}_U\leftarrow \text{Hash}(h(\cdot );\text{XOR}(p{w}_U,b_U))$
(U6) h(x⊕y) ← XOR( ${\overline{pw}}_U$, AU)	(U24) B′U ← Hash(h(·); Concat (IDU, ${\overline{pw}}_U$, h(x(⊕ y))
(U7) B′U ← Hash(h(·);Concat (IDU, ${\overline{pw}}_U$, h(x⊕y)))	(U25) Check(B′U, BU)
(U8) Check(B′U, BU)	(U26) KU ← XOR(Hash(h(·); Concat(IDU, ${\overline{pw}}_U$)), WU)
(U9) KU ← XOR(Hash(h(·); Concat(IDU, ${\overline{pw}}_U$)),WU)	(U27) h(x(⊕ y) ← XOR( ${\overline{pw}}_U$, AU)
(U10) Generate-nonce(rU)	(U28) ${\overline{pw\prime }}_U\leftarrow \text{Hash}(h(\cdot );\text{XOR}(pw{\prime }_U,A_U))$
(U11) X ← Scalar-multiplication(rU, P)	(U29) A′U ← XOR ( ${\overline{pw}}_U$, h(x ⊕ y))
(U12) X′ ← Scalar-multiplication(rU, KU)	(U30) B′U ← Hash(h(·); Concat(IDU, ${\overline{p{w}^{\prime }}}_U$, h(x(⊕ y)))
(U13) ω ← Hash(h(·); Concat(IDU, Hash(h(·);IDSn, h(x ⊕ y)), TU))	(U31) W′U ← XOR(Hash(h(·); Concat(IDU, ${\overline{p{w}^{\prime }}}_U$)), KU)
	(U32) AU ← A′U
(U14) α ← Hash(h(·); Concat(IDU, IDSn, X, X′, TU,ω))	(U33) BU ← B′U
(U15) Send(Sn, {IDU, IDSn, X, TU, α, ω})	(U34) WU ← W′U