Normal ： Simi-fuctional ：. Normal ： Simi-fuctional ：

Normal：

Simi-fuctional：

Normal：

Simi-fuctional：

• Gamereal:Real security game

• Gamerestrictedrestricted:

Real security game except that the attacker cannot ask for keys for identities which are equal to the challenge identity modulo p2.

• Gamek:This is like the restricted security game, except that the ciphertext given to the attacker is semi-functional and the first k keys are sem

i- functional. The rest of the keys are normal.

• Gamefinal:same as Gameq except that the ciphertext is a semi-functional encrypt

ion of a random message, not one of the two messages requested by the attacker.

• Gamereal Gamerestrictedrestricted

C B A

g, X1X2, X3, Y2Y3, TPK = {N,u=ga,g,h=gb,e(g,g)α}

K1=grX3t K2=gα(uIDh)rX3

t`

ID

(M0,M1,ID*)

C0=Mβe(g,g)αs, C1=((uID*h)s), C2=gs

• Gamerestrictedrestricted Game00

C B A

g, X3, T

PK = {N,u=ga,g,h=gb,e(g,g)α}

K1=grX3t K2=gα(uIDh)rX3

t`

ID

(M0,M1,ID*)

C0=Mβe(T,g)αs, C1=TaID*+b, C2=T

β`

b`

• Gamek-1k-1 Gamek

C B Ag, X1X2, X3, Y2Y3, T

PK = {N,u=ga,g,h=gb,e(g,g)α}

if i>k K1=griX3ti K2=gα(uIDih)riX3

ti`

IDi

(M0,M1,ID*)

C0=Mβe(X1X2,g)αs, C1=(X1X2)aID*+b, C2=X1X2

β`b`

if i>k K1=gri(Y2Y3)ti K2=gα(uIDih)ri(Y2Y3)ti`

if i=k K1=T K2=gαTaIDk+bX3ti`

• Gameq Gamefinalfinal

C B A

g, gαX2, X3, gsY2, Z2, TPK = {N,u=ga,g,h=gb,e(g,g)α=e(gαX2,g)}

K1=gr Z2θX3

t K2=gα(uIDh)rZ2θ`X3

t`

ID

(M0,M1,ID*)

C0=MβT, C1=(gsY2)aID*+b, C2=gsY2