Cryptography Reference
In-Depth Information
i
pub
()
3. For i = 1, 2,…, n , compute
P
and P pub = sP .
f i P
()
G
1
Before requesting his private share, each player can check that
å ()
i
i b
LP
=
P
(4.51)
b
iS
Î
for any subset Ì {1,..., }
S
n such that | S | = t , where L i denotes the appropriate
Lagrange coefficient.
Key-Gen : Given a user's identity ID , the PKG plays the role of the trusted dealer.
For i = 1,…, n it delivers =
d
f i QG of player i . After receiving d ID ,
()
Î 1
ID
ID
i
player i checks
i
pub
()
=
(4.52)
eP
(
,
Q
)
ePd
(
,
)
ID
ID
i
If verification fails, he complains to the PKG, which issues a new share.
Encrypt : Given messages m and identity ID,
1. Compute Q ID = H 1 ( ID ).
2. Choose a random Î q
rZ .
3. Set the cipher text to be
) r
CrPmHe PQ
2
,
( (
,
(4.53)
pub
ID
Decrypt : Upon receiving
UV , player i computes his decryption share
eU d
(,
)
and gives it to the recombiner, who may be a designated player.
ID
i
Recombination : The recombiner selects a set Ì {1,..., }
S
n of t acceptable shares
eU d
(,
)
and computes
ID
i
= (,
L
g
e Ud
) i
(4.54)
ID
i
Î
iS
Once he has g , he recovers the plain text
mV Hg
2 ()
(4.55)
The correctness of the scheme is easy to verify since
æ
ö ÷
ç
å
÷
ç
r
ge P
=
,
Ld
,
=
e PsQ
(
,
)
=
ePQ
(
,
)
(4.56)
÷
ç
÷
ç
i
ID
ID
b ID
÷
ç è
i
ø
iS
Î
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