Cryptography Reference
In-Depth Information
i
pub
()
3. For
i
= 1, 2,…,
n
, compute
P
=Î
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
=Å
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
=Å
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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