Broadcast encryption method and broadcast decryption method thereof
A broadcast encryption method and a broadcast decryption method. The broadcast encryption method includes generating a message encryption key using a public key and a secret key generated by using a Strong Diffie-Hellman tuple; encrypting a message by the message encryption key; and generating a message header using a sum of a plurality of Strong Diffie-Hellman tuples corresponding to an authorized user group. Accordingly, a transmission amount and a storage amount can be reduced when a broadcast encryption message is transmitted.
1. A broadcast encryption method comprising:
generating at least one Strong Diffie-Hellman tuple in accordance with the equation (1/(x+1))G, (1/(x+2))G, . . . , and (1/(x+n))G, where G denotes a generator of a group having an order p on an elliptic curve, x denotes an arbitrary element selected from a group Zp of order p, and n denotes a total number of users;
generating a message encryption key using a public key and a secret key generated using the Strong Diffie-Hellman tuple;
encrypting a message using the message encryption key;
generating a message header using a sum of a plurality of Strong Diffie-Hellman tuples corresponding to an authorized user group; and
transmitting the message header, the encrypted message, and the Strong Diffie-Hellman tuple to at least one receiver.
2. The broadcast encryption method of claim 1 , further comprising generating no more than n+2 public keys and n secret keys using the Strong Diffie-Hellman tuple,
wherein the message encryption key is generated using at least one public key from the no more than n+2 public keys and at least one secret key from the n secret keys; and
wherein n denotes a total number of users.
3. The method of claim 2 , wherein the n+2 public keys are in accordance with Equation 1 and the n secret keys are in accordance with Equation 2:
PK
=
(
G
,
H
,
1
x
+
1
G
,
1
x
+
2
G
,
…
,
1
x
+
n
G
)
x
∈
Z
p
,
and
[
Equation
1
]
D
i
=
W
+
1
x
+
i
H
+
1
(
x
+
i
)
2
G
,
[
Equation
2
]
where PK denotes the public key, Di denotes a secret key of a user i, G denotes a generator of a group having an order p on an elliptic curve, W and H denote arbitrary elements in the group generated by G, x denotes an arbitrary element selected from a group Zp, i denotes a natural number in a range of 1≦i≦n, and n denotes a total number of users.
4. The method of claim 3 , wherein the generating of the message encryption key comprises generating the message encryption key using Equation 3:
K
=
e
(
W
,
G
)
s
,
s
∈
Zp
e
(
W
,
G
)
=
e
(
D
i
,
G
)
e
(
H
,
G
i
)
e
(
G
i
,
G
i
)
,
[
Equation
3
]
where K denotes the message encryption key, e( ) denotes a pairing function defined on an elliptic curve, s denotes an arbitrary element selected from the group Zp, and Gi=(1/(x+i))G.
5. The method of claim 4 , wherein the generating of the message header comprises generating the message header using Equation 4:
Hdr
=
(
s
(
H
+
∑
j
∈
S
G
j
)
,
sG
)
,
[
Equation
4
]
where Hdr denotes the message header, Gj=(1/(x+j))G, and S denotes the authorized user group.
6. The method of claim 2 , wherein the message header, the encrypted message, and the Strong Diffie-Hellman tuple are transmitted to each small group into which an entire user group is divided.
7. The method of claim 1 , wherein the message header, the encrypted message, and the Strong Diffie-Hellman tuple are transmitted to each small group into which an entire user group is divided;
each small group comprises five users; and
the generating of the message header comprises generating the message header using Equation 5:
Hdr
=
[
s
(
H
1
+
∑
j
∈
S
1
G
j
)
,
s
(
H
2
+
∑
j
∈
S
2
G
j
)
,
s
(
H
3
+
∑
j
∈
S
3
G
j
)
,
s
(
H
4
+
∑
j
∈
S
4
G
j
)
,
s
(
H
5
+
∑
j
∈
S
5
G
j
)
,
sG
]
,
[
Equation
5
]
where Hdr denotes a message header, G denotes a generator of a group having an order p on an elliptic curve, H 1 through H 5 denote arbitrary elements of the group generated by G, s denotes an arbitrary element selected from a group Zp, Gj=(1/(x+j))G, and S 1 through S 5 denote small groups.
8. A broadcast decryption method comprising:
receiving, by a user included in an authorized user group, a Strong Diffie-Hellman tuple corresponding to the group from within a public key, a message header, and an encrypted message;
calculating a value using the Strong Diffie-Hellman tuple and the message header in accordance with Equation 6:
∑
j
∈
S
,
j
≠
i
a
ij
G
i
+
b
ij
G
j
,
[
Equation
6
]
where a ij =1/(j−i), b ij =−1(j−i), S denotes the authorized user group, Gi=(1/(x+i))G, Gj=(1/(x+j))G, G denotes a generator of a group having an order p on an elliptic curve, i denotes an index corresponding to the user receiving the encrypted message, and j denotes an index of a member included in the authorized user group other than the user i;
performing a pairing operation using the calculated value and restoring an encryption key; and
decrypting the encrypted message using the encryption key.
9. The method of claim 8 , wherein the calculating of the value comprises:
calculating a ij and b ij using Equation 7 with respect to all users in the group except for the user i:
1
(
x
+
i
)
(
x
+
j
)
=
a
ij
x
+
i
+
b
ij
x
+
j
→
aij
=
1
(
j
-
i
)
,
bij
=
-
1
(
j
-
i
)
,
[
Equation
7
]
where x denotes an arbitrary element selected from a group Zp; and
calculating a value in accordance with Equation 6 using a ij and b ij .
10. The method of claim 9 , wherein the restoring of the encryption key comprises restoring the encryption key using Equation 8:
K
=
e
(
D
i
+
∑
j
∈
S
,
j
≠
i
a
ij
G
i
+
b
ij
G
j
,
B
)
e
(
A
,
G
i
)
=
e
(
W
,
G
)
S
Hdr
=
(
s
(
H
+
∑
j
∈
S
,
j
≠
i
G
j
)
,
sG
)
=
(
A
,
B
)
,
[
Equation
8
]
where K denotes an encryption key, e( ) denotes a pairing function defined on an elliptic curve, Hdr denotes the message header, H and W denote arbitrary elements in a group generated by G, s denotes an arbitrary element selected from the group Zp, and Di denotes a secret key of the user i.
11. The method of claim 8 , wherein the Strong Diffie-Hellman tuple corresponding to the group is a set of Gi's related to the group.
12. The method of claim 8 , wherein the message header, the encrypted message, and the Strong Diffie-Hellman tuple are transmitted to each small group into which an entire user group is divided.
13. A non-transitory computer-readable recording medium storing a program to implement a broadcast encryption method, the method comprising:
generating at least one Strong Diffie-Hellman tuple in accordance with the equation (1/(x+1))G, (1/(x+2))G, . . . , and (1/(x+n))G, where G denotes a generator of a group having an order p on an elliptic curve, x denotes an arbitrary element selected from a group Zp of order p, and n denotes a total number of users;
generating a message encryption key using a public key and a secret key generated using the Strong Diffie-Hellman tuple;
encrypting a message using the message encryption key; and
generating a message header using a sum of a plurality of Strong Diffie-Hellman tuples corresponding to an authorized user group.