Cryptography Reference
In-Depth Information
L
m
be a message. We compute
A
=
A
−
1
(
M
),
B
=
G
−
1
(
A
)and
C
=
A
−
2
(
B
) in the same order. The signature of the message
is
C
Signature Generation.
Let
M
∈
L
n
.Notethat
B
=
G
−
1
(
A
) can easily be computed by the above prop-
erty of
G
.
∈
Verification.
If
F
(
C
)=
M
then the signature is accepted, otherwise it is rejected.
This scheme is denoted by Rainbow(
L
;
v
1
,o
1
,...,o
t
), and we call
v
1
,o
1
,...,o
t
a parameter of Rainbow.
6.2
Reduction of HS Scheme to Rainbow
G
Uchiyama and Ogura wrote down
φ
−n
+1
φ
n
for HS(
◦
◦
Z
/N
Z
, n
)andshowed
the following [22].
F
Proposition 2.
Let
R
be a non-commutative ring over
Z
/N
Z
of rank
r
.Let
F
be a public key of HS
(
R
;
n
)
.Then
becomes a public key of Rainbow
(
Z
/N
Z
;
n
r,...,r
)
.
Remark 2.
The above proposition defines correspondence between signature
schemes,
n
r,...,r
)
HS(
R
;
n
)
Rainbow(
Z
/N
Z
;
Secret Key: (
A
1
, G, A
2
)
G
(
A
1
,φ
−n
+1
φ
n
,A
2
)
→
◦
◦
F
F.
Public Key:
→
Using this notation, we have the following correspondence:
OSS scheme
Rainbow(
Z
/N
Z
;1
,
1)
,
Birational Permutation scheme
Rainbow(
Z
/N
Z
;1
,...,
1)
,
Z
/N
Z
;4
,
4)
.
Sato-Araki scheme
Rainbow(
The argument of Uchiyama and Ogura in [22] is also valid for the case of HS
scheme defined over field
K
. Therefore, we have
F
Proposition 3.
Let
R
be a non-commutative ring over
K
of dimension
r
.Let
n
r,...,r
)
.
F
becomes a public key of Rainbow
(
K
;
be a public key of HS
(
R
;
n
)
.Then
Remark 3.
The above proposition shows that HS scheme is another way of con-
struction of the uniformly-layered Rainbow. Here, “uniformly-layered” means
that all components in the parameter of Rainbow are equal.
