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Research Program of Chinese Academy of Sciences
(No.XDA06010701), and the Foundation of Institute
of Information Engineering for Cryptography.
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APPENDIX (QLH-Ensemble)
Let q be a prime number and X be a subset of Z
q
.
Then the QLH-function ensemble is a family of func-
tions F
q,n
:= { f : X
n
→ Z
N
} and each function f ∈
F
q,n
is defined as
f (x
1
,··· ,x
n
) =
∑
t
α
t
∏
i6= j,i, j∈[n]
(x
i
− x
j
)
a
i, j,t
mod q,
where α
t
∈ Z
q
and a
i, j,t
∈ N.
Specific to the tailored CS-scheme, we can repre-
sent functions from the QLH-ensemble as
f (sk
1
,··· ,sk
n
)
=
∑
t
1
,t
2
,t
3
α
t
1
,t
2
,t
3
∏
i> j,i, j∈[n],s
1
,s
2
,s
3
∈{1,2}
[(x
i,s
1
− x
j,s
1
)
b
i, j,t
1
· (y
i,s
1
− y
j,s
1
)
b
i, j,t
2
· (z
i,s
1
− z
j,s
1
)
b
i, j,t
3
] (mod q),
where sk
i
= (x
i1
,x
i2
,y
i1
,y
i2
,z
i1
,z
i2
) is the secret key
for the ith user, α
t
1
,t
2
,t
3
∈ Z
q
, b
i, j,t
1
,b
i, j,t
2
and b
i, j,t
3
∈
N.
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