General Constructions for Threshold Multiple-Secret Visual Cryptographic Schemes

A conventional threshold (
k
out of
n
) visual secret sharing scheme encodes one secret image
P
into
n
transparencies (called shares) such that any group of
k
transparencies reveals
P
when they are superimposed, while that of less than
k
ones cannot. We define and develop general constructions for threshold multiple-secret visual cryptographic schemes (MVCSs) that are capable of encoding
s
secret images
P
₁
,
P
₂
,...,
P
_s
into
n
shares such that any group of less than
k
shares obtains none of the secrets, while 1) each group of
k
,
k
+1,...,
n
shares reveals
P
₁
,
P
₂
, ...,
P
_s
, respectively, when superimposed, referred to as (
k
,
n
,
s
)-MVCS where
s
=
n
-
k
+1; or 2) each group of
u
shares reveals P(r
_u
) where
r
_u
∈ {0,1,2,...,
s
} (
r
_u
=0 indicates no secret can be seen),
k
≤
u
≤
n
and 2 ≤
s
≤
n
-
k
+1, referred to as (
k
,
n
,
s
,
R
)-MVCS in which
R
=(
r
_k
,
r
_k+1
, ...,
r
_n
) is called the revealing list. We adopt the skills of linear programming to model (
k
,
n
,
s
) - and (
k
,
n
,
s
,
R
) -MVCSs as integer linear programs which minimize the pixel expansions under all necessary constraints. The pixel expansions of different problem scales are explored, which have never been reported in the literature. Our constructions are novel and flexible. They can be easily customized to cope with various kinds of MVCSs.