Cryptography Reference
In-Depth Information
2.5 Schemes for the ND Model
In this section we describe schemes that work for the ND model. This model
has been considered only in [4] where a construction for c-color (k;n)-threshold
schemes is presented.
In order to have pixels with exactly the same color as the original one the
schemes of [4] have the property that in any shares superposition at most one
pixel is colored; all other pixels have one of the two special colors or .
The construction uses as a building block a black and white (k1;k1)-
threshold scheme.
Construction 3 Let S
k1
and S
k1
be the basis matrices of a (k1;k1)-
threshold scheme and let m
0
be the pixel expansion of such a scheme. Denote
the rows of S
k1
and S
k1
with wi
i
and b
i
, respectively:
2
4
3
5
2
4
3
5
w
1
w
2
:::
:::
w
k1
b
1
b
2
:::
:::
b
k1
S
k1
=
S
k1
=
;
:
Let S
1
= [] and S
1
= []. Then let F
k;n
(i;S
k1
), where i 2 f1; 2;:::;cg
and 2f;g be the n
k
m
0
matrix constructed by
k
submatrices, called
"blocks," with dimension nm
0
each consisting of the following rows: nk
("black") rows of m
0
elements ; Each block diers from the others in the
choice of the nk "black" rows; The remaining k rows are lled with one row
of elements equal to i followed in order by the k 1 rows of S
k1
.
Base matrix for color i, for i 2f1; 2;:::;cg, is given by
B
i
= F
k;n
(1;S
k1
) + ::: + F
k;n
(i 1;S
k1
) + F
k;n
(i;S
k1
) +
F
k;n
(i + 1;S
k1
) + ::: + F
k;n
(c;S
k1
)
where + denotes the concatenation of the matrices.
An example will clarify the above construction. Let k = 3 and n = 4 and
consider the matrices S
k1
and S
k1
given by the Naor and Shamir (2; 2)-
threshold scheme [8], that is,
S
2
=
S
2
=
;
:
The F matrices will have
k
= 4 blocks, since we have to place 1 black row
in each of 4 possible positions. Hence, we have
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