Two-layered structure for optimally essential secret image sharing scheme

J. Vis. Commun. Image R. 38 (2016) 595–601

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Two-layered structure for optimally essential secret image sharing scheme q Chien-Chang Chen ⇑, Shih-Chang Chen Department of Computer Science and Information Engineering, Tamkang University, No. 151, Yingzhuan Rd., Tamsui Dist., New Taipei City 25137, Taiwan

a r t i c l e

i n f o

Article history: Received 23 November 2015 Revised 18 February 2016 Accepted 6 April 2016 Available online 8 April 2016 Keywords: Secret image sharing Essential Non-essential Two-layered structure Optimal sharing ratios

a b s t r a c t This paper presents a two-layered structure for optimally sharing a secret image among s essential and n
s non-essential shared shadows using the (t, s, k, n) essential thresholds, that t essential shared shadows and totally k shared shadows are needed to recover the secret image. The presented two-layered structure includes one user-defined parameter m to determine different kinds of optimal results. m = 1 leads to minimum size of total shared shadows (ST) and size of an essential shared shadow is close to size of a non-essential shared shadow. On the other hand, m = t leads to size of an essential shared shadow being twice of size of a non-essential shared shadow to signify the importance of an essential shared shadow. Moreover, the proposed structure overcomes the threshold fulfillment problem in Chen’s scheme (Chen, 2016). Theoretical analyses and experimental results show that the proposed scheme exhibits secure with optimal sharing ratios among related works. Ó 2016 Elsevier Inc. All rights reserved.

1. Introduction Multimedia contents are popularly used over network and the security problems in storing or transmitting multimedia files are more and more important. Secret image sharing techniques share a secret image among shared shadows and then recover the secret image from shared shadows to protect the image secretly. The first secret image sharing scheme (SIS) is presented by Thien and Lin [1], in which the Shamir–Lagrange technique is adopted to share an image secretly. Many researchers have then proposed other SIS for sharing a secret image with different properties, including sharing secret image among host images [2–6], sharing both with secret image sharing and visual cryptography [7], scalable sharing [8,9], cheater identification [10], and essential sharing [11–13]. Moreover, many different techniques, including Shamir–Lagrange [1], Blakley geometry [14], Chinese Remainder Theorem [15], cellular automata [16,17], combination theory [18], and Boolean operations [19–21], have been presented for sharing a secret image. Among these studies, an essential secret image sharing scheme (ESIS) [11–13] generates shared shadows including essential and non-essential ones. Since two kinds of shadows meet the

q

This paper has been recommended for acceptance by M.T. Sun.

⇑ Corresponding author.

E-mail address: [email protected] (C.-C. Chen). http://dx.doi.org/10.1016/j.jvcir.2016.04.004 1047-3203/Ó 2016 Elsevier Inc. All rights reserved.

requirement of participants with different privileges, the essential secret image sharing problem merits our study. A (t, s, k, n) ESIS shares the secret image among n shared shadows, including s essential shared shadows and n
s non-essential shared shadows. Two threshold requirements are needed to recover the secret image, the first threshold requirement is collecting t essential shared shadows and the other threshold requirement is collecting totally k shared shadows including essential and non-essential shared shadows. In the study of ESIS, Li et al. first present the concept of (t, s, k, n) ESIS [11]. Yang et al. [12] then modified Li et al.’s scheme [11] to reduce total size of shared shadows. In 2016, Chen adopted two SIS with different thresholds to share a secret image among essential and non-essential shared shadows [13]. Among these ESIS schemes, Chen’s scheme acquires presently optimal sharing ratios on size of essential shared shadows (SE), size of non-essential shared shadows (SN), size of total shared shadows (ST), and size of required shared shadows to recover the secret image (SR). However, Chen’s scheme exhibits a threshold fulfillment problem that satisfying only one threshold requirement partially recovers the secret image. This paper improves Chen’s scheme to a two-layered scheme to overcome the threshold fulfillment problem but keeps at least two kinds of optimal sharing ratios. A pre-defined parameter m determines the selection of optimal sharing ratio. The first one is m = 1 that acquires nearly sizes between essential and non-essential shared shadows with smallest ST value. The second one is m = t that obtains size of an essential

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shared shadow being twice of size of a non-essential shared shadow to signify the importance of an essential shared shadow. Moreover, the second one has the same optimal results presented by the Chen’s scheme [13]. This paper is organized as follows. Section 2 briefly introduces three related literatures [11–13] with discussions of their properties. Section 3 presents our proposed ESIS. Section 3.1 introduces the sharing strategy over Galois field GF(28). Section 3.2 introduces algorithm of the proposed scheme and Section 3.3 theoretically compares the sharing ratios between the proposed scheme and related works [11–13]. Section 4 presents experimental results of our proposed scheme. Section 5 provides a conclusion and suggestions for future research. 2. Literature reviews This section reviews three important literatures [11–13] of essential secret image sharing schemes (ESIS). A conventional (k, n) secret image sharing scheme (SIS) shares a secret image among n shared shadows and gathering k shared shadows recovers the secret image. A (t, s, k, n) ESIS shares the secret image among n shared shadows, in which s shared shadows are essential and n
s shared shadows are non-essential. Collecting t essential shared shadows and collecting totally k shared shadows are two threshold requirements to recover the secret image. Three important ESIS schemes are briefly reviewed in this section. Section 2.1 introduces the first (t, s, k, n) ESIS that is presented by Li et al. [11]. Section 2.2 introduces Yang et al.’s work [12] that improves Li et al.’s scheme [11] to reduce total size of shared shadows. Section 2.3 introduces Chen’s ESIS to adopt two SIS with different thresholds for sharing a secret image [13]. 2.1. Review of the Li et al.’s scheme [11] Li et al. used a (k, s + k
t) SIS and a series of k
t SIS to construct the ESIS. Their scheme first adopted a (k, s + k
t) SIS to partition the secret image among s + k
t shadows. The first s shadows are essential shared shadows. The remaining k
t shadows are then applied to a series of k
t SIS with thresholds (1, n
s), (2, n
s), . . . , (k
t, n
s), respectively. A non-essential shared shadow is then acquired from concatenating one generated shadow from each SIS. Sizes of essential and non-essential shared

non-essential shared shadow Si(s 6 i 6 s + k
t
1) is only acquired from the concatenation of sx,i(s 6 x 6 s + k
t
1). Therefore, sizes of essential and non-essential shared shadows are 2k
t  z and k
t  z, respectively. k2 k2 2.3. Review of Chen’s scheme [13] Chen used two SIS to construct a new structure for reducing total size of essential and non-essential shared shadows. His scheme first permutes the secret image to obtain a permuted secret image. The permuted image is then partitioned to two parts t k of sizes proportional to tþk and tþk of the secret image. These two parts are then applied to thresholds (t, s) and (k, n) for generating two sets of shadows. Combining two shadows of one from the first set of shadows and another from the other set of shadows acquires an essential shared shadow. Since k > t, each remaining (k, n) shadow forms a non-essential shared shadow. The size of essential 2 1 and non-essential shadows are tþk  z and tþk  z, respectively. 2.4. Properties discussion on three works [11–13] Li et al. [11] and Yang et al. [12] both required 1 + k
t SIS to share a secret image. The modified method of Li et al.’s scheme requires only two SIS. Chen [13] also used two SIS to reduce sizes of all shared shadows. Chen’s scheme is simple and efficient on sharing a secret image among essential and non-essential shared shadows. Since two thresholds have to be both required in a (t, s, k, n) ESIS, Chen’s scheme adopts two SIS independently leads to a threshold fulfillment problem on only meeting one of these two thresholds. Fig. 1 shows two examples of threshold fulfillment problem on only meeting 3 essential shared shadows requirement or totally 5 shared shadows requirement in the Chen’s (3, 5, 5, 8) ESIS. Experimental results in Fig. 1 show that part of the secret image can be recovered. The problem can be perfectly overcome by adding an extra layer to construct a two-layered structure. Our proposed scheme, introduced in Section 3, shows that the presented two-layered scheme also exhibits optimal shared ratios among (t, s, k, n) ESIS [11–13]. 3. Proposed scheme

shadows are 1k  z and Hk
t  z, respectively, where Hk
t is the k harmonic number and z is size of the secret image. In order to decrease size of a non-essential shared shadow under the restriction of s + 1 6 k, they further present a modified version that first concatenate shadows s + 1, s + 2, . . . , s + k
t to acquire an intermediate shadow and then apply the intermediate shadow to a (k
s, n
s) SIS for generating non-essential shared shadow. Size of the modified non-essential shared shadow is

This section presents the proposed (t, s, k, n) ESIS. Section 3.1 briefly introduces secret sharing calculation over GF(28). Section 3.2 introduces the proposed two-layered (t, s, k, n) ESIS. Section 3.3 discusses sharing ratios of our proposed scheme. Theoretically analyses with three works [11–13] are also compared.

ð1þHk
t
Hk
s Þ k

This section briefly introduces a (k, n) SIS to share k secret numbers ci(0 6 i 6 k
1) among n shared numbers f(xj) with keys xj (0 6 j 6 n
1), respectively. The sharing calculation is performed by the following equation

 z.

2.2. Review of Yang et al.’s scheme [12] Yang et al. improved previous Li et al.’s scheme [11] to reduce total size of the shared shadows. Their scheme first applied a (k, s + k
t) SIS on the permuted secret image to acquire 1stlayered shadows sx(0 6 x 6 s + k
t
1). Like the method in previous Li et al.’s scheme, all these 1st-layered s + k
t shadows sx are partitioned to two groups sx(0 6 x 6 s
1) and sx(s 6 x 6 s + k
t
1). Each of the second group of 1st-layered shadows sx(s 6 x 6 s + k
t
1) is applied to a (k, n) SIS to acquire 2nd-layered shadows sx,y(s 6 x 6 s + k
t
1, 0 6 y 6 n
1). The essential shared shadow Si(0 6 i 6 s
1) is acquired from concatenating si(0 6 i 6 s
1) and sx,i(s 6 x 6 s + k
t
1). On the other hand, a

3.1. Secret sharing over Galois field GF(28)

f ðxj Þ ¼ c0 þ c1 xj þ c2 x2j þ    þ ck
1 xk
1 j Numbers are represented through polynomials. For example, a number (193)10 = (11000001)2 is represented by x7 + x6 + 1. Mathematical calculations are also performed through polynomial processing. For example, (5)10 + (20)10 acquires (17)10 by the processing of (x2 + 1) + (x4 + x2) = x4 + x2 + x2 + 1 = x4 + 1 and (5)10  (20)10 acquires (68)10 by the processing of (x2 + 1)  (x4 + x2) = x6 + x4 + x4 + x2 = x6 + x2. The polynomial calculation is processed under modulus result of an irreducible polynomial like x8 + x4 + x3 + x + 1. In recovering calculation, all k secret numbers ci(0 6 i 6 k
1) can

C.-C. Chen, S.-C. Chen / J. Vis. Commun. Image R. 38 (2016) 595–601

(a)

597

(b)

Fig. 1. Two examples of threshold fulfillment problem in the Chen’s (3, 5, 5, 8) ESIS [13], (a) only meeting 3 essential shadows requirement, (b) only meeting totally 5 shadows requirement.

be acquired through k shared numbers f(xj) and corresponding keys xj by Lagrange interpolation polynomial. More discussions of secret sharing over the GF(28) can be found in literatures [13,22,23]. 3.2. Algorithm of the proposed two-layered (t, s, k, n) ESIS This section introduces our proposed two-layered (t, s, k, n) ESIS. Note that all calculation in the proposed scheme is based on GF(28). The proposed scheme uses 1st-layer SIS to partition the permuted secret image to two 1st-layered intermediate shadows. Each 1stlayered intermediate shadow denotes one threshold requirement for recovering the secret image. Two intermediate shadows are then applied to a (t, s) SIS and a (k, n) SIS respectively to generate 2nd-layered shadows for composing the essential and nonessential shared shadows. Sharing algorithm of the proposed (t, s, k, n) ESIS is introduced as follows. 1. Permute the input secret image I by a permutation key pk to acquire a permutated secret image PI. 2. Apply PI to a (t + k, t + k) SIS for acquiring t + k 1st-layered shadows PIi(0 6 i 6 t + k
1). 3. For i = 0 to m
1, Concatenate 1st-layered shadows PIi to acquire the first intermediate shadow I1. 4. For i = m to t + k
1, Concatenate 1st-layered shadows PIi to acquire the second intermediate shadow I2. 5. Apply the intermediate shadow I1 to a (t, s) SIS to acquire s 2nd-layered shadows I1,i (0 6 i 6 s
1). 6. Apply the intermediate shadow I2 to a (k, n) SIS to acquire n 2nd-layered shadows I2,i (0 6 i 6 n
1). 7. For i = 0 to s
1, Concatenate a pair of 2nd-layered shadows I1,i and I2,i to acquire an essential shared shadow Si. 8. For i = s to n
1, Assign 2nd-layered shadow I2,i to the non-essential shadow Si. The 1st-layered SIS meets the requirement of both two intermediate shadows I1 and I2 being necessary in recovery. Therefore, the threshold fulfillment problem is overcome in our proposed scheme. Fig. 2 depicts the sharing procedure of the proposed scheme. Sizes of all 1st-layered shadows PIi are identical. Therefore, sizes of intermediate shadows I1 and I2 are determined by the parameter

m. A small m value leads to small sized I1. Moreover, sizes of all 2nd-layered shadows Ii are also up to parameter m. Consequently, when m = s, sizes of all 2nd-layered shadows I1,i(0 6 i 6 s
1) and I2,i(0 6 i 6 n
1) are all identical and size of an essential shared shadow is twice of size of a non-essential shared shadow. On the contrary, m = 1 leads to size of an essential shared shadow only a little larger than size of a non-essential shared shadow. The recovering algorithm needs t essential shared shadows, k non-essential shared shadows, two parameters m and pk to recover the secret image. Recovering algorithm of the proposed (t, s, k, n) ESIS is introduced as follows. 1. Gather t essential shared shadows ESi(0 6 i 6 t
1) and k
t non-essential shared shadows NSi(0 6 i 6 k
t
1). 2. Split each essential shared shadows ESi(0 6 i 6 t
1) to two shadows ES1,i and ES2,i. 3. Recover intermediate shadow I1 using ES1,i(0 6 i 6 t
1). 4. Recover intermediate shadow I2 using ES2,i(0 6 i 6 t
1) and NSi(0 6 i 6 k-t
1). 5. Split the intermediate shadow I1 to shadows PIi(0 6 i 6 m
1). 6. Split the intermediate shadow I2 to shadows PIi(m 6 i 6 t + k
1). 7. Recover the permuted secret image PI using PIi(0 6 i 6 t + k
1). 8. Apply inverse permutation with the permutation key pk to recover the secret image I. The recovering algorithm applies the inverse procedure of the sharing algorithm. Section 3.3 proves the correctness and discusses sharing ratios of the proposed scheme. 3.3. Properties discussion This section discusses properties of the proposed scheme and calculates sharing ratios of the proposed scheme. The sharing ratios include size of an essential shared shadow (SE), size of a non-essential shared shadow (SN), size of total shared shadows (ST), and size of required shared shadows to recover the secret image (SR). Since the proposed scheme shares using Shamir method and recovers using Lagrange method, the following three equations (Eqs. (1)–(3)) are used to explain properties of the proposed scheme. Eq. (1) represents sharing the secret image I among shared

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C.-C. Chen, S.-C. Chen / J. Vis. Commun. Image R. 38 (2016) 595–601

Secret Image

I

permutation Permuted Secret Image

PI

(t+k,t+k) SIS

1st-layered shadows

… PI0 PI1

1st-layered shadows

… PIm-1 PIm+1

PIt+k-2 PIt+k-1

m shadows

intermediate shadow

t+k-m shadows

I1

I2

intermediate shadow

(k, n) SIS

(t, s) SIS

…

…

2nd-layered shadows I1,0

I1,1

I1,s-1

I2,0

I2,1

… I2,s-1

I2,s

I2,n-1

…

…

… S0

S1

2nd-layered shadows

… Ss-1

Ss

Sn-1

non-essential shared shadows

essential shared shadows

Fig. 2. Sharing procedure of the proposed two-layered ESIS.

shadows S0, S1, . . . , Sn
1 using (t, n) Shamir method. Size of each shared shadow is 1t  z, where z denotes size of the secret image. Eq. (2) represents recovering the secret image I using Lagrange method through t different shared shadows Si, where |Si| denotes number of different shadows Si. Eq. (3) shows that any modified Si (S00 ) fails to recover the secret image I.

ShamirðI; t; nÞ ) S0 ; S1 ; . . . ; Sn
1 with size LagrangeðSi ; jSi j ¼ tÞ ) I

1 z t

LagrangeðS00 ; S1 ; . . . ; St
1 Þ;I; where S0 –S00

ð3Þ st

ð1Þ

Eq. (4) shows that our proposed scheme uses 1 -layered (t + k, t + k) SIS to share the secret image I among t + k shadows. The first intermediate shadow I1 is acquired by concatenating first m shadows as shown in Eq. (5). Concatenating the remaining shadows acquires the second intermediate shadow I2. Choosing m = 1 or m = t acquires smallest ST value or optimizing SE, SN, ST, and SR values, respectively.

ð2Þ

ShamirðI; t þ k; t þ kÞ ) PI0 ; PI1 ; . . . ; PItþk
1 with size

1 z tþk

ð4Þ

599

C.-C. Chen, S.-C. Chen / J. Vis. Commun. Image R. 38 (2016) 595–601

m z tþk tþk
m ) I2 with size z tþk

to that size of required shared shadows to recover the secret image
 2 þkt
mt (SR) is t  mkþt  z ¼ z. þ ðk
tÞ  tþk
m tkðtþkÞ kðtþkÞ

PI0 jjPI1 jj . . . jjPIm
1 ) I1 with size PIm jjPImþ1 jj . . . jjPItþk
1

ð5Þ

Table 1 lists theoretical sharing ratio comparisons between the proposed scheme and three literatures [11–13]. In our proposed scheme, m is a pre-determined parameter. The proposed scheme exhibits the same ratios to Chen’s work [13] under the selection of m = t to acquire optimized SE, SN, ST, and SR among these works. Moreover, the selection of m = 1 leads to smallest ST value with the
2 
 property of SE value t þktþðk
tÞ  z close to SN value tþk
1 z . tkðtþkÞ kðtþkÞ

Two intermediate shadows I1 and I2 are then applied to different thresholds (t, s) and (k, n), respectively. Eqs. (6)–(8) show that sizes of essential shared shadows (SE) and non-essential shared
 2 þkt
mt m shadows (SN) are tðtþkÞ  z ¼ mkþt þ tþk
m  z and tþk
m  z, kðtþkÞ tkðtþkÞ kðtþkÞ respectively. Moreover, size of total shared shadows (ST) is 2

2

þkt
mt þnkt
nmt ðs  mkþt þ ðn
sÞ  tþk
m Þ  z ¼ smkþntktðtþkÞ  z. tkðtþkÞ kðtþkÞ

ShamirðI1 ; t; sÞ ) I1;0 ; I1;1 ; . . . ; I1;s
1 with size

ShamirðI2 ; k; nÞ ) I2;0 ; I2;1 ; . . . ; I2;n
1

m z tðt þ kÞ

ð6Þ

tþk
m z with size kðt þ kÞ

4. Experimental results and discussion This section presents experimental results of the proposed scheme. Fig. 3 shows shared shadows of a (3, 5, 5, 8) ESIS with m = t = 3. Fig. 3(a) shows the secret image Slum, with size 480  480. Fig. 3(b) shows the permutation of the secret image. Two intermediate shadows I1 with size 480  180 and I2 with size 480  300 are shown in Fig. 3(c) and (d), respectively. Five essential shared shadows with size 480  120 and three non-essential shared shadows with size 480  60 are shown in Fig. 3(e)–(i) and (j)–(l), respectively. Fig. 4 shows another example of the same threshold (3, 5, 5, 8) ESIS. The assignment of m = 1 leads to smallest intermediate shared shadow I1 with size 480  60 as shown in Fig. 4(c). Size of the essential shared shadows are close to size of non-essential shared shadows as exhibiting 480  104 and 480  84, respectively. Fig. 3 shows that the selection of m = 3 in a (3, 5, 5, 8) SIS leads to two 1st-layered shadows being 480  180 and 480  300, which is proportional to t = 3 and k = 5. Therefore, the 2nd-layered (3, 5) SIS and (5, 8) SIS acquire 2nd-layered shadows being 480  60. Consequently, under the assignment of m = 3, sizes of essential shared shadows and non-essential shared shadows are then 480  120 and 480  60, respectively. On the other hand, the selection of m = 1 in a (3, 5, 5, 8) SIS leads to two 1st-layered shadows being 480  60 and 480  420. The 2nd-layered (3, 5) SIS acquires shared shadows with size 480  20, which is the difference between an essential shared shadow and a non-essential shared shadow. In the recovery, collecting enough essential and non-essential shared shadows reconstruct the original secret image. Fig. 5 shows recovery examples of using all correct shared shadows and an attacked essential shared shadow with other correct shared shadows. Fig. 5(a)–(h) demonstrate the correct secret image recovery and Fig. 5(i)–(p) demonstrate the secret image recovery under attacking one essential shared shadow. Fig. 5(a)–(c) show three correct essential shared shadows as shown in Fig. 3(e)–(g) and Fig. 5(d)–(e) show two correct non-essential shared shadows as shown in Fig. 3(j)–(k). Since these five shared shadows are correct and fit the (3, 5, 5, 8) thresholds, two recovered intermediate shadows as shown in Fig. 5(f)–(g) and the recovered

ð7Þ

essential shared shadows : I1;i jjI2;i ) Si ð0 6 i 6 s
1Þ with size

mk þ t 2 þ kt
mt z tkðt þ kÞ

ð8Þ

non
essential shared shadows : I2;i ) Si ðs 6 i 6 n
1Þ with size

tþk
m z kðt þ kÞ

In the recovery, gathering t essential shared shadows is needed to acquire t 2nd-layered shadows I1,i for recovering the 1st-layered intermediate shadow I1. Moreover, gathering k shared shadows, including essential and non-essential ones, is also needed to acquire k 2nd-layered shadows I2,i for recovering the 1st-layered intermediate shadow I2. The secret image I can be then acquired from two intermediate shadows I2 and I2. However, an attacked essential shared shadow S00 fails to recover two intermediate shadows I1 and I2 as shown in Eq. (9). Therefore, Eq. (10) shows that the attacked essential shared shadow also fails to recover the original secret image I. Any modified shared shadow S0i ð0 6 i 6 n
1Þ acquires the same results like Eqs. (9) and (10).

LagrangeðI01;0 ; I1;1 ;    ; I1;t
1 Þ ) I01 –I1 ;

)

LagrangeðI02;0 ; I2;1 ;    ; I2;k
1 Þ ) I02 –I2 ; where

S00 –S0 ðI01;0 –I1;0

and

ð9Þ

I02;0 –I2;0 Þ

LagrangeðPI00 ; PI01 ;    ; PI0tþk
1 Þ ) I0 –I; ( I01 –I1 ) PI0i –PIi ð0 6 i 6 m
1Þ where I02 –I2 ) PI0i –PIi ðm 6 i 6 t þ k
1Þ

ð10Þ

Without loss of generality, t essential shared shadows and k
t non-essential shared shadows leads to optimal size of shared shadows for recovering the secret image. This assumption leads

Table 1 Theoretical sharing ratio comparisons between the proposed scheme and three (t, s, k, n) ESIS schemes [11–13]. Schemes

Sharing ratios SE

Li et al. [11] Modified Li et al. [11] Yang et al. [12] Chen [13]

1 k 1 k

z z

2k
t 2  z k 2 z tþk mkþt 2 þkt
mt tkðtþkÞ

Proposed scheme (m = 1)

t þktþðk
tÞ tkðtþkÞ 2 z tþk

Proposed scheme (m = t)

ST

Hk
t z k 1þHk
t
Hk
s k

sþðn
sÞHk
t z k sþðn
sÞð1þHk
t
Hk
s Þ k

k
t 2 k 1 tþk

Proposed scheme

2

SN

z

z

z

z

t þkt
t tkðtþkÞ 2

1 tþk

skþnk
nt 2 k sþn z tþk

z

tþk
m kðtþkÞ

z

z

SR

z

z

skþnt þnkt
nt ktðtþkÞ sþn z tþk

tþðk
tÞHk
t z k tþðk
tÞð1þHk
t
Hk
s Þ k

z z

smkþnt2 þnkt
nmt ktðtþkÞ 2

z

z

z

z z z

z

600

C.-C. Chen, S.-C. Chen / J. Vis. Commun. Image R. 38 (2016) 595–601

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j) (k) (l)

Fig. 3. Examples of the proposed (3, 5, 5, 8) ESIS and m = t = 3, (a) 480  480 secret image Slum, (b) permutation image, (c) 480  180 intermediate shadow I1, (d) 480  300 intermediate shadow I2, (e)–(i) five 480  120 essential shared shadows, (j)–(l) three 480  60 non-essential shared shadows.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

Fig. 4. Examples of the proposed (3, 5, 5, 8) ESIS and m = 1, (a) 480  480 secret image Slum, (b) permutation image, (c) 480  60 intermediate shadow I1, (d) 480  420 intermediate shadow I2, (e)–(i) five 480  104 essential shared shadows, (j)–(l) three 480  84 non-essential shared shadows.

(a)

(b)

(c)

(d) (e)

(f)

(g)

(h)

(i)

(j)

(k)

(l) (m)

(n)

(o)

(p)

Fig. 5. Example of attacked recovery under the proposed (3, 5, 5, 8) ESIS and m = t = 3, (a)–(c) three 480  120 correct essential shared shadows, (d) and (e) two 480  60 correct non-essential shared shadows, (f) recovered 480  180 intermediate shadow, (g) recovered 480  300 intermediate shadow, (h) recovered secret image, (i) attacked essential shared shadows of (a), (j) and (k) two correct essential shared shadows, (l) and (m) two correct non-essential shared shadows, (n) recovered 480  180 intermediate shadow, (o) recovered 480  300 intermediate shadow, (p) recovered secret image.

secret image of Fig. 5(h) are all correct. Relatively, one attacked essential shared shadow as shown in Fig. 5(i) leads to false recovery on two intermediate shadows as shown in Fig. 5(n) and (o). Therefore, the recovered secret image as shown in Fig. 5(p) is also incorrect. Moreover, the incorrect recovered secret image always looks like a noise image.

Table 2 compares the sharing ratios of (3, 5, 5, 8) ESIS between three works [11–13] and our proposed scheme. Table 2 shows that the proposed scheme and Chen’s scheme [13] exhibit the optimal sharing ratios of SE, SN, ST, and SR. The requirement of the modified Li et al.’s work is s < k [11]. Therefore, no SN, ST and SR values can be measured in a (3, 5, 5, 8) ESIS. Moreover, m = 1 leads

C.-C. Chen, S.-C. Chen / J. Vis. Commun. Image R. 38 (2016) 595–601 Table 2 Sizes of different kinds of shared images between the proposed scheme and three (3, 5, 5, 8) ESIS [11–13]. Schemes

SE

Li et al. [11]

1 5z 1 5z

Modified Li et al. [11] Yang et al. [12] Chen [13] Proposed scheme (m = 1) Proposed scheme (m = 3)

SN 3 10 z

¼ 0:2z

–

–

2 25 z ¼ 0:08z 1 8 z ¼ 0:125z 7 40 z ¼ 0:175z

41 25 z ¼ 1:64z 13 8 z ¼ 1:625z 193 120 z ¼ 1:608z

¼ 0:25z

1 8z

¼ 0:3z

SR

¼ 0:2z

7 25 z ¼ 0:28z 2 8 z ¼ 0:25z 13 60 z ¼ 0:2167z 2 8z

ST

¼ 0:125z

19 10 z

13 8 z

¼ 1:9z

¼ 1:625z

6 5 z ¼ 1:2z –

z z z

Table 3 Sizes of different kinds of shared images between the proposed scheme and three (2, 3, 5, 7) ESIS [11–13]. SE

SN

ST

SR

Li et al. [11]

1 5 z ¼ 0:2z 1 5 z ¼ 0:2z 8 25 z ¼ 0:32z 2 7 z ¼ 0:286z 17 70 z ¼ 0:243z

11 30 z ¼ 0:367z 8 30 z ¼ 0:267z 3 25 z ¼ 0:12z 1 7 z ¼ 0:143z 6 35 z ¼ 0:171z

31 15 z ¼ 2:067z 5 3 z ¼ 1:67z 36 25 z ¼ 1:44z 10 7 z ¼ 1:429z 99 70 z ¼ 1:414z

3 2z 6 5z

Modified Li et al. [11] Yang et al. [12] Chen [13] Proposed scheme (m = 1) Proposed scheme (m = 2)

2 7z

¼ 0:286z

1 7z

¼ 0:143z

10 7 z

Acknowledgment This paper was partially supported by the National Science Council of the Republic of China under contract NSC 103-2221-E032-051. References

z

Schemes

601

¼ 1:5z ¼ 1:2z

z z z

¼ 1:429z z

to smallest ST ratio among these related works. Table 3 compares the sharing ratios of (2, 3, 5, 7) ESIS between three works [11–13] and our proposed scheme. The proposed 1st-layered SIS prevents Chen’s threshold fulfillment problem efficiently to become a secure (t, s, k, n) ESIS. The optimal sharing ratios in Tables 2 and 3 demonstrate the optimal sharing ratios of the proposed scheme. 5. Conclusion This paper presents a two-layered scheme for solving the (t, s, k, n) ESIS problem. The proposed scheme overcomes previous work’s drawback [13] on threshold fulfillment problem but also exhibits optimal sharing ratios among the related works. The usage of Shamir–Lagrange and Galois field(28) exhibit secure threshold recovery and perfect recovery. One parameter m is presented for acquiring different optimal sharing ratios. Experimental results show that our proposed scheme shares a secret image with optimal sharing ratios and the threshold fulfillment drawback is perfectly overcome. Minimizing the size difference between essential and non-essential shared shadows merits our future study.

[1] C.C. Thien, J.C. Lin, Secret image sharing, Comput. Graph. 26 (2002) 765–770. [2] C.C. Chang, Y.H. Chen, H.C. Wang, Meaningful secret sharing technique with authentication and remedy abilities, Inf. Sci. 181 (2011) 3073–3084. [3] C. Guo, C.C. Chang, C. Qin, A hierarchical threshold secret image sharing, Pattern Recogn. Lett. 33 (2012) 83–91. [4] T.H.N. Le, C.C. Lin, C.C. Chang, H.B. Le, A high quality and small shadow size visual secret sharing scheme based on hybrid strategy for grayscale images, Digit. Signal Process. 21 (2011) 734–745. [5] P.Y. Lin, C.S. Chan, Invertible secret image sharing with steganography, Pattern Recogn. Lett. 31 (2010) 1887–1893. [6] G. Ulutas, M. Ulutas, V. Nabiyev, Secret image sharing scheme with adaptive authentication strength, Pattern Recogn. Lett. 34 (2013) 283–291. [7] S.J. Lin, J.C. Lin, VCPSS: a two-in-one two-decoding-options image sharing method combining visual cryptography (VC) and polynomial-style sharing (PSS) approaches, Pattern Recogn. 40 (2007) 3652–3666. [8] Y.Y. Lin, R.Z. Wang, Scalable secret image sharing with smaller shadow images, IEEE Signal Process. Lett. 17 (2010) 316–319. [9] R.Z. Wang, Y.F. Chien, Y.Y. Lin, Scalable user-friendly image sharing, J. Vis. Commun. Image R. 21 (2010) 751–761. [10] C.C. Chen, C.A. Liu, Tamper-proof secret image sharing scheme for identifying cheated secret keys and shared images, J. Electron. Imag. 22 (2013) 013008. [11] P. Li, C.N. Yang, C.C. Wu, Q. Kong, Y. Ma, Essential secret image sharing scheme with different importance of shadows, J. Vis. Commun. Image R. 24 (2013) 1106–1114. [12] C.N. Yang, P. Li, C.C. Wu, S.R. Cai, Reducing shadow size in essential secret image sharing by conjunctive hierarchical approach, Signal Process.: Image Commun. 31 (2015) 1–9. [13] S.K. Chen, Essential secret image sharing with increasable shadows, Opt. Eng. 55 (2016) 013103. [14] H.K. Tso, Sharing secret images using Blakley’s concept, Opt. Eng. 47 (2008) 077001. [15] R.Y.V. Subba, B. Chakravarthy, CRT based threshold multi secret sharing scheme, Int. J. Network Security 16 (4) (2014) 249–255. [16] Z. Eslami, J.Z. Ahmadabadi, A verifiable multi-secret sharing scheme based on cellular automata, Inf. Sci. 180 (2010) 2889–2894. [17] X. Wu, D. Ou, Q. Liang, W. Sun, A user-friendly secret image sharing scheme with reversible steganography based on cellular automata, J. Syst. Software 85 (2012) 1852–1863. [18] Z. Liu, M.A. Ahmad, S. Liu, Image sharing scheme based on combination theory, Opt. Commun. 281 (2008) 5322–5325. [19] T.H. Chen, C.S. Wu, Efficient multi-secret image sharing based on Boolean operations, Signal Process. 91 (2011) 90–97. [20] C.C. Chen, W.J. Wu, A secure boolean-based multi-secret image sharing scheme, J. Syst. Software 92 (1) (2014) 107–114. [21] C.N. Yang, C.H. Chen, S.R. Cai, Enhanced Boolean-based multi secret image sharing scheme, J. Syst. Software (2016) (in press). [22] R.Z. Wang, C.H. Su, Secret image sharing with smaller shadow images, Pattern Recogn. Lett. 27 (2006) 551–555. [23] W. Stalling, Cryptography and Network Security: Principles and Practices, Pearson Education Inc., 2003. pp. 126–134.