Applied Mathematical Sciences, Vol. 11, 2017, no. 56, 2807-2815 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.710309 Some Algebraic (n, n)-secret Image Sharing Schemes Selda Çalkavur Mathematics Department, Kocaeli University, Kocaeli, Turkey Copyright c 2017 Selda Çalkavur. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We have presented some algebraic (n, n)- threshold schemes in [3]. In this paper we proposed the algebraic (n, n)- secret image sharing schemes. Keywords: Finite fields, secret sharing, multisecret sharing, secret image sharing scheme 1 Introduction Secret sharing schemes were examined by Shamir in 1979 [10]. Shamir s scheme is a (t, n)- threshold secret sharing scheme and this scheme was based on polynomial interpolation. A (t, n)- secret sharing scheme is a method of distribution of information among n participants such that t > 1 can reconstruct the secret but t 1 can not. In a secret sharing scheme there are participants and a dealer. The dealer has a secret and distributes it to the other participants. In a minimal t- subset of participants recover the secret while combining their shares.these subsets are called the minimal access sets. The access structure of a secret sharing scheme is the set of all minimal access sets. Another secret sharing scheme is the multi-secret sharing scheme. This scheme was proposed [5], [6], [8], [9], [13], [4]. In the multi-secret sharing scheme [8], [9], [2] there is a set of p secrets can be shared at once and each participant needs to keep one share is called secret share. In this scheme all p secrets are recovered at once or all p secrets can not reconstruct. To recover
2808 Selda Çalkavur the secrets the participants need to submit a pseudo share computed from their secret share instead of the secret share itself. In a (t, n)- secret image sharing scheme, the secret image is used to generate n image shares (shadows) as follows. the secret image can be recovered by combining any t image shares (t n) the secret image can not be recovered by combining any t 1 or fewer image shares. Thien and Lin [11] proposed a new lossless (t, n)- secret image sharing scheme based on Shamir s secret sharing in 2002. Then Li Bai [2] proposed an image secret sharing scheme combining matrix projection and Shamir s secret sharing schemes. Wang and Su [12] proposed a secret image sharing method using Huffman coding. In this work we propose a new (n, n)- secret image sharing scheme. In fact we generalise the results of [3]. The rest of this paper is organized as follows. The next section gives the basic preliminaries used in the paper. Section III presents the proposed scheme. Section IV explains its analysis and discussions and the last section concludes the paper. 2 Background and Preliminaries In this section we give the basic preliminaries and some necessary mathematical information used in this work. 2.1 Representation of elements of finite fields It is known that there are three different ways to represent the elements of a finite field F q with q = p n elements, where p is the characteristic of F q. One of them is given by means of matrices. The companion matrix of a monic polynomial f(x) = a 0 + a 1 x +... + a n 1 x n 1 + x n of positive degree n over a field is defined to be n n matrix 0 0 0... 0 a 0 1 0 0... 0 a 1 A = 0 1 0... 0 a 2....... 0 0 0... 1 a n 1
Some algebraic (n, n)-secret image sharing schemes 2809 It is clear that A satisfies the equation f(a) = 0. This means a 0 I + a 1 A +... + a n 1 A n 1 + A n = 0, where I is the n n identity matrix. So, if A is the companion matrix of a monic irreducible polynomial over F p of degree n, then f(a) = 0. Hence A can play the role of a root of f [7]. 2.2 (t, n) Multi-secret image sharing schemes A (t, n) multi-secret sharing scheme has the following properties. there is a set of p secrets can be shared at once, participants only need to pool their pseudo shares instead of disclosing their secret shares when recovering secret images, each participant can share many secret images by holding only one secret share [1]. 2.3 Image secret sharing An image I is defined by c number of colors and d l pixels m i,j, 1 i d, 1 j l, which form a matrix M with coefficients in Z c such that if I is a black and white image, then M is an d l matrix, where m i,j = 1 if the corresponding pixel is black and m i,j = 0 if the corresponding pixel is white, m i,j Z 2 for 1 i d, 1 j l [1]. 3 The scheme In this section we present an (n, n)- secret image sharing scheme that combines of (n, n)- threshold scheme. 3.1 Proposed method Assume the finite field F 2 d be the secret image space and f be an irreducible polynomial of degree d in F 2 [x]. We construct a secret image sharing scheme based on this ring. Let be N = 2 d. The secret image share dealing protocol as follows. 1) The dealer picks s elements e i amongst N at random such that N > s > 1, 2) the secret image is S = s i=1 e i, 3) he distributes the remaining N s images of F 2 d to the N s users.
2810 Selda Çalkavur 3.2 Secret image distribution We know that the elements of F 2 d can be represented by the d d matrices. So, each element of this finite field has d d pseudo shares. This means the secret image has also d d pseudo shares. That is the secret image is divided by d d parts. Moreover, a pseudo image share is distributed instead of the secret image share itself. 3.3 Secret image reconstruction Since the finite field F 2 d is zero-sum set, then the sum of all elements of F 2 d is zero, but without smaller size zero-sum sets. It is clear that the entries of matrices will be 0 or 1. If it is 1, then the corresponding pixel is black and if it is 0, then the corresponding pixel is white. The black and white pictures are transmitted and by using the properties of zero-sum sets the colour picture is reconstructed. That is the secret image can recover in this way. Let f denote a polynomial of degre d in Z 2 [x] and consider the finite field F 2 d. The recommended values of the parameters are f irreducible, 2 d 1 prime, d > 1. Proposition 1. With the above condition the finite field F 2 d determines a (2 d s, 2 d s)- secret image sharing scheme for all integers 1 < s < 2 d. Proof. It is clear that any coalition of size 2 d can recover the secret image because of the trivial zero-sum set of size 2 d. Proposition 2. In the (2 d s, 2 d s)- secret image sharing schemes there are (2 d s).(d d) secret image sharings in the minimal access set. Proof. In these secret image sharing schemes there are (2 d s) images can recover the secret image by combining their image shares. Each image has a pseudo-share with d d parts. So, (2 d s) images have the pseudo-shares with (2 d s).(d d) parts. This is also the number of minimal access set. Example: Let F 2 3 be the secret image space. Consider the polynomial f(x) = x 3 + x + 1 which is irreducible over F 2, where a 0 = 1, a 1 = 1, a 2 = 0. The companion matrix of f(x) is as follows.
Some algebraic (n, n)-secret image sharing schemes 2811 that is So the set of elements of F 2 3 A = A = is 0 0 a 0 1 0 a 1 0 1 a 2 0 0 1 1 0 1 0 1 0. {0, I, A, A 2, I + A, I + A 2, A + A 2, 1 + A + A 2 }, where I is the 3 3 identity matrix. We write these elements explicitly. 0 0 0 0 = 0 0 0 0 0 0 1 0 0 I = 0 1 0 0 0 1 0 1 0 A 2 = 0 1 1 1 0 1 1 0 1 I + A = 1 1 1 0 1 1 1 1 0 I + A 2 = 0 0 1 1 0 0 0 1 1 A + A 2 = 1 1 0 1 1 1 1 1 1 1 + A + A 2 = 1 0 0 1 1 0 Now, we construct a secret images sharing scheme based on F 2 3. It is clear that in this scheme with s = 2
2812 Selda Çalkavur there are 2 3 2 = 6 image shares, 2 elements whose sum is the secret image s, each element has 3 3 = 9 pseudo share images. So, the secret image has also 9 pseudo share images. Let A 5 = A + A 2 be the secret image and it is distributed to the other participants. Then N s = 2 3 2 = 6 participants can recover the secret by combining their image shares. Now, we explain how the secret image reconstruct can. Since for the entry 1 the corresponding pixel is black and for the entry 0 the corresponding pixel is white, then every element of F 2 3 can be represented as follows. 0 = I = A 2 = I + A = I + A 2 = A + A 2 =
Some algebraic (n, n)-secret image sharing schemes 2813 I + A + A 2 = Since the finite field F 2 3 is zero-sum set, then the sum all of nonzero images must be zero. I + A + A 2 + (I + A) + (I + A 2 ) + (I + A + A 2 ) + s= In the left side of equation if it is written the images of these elements, then it is obtained s =. It is easily seen that this scheme is also a (6, 6)-secret image sharing scheme. 4 Analysis and Discussions 4.1 Security analysis We discuss the security of the proposed (2 d s, 2 d s)- secret image sharing method is as folllows. Since the public values are the elements of F 2 d is zerosum set, then the sum of all elements is zero, but without smaller size zero-sum sets. Every public value has d d pseudo share images. However none of the public values can be determined using pseudo share images by himself. All of nonzero public values while combining their image shares can recover the secret image. So this scheme is very reliable. 4.2 Performance analysis The size of each image share does not depend on the size of the secret image in the proposed secret image sharing method. This is an important property
2814 Selda Çalkavur for the another process of the image shares. Each participant only needs to hold one secret share in order to secret image can share. The proposed scheme does not generate shadow images which are difficult to identify. 5 Conclusion In this paper we propose a new algebraic (n, n)-secret image sharing scheme using the results the work [3]. It is realized by combining the finite field F 2 d and the algebraic (n, n)- threshold scheme. Each participant can be represented as a matrix and only needs to hold one secret share in order to the secret image can share. Morever, the proposed scheme does not generate shadow images which are difficult to identify. The scheme is also effective, secure, reliable and suitable for network applications. In the current research we are explaining the method for the general access structure. References [1] T. Alexandrova, Y. Suzuki, K. Okubo, N. Tagawa, Secret Images Sharing Scheme Using Two-Variable One-Way Functions, 2010 IEEE International Conference on Wireless Communications, Networking and Information Security, (2010), 553-557. https://doi.org/10.1109/wcins.2010.5541840 [2] Li Bai, A Reliable (k, n) Image Secret Sharing Scheme, Proc. of the 2nd International Symposium on Dependable, Autonomic and Secure Computing DASC 06, (2006), 1-6. https://doi.org/10.1109/dasc.2006.11 [3] S. Çalkavur, P. Solé, Some algebraic (n, n)- threshold schemes, Conference Paper, International Conference on Modeling, Simulation and Applied Optimization (ICMSAO 15), (2015). [4] S. Çalkavur, P. Solé, Multisecret sharing schemes and bounded distance decoding of linear codes, International Journal of Computer Mathematics, 94 (2017), no. 1, 107-114. https://doi.org/10.1080/00207160.2015.1091071 [5] L. Harn, Comment: Multistage secret sharing based on one-way function, Electronics Letters, 31 (1995), no. 4, 262. https://doi.org/10.1049/el:19950201
Some algebraic (n, n)-secret image sharing schemes 2815 [6] J. He and E. Dawson, Multistage secret sharing based on one-way function, Electronic Letters, 30 (1994), no. 19, 1591-1592. https://doi.org/10.1049/el:19941076 [7] R. Lidl, H. Niederreiter, Finite Fields, vol. 20, Cambridge University Press, 1996. https://doi.org/10.1017/cbo9780511525926 [8] H. -X. Li, C. -T. Cheng and L. -J. Pang, A New (t, n) Threshold Multi- Secret Sharing Scheme, Computational Intelligence and Security, Lecture Notes in Computer Science, vol. 3802, Springer-Verlag Berlin Heidelberg, 2005, 421-426. https://doi.org/10.1007/11596981 61 [9] L. -J. Pang and Y. -M. Wang, A new (t, n) multi-secret sharing scheme based on Shamir s secret sharing, Applied Mathematics and Computation, 167 (2005), 840-848. https://doi.org/10.1016/j.amc.2004.06.120 [10] A. Shamir, How to share a secret, Comm. of the ACM, 22 (1979), 612-613. https://doi.org/10.1145/359168.359176 [11] C. -C. Thien and J. C. Lin, Secret image sharing, Computers and Graphics, 26 (2002), 765-770. https://doi.org/10.1016/s0097-8493(02)00131-0 [12] R. -Z. Wang and C. -H. Su, Secret image sharing with smaller shadow images, Pattern Recognition Letters, 27 (2006), 551-555. https://doi.org/10.1016/j.patrec.2005.09.021 [13] C. -C. Yang, T. -Y. Chang, M. -S. Hwang, A (t, n) multi-secret sharing scheme, Applied Mathematics and Computation, 151 (2004), 483-490. https://doi.org/10.1016/s0096-3003(03)00355-2 Received: October 29, 2017; Published: November 14, 2017