Reviewing Multiple Secret Image Sharing Scheme based on Matrix Multiplication

Reviewing Multiple Secret Image Sharing Scheme based on Matrix Multiplication Fereshte Sheikh Sang Tajan Massoud Hadian Dehkordi Abdolrasoul Mirghadri Faculty and Research Center of Communication and Information Technology, IHU Tehran, Iran Department of Mathematical Sciences Iran University of Science and Technology Narmak, Tehran, 16844 Iran Faculty and Research Center of Communication and Information Technology, IHU Tehran, Iran Abstract Information security is one of the most important defensive issues to prevent the attackers' assault to the privacy of organizations' valuable information The secret sharing scheme is an attractive branch of advanced cryptography that has played crucial role in passive defense and it would be employed in protecting important information and documents against dangers like robbery and illegal accesses In this article a scheme based on the method of matrix multiple secret sharing is provided In which the shareholders reveal the artificial image (shadow image) instead of their genuine share to recover the image of secret In other word, each images sharing that has been efficiently and shareholder with having only one portion of secret image can share it with others in many of secret images Unlike other known secret sharing schemes, in this scheme the threshold images will not be produced in a way that are difficult to identification and control We also could create better speed and security through making change in this scheme by using row and column round in generator function The considered plan is a scheme of multiple secret practically performed with more appropriate speed Key words: Matrix multiplication, secret sharing, divider, multiple secret, artificial images, shadow 494

1 Introduction Cryptography has been founded when people was communicating with each other while their enemy could not be aware of the exchanged information So, if breaking the code of encrypted systems for attackers was more difficult and complicated they would have been able to provide the better security for their consumers, it is cleared that such systems have high value The security of a cryptography system has dependent on its key It may be supposed the best way to manage the key is putting it on a safe place where it has protected; but this way is not reliable because an unexpected bad event like intentional or accidental sabotages can lead information to be inaccessible A simple approach to prevent this problem can be storing several copies of key (secret) in different places; however it will reduce the key security Sometime the problem may be easier than occurrence of an unexpected bad event We tend to define precisely which subset of shareholders is able to determine the key and which cannot do that has been considered as a set of subsets, in fact participance subsets in are supposed as shareholders subsets which must be able to calculate the key Also has been called access structure Subsets of that are not able to key recovery has been called forbidden sets In 1979, secret sharing scheme was offered by Shamir and Blakley separately The secret sharing scheme has employed for identified secret information sharing between individuals of a specified group, as none of these people with certain share that is allocated to them, be unable to access the confidential information and the order of access to secret can be possible only through using shares of known sets of people Thus, secret sharing has increased reliability of a cryptography system Karnin and et al in 1983 {3} had introduced perfect secret sharing scheme and then they offered a type of secret sharing scheme based on multiple matrix for better security Naor and Shamir {4, 5} during Eurocrypt conference 1994 had offered a new type of sharing scheme which has been called visual cryptographyin this kind of cryptography, the distributer shares out a secret image on n transparency sheets among n people which subsets of the people would be able to recover the secret image by putting their transparencies on each other Shareholders of a participance subset would be able to recover original image through only storing their shares without any information about cryptography and without cryptography computing; a subset of them cannot obtain any information of secret by putting their transparencies on each other and then they will not be permitted to observe the secret image Hu & Gao in 2008 introduced the secret image sharing scheme which was based on multiple matrix; in order to create a share on secret image and distribute it to each shareholder, shares of new secret should be computed and distributed in each time Threshold images have been produced in most of image sharing schemes like {6, 9, 10, 12, 13 and 14} which their control, identification and distribution were difficult The existing visual secret sharing scheme has limited the number of secret images All mentioned schemes were one time scheme By expanding the number of secret images, a subject on new scheme of secret sharing has been considered for hiding multiple secret images which each shareholder must receive and retain its share in a multiple secret sharing scheme In this scheme, many secrets could have been shared independently and without redistribution In order to recover a secret, a shareholder receives and presents an unreal share obtained from original share in place of its genuine share; recovering a secret has not threatened remaining tools 495

(recovery other secrets) which has not recovered yet The article is structured as follows: Definitions and concepts have been brought in part 2 which may be required for some of the secret sharing schemes The multiple secret sharing scheme of Hu Zengfeng & Gao has been explained in part 3; we have introduced our proposal in part 4; an analysis of the reformed proposed scheme in part 5 and finally, conclusion of the article can be seen in part 6 2 Multiple secret sharing scheme A multiple secret shadow sharing scheme is a special multiple secret sharing scheme which basis have been defined through shadow It seems that every access structure of threshold multiple secret sharing can receive a scheme that is similar to Shamir's scheme 11 Cumulative threshold of image sharing scheme in matrix multiplication In 2008 Hu Zengfeng and et al {9} had proposed a compressed image sharing scheme that was based on Karnin scheme In the plan, secret image has been divided into several uncommon square blocks instead of single pixel and each block is considered as a secret matrix This plan could reduce shadow image to secret image on the base of coefficient matrix features and it did not require to permutation in image sharing phase However, the process was belonging to one time secret sharing scheme because the portion of shareholder would be revealed after recovering secret image On the other hand, in order to share another secret image, new secret shares must be computed and distributed again among shareholders 12 Multiple secret images sharing scheme based on matrix multiplication Hu Zengfeng & et al {10} in reforming their proposed scheme at {9} had presented a plan in 2009 which was derived from multiple secret image sharing based on matrix multiplication; in this plan shareholders was storing artificial images rather than their original share at the time of recovering secret images So each shareholder with having only one portion of secret was able to share it with others in many of secret images Unlike other known secret sharing schemes, in this scheme the threshold images which were difficult to be identified and controlled were not produced Their scheme was a plan of efficient and practical multiple secret image sharing scheme The mentioned plan is a secure multiple secret sharing scheme (t, n) that has the following features: 1 It has ability to perform parallel recovery that means it can recover several secrets simultaneously; 2 The intermediate can compute number of distributed secrets dynamically; 3 A scheme which is used several times has been employed in several secret sharing courses without distributing new shares The mentioned scheme of multiple secret images sharing has been derived from a complex scheme, as concealed shares of shareholders would not be revealed at the time of recovering secret image The scheme will emerge only some general information rather than producing shadow images Advantages of this scheme have possessed following characteristics in comparison with other secret sharing schemes: Each shareholder with having only one portion of secret can share it with others in many of secret images On the other hand, each secret portion of any shareholder will not be revealed in recovery phase The present scheme has created shadow images in secret image sharing 496

phase that can protect it from problems like distribution images, control and identification In image recovery phase, shareholders cooperating with each others can recover secret image completely through general information and their concealed share 3 Describing Hou Zhengfeng & et al scheme a Initiative phase: In this phase has been considered as a one-way function of two variables Divider (D) has dealt out a number of secret images and a set of matrix on that has been made as a general determiner of shareholders and then divider has supposed - matrix on as general information which they consisted each dimension of matrix In this scheme, every of should have full rank and every of and of must be full rank; here the amount of would be Then, divider has chosen which as the concealed share of the corresponding shareholders b Secret image sharing phase: Size of in secret image is that divider has shared it as follows 1 in secret image has been shared in several uncommon blocks of with size of and generating image has been chosen which is corresponded with and has similar size of and then would be shared in several uncommon blocks of with size of, 2 Amount of has been calculated and values of in matrix would be changed such as pseudo shadow with size of that the value of and are placed in and 3 The equation of has been considered as following relations: (1-3-3) (2-3-3) Then value of would be computed ( is block of that the value of and are, ) 4 Value of would be computed that is j block of as general image with the amount of that is 5 Relationship between, general image and of generating image has been presented that c Recovering secret image phase: 1 By creating the image of, each shareholder can obtain and compute the value of through presenting image of its shadow and then it would be able to reflect them in a matrix with form of with and 2 With general information of, shareholders can calculate on the base of the following equation: B1, Uj Bn t(mod 251) (3-3-3) In this equation the value of counted as follows: would be (4-3-3) 497

The value of is 3 Shareholders cooperating with each other are able to transform to and calculate the value of then all would be combined to create secret image that d Sharing Other image: At the time of sharing or other image, divider must choose generator image different from and repeat the secret image sharing phase in subsection (b) Similarly shareholders cooperating with each other can acquire image through repeating the step of recovering secret image that was explained in subsection (c) 31 Our proposed scheme; Multiple secret images sharing scheme is derived from complicated plan In this paper we have improved the scheme of Hou Zhengfeng and Gao Hanjun [11] with a change in method of making shares of generating image It is cleared that the improved scheme has created better speed and security As it was mentioned the scheme has shown only some of the general information rather than creating threshold images and also concealed portions of shareholders would not be revealed at the time of recovering secret image a Initiative phase: First one-way function of two variables has been considered as a special type of hash function and then would be proposed (in terms of better speed and security in comparison with the presented function in [29] scheme) Divider D must share certain numbers of secret images Divider has made a set of n matrix in as a general determiner of shareholders then supposed - matrix in as general information; each dimension of matrix is Every of has full rank and every of and any of have possessed full rank that After that, the divider has chosen which would be as the concealed share of the corresponding shareholders b Secret image sharing phase: Size of in secret image is that divider D has shared it as follows 1 Secret image has been shared in several uncommon blocks of with size of and then corresponding generator image with similar size of has been computed through function of and in this function 2 By calculating the amount of and then transforming it to matrix, shareholder of pseudo shadow with size of would be created with and The equation of has been established as following relations: And the value of would be computed ( is block of that and ) The equation has been calculated that is j block of general image with the amount of that is Relationship between, general image and of generating image has been presented which c Recovering secret image phase: 498

1 By creating the image of, each shareholder can obtain and compute generating function of and then it would be able to put them in a matrix with form of with and the amount of that is If, the row round will be which 2 With general information of, shareholders can calculate and consider the following equation: B1, Uj Bn t(mod 251) The can be calculated as follows: And it may be possible to see input of as the following square matrix They are parameters which are given to shareholders With the value of that is 3 Shareholders cooperating with each other are able to divide to and calculate the value of then all would be stored to reveal secret image with d Sharing another image: At the time of sharing another image, the divider only requires to choose a generator image different from and repeat the secret image sharing phase in subsection (b) Similarly shareholders cooperating with each other can acquire image through repeating the step of recovering secret image that was explained in subsection (c) In row round function, matrix rows have been ordered into quarter round in parallel way with permutation of each row Amounts of,, and respectively have been put in the first row of the row round function Amounts,, and would be placed in the second row of the row round function orderly In the third row of the row round function, amounts of,, and have been arranged respectively and the amounts of,, and have been put orderly in the fourth row of the row round function 2 Column round function If, the column round will be which 311 Function of is operated as follows: 1 Row round function 499

And it may be possible to see input of as the following square matrix In column round function, matrix column have been ordered into quarter round in parallel way with permutation of each row Amounts of,, and respectively have been put in the first column of the column round function Amounts of,, and would be placed in the second column of the column round function orderly In the third column of the column round function, amounts of,, and have been arranged respectively and the amounts of,, and have been put orderly in the fourth column of the column round function If input function of is 16 the output of it will be 16 312 Defining the relations of proposed function: Cryptography function of is a long chain including three simple operations on 32 bit words: Sum of 32-bit: a b (mod32), a and b all of them are 32-bit 32-bit XOR: axor b, a and b all of them are 32-bit Rotation with 32-bit fixed distance: a b, 32-bit word and a, b bit have shifted towards left It should be mentioned B is fix Above operation can be created and implemented through any circuit The objective of this scheme is to increase speed along with a good security Figure (1-5): Function between x and y in core of function =qarterround We have performed MATLAB programming in scheme 1-5 through an example which reforming the executable scheme has been implemented with high speed and security 4 Security-efficiency analysis The mentioned scheme is multiple secret images sharing scheme Divider (D) needs only to choose new generator image for each secret image and repeat the steps of secret image sharing at the time of different secret images sharing We have employed the one-way hash function of two variables to improve the scheme and we observe that speed and security of the scheme have increased through using function in computation Shareholders require only to combine in pseudo shadow image rather than revealing their secret share (x) Thus, the secret share (x) of each shareholder can be used several times This scheme has been employed effectively in most cases Therefore, the scheme not only can be shared in secret image without creating shadow images and is able to prevent from other problems of multiple secret images sharing schemes; but also the time, memory complexity, consumption time of memory status and implementation plan of the scheme have been considered 500

5 Conclusion In this article east t shareholder has recovered multiple secret images and each shareholder keeps only one share In the scheme of Hou Zhengfeng and Gao Hanjun a number of information simultaneously has been created instead of shadow images which includes identification, distribution and control information Accordingly, the multiple secret images sharing scheme is an effective and practical scheme, whereas a new and effective generator function has been employed in our reformed scheme that has caused speed and security improvement Computation time is so complicated 7Reference [1] Shamir A How to share a secret[j] CommunACM, 1979 24(11):612-613 [2] Blakley G R Safeguarding cryptographic keys[c] Proc of AFIPS 1979 National Computer Conference, 1979:313-317 [3] EDKarnin,JWGreene,MEHellman On Sharing Secret Systems[J]IEEE Transactions on Information Theory,1983, 29(1):35-41 [4] M Naor, A Shamir Visual cryptography[c] Advances in Cryptology-Eurocrypt 94, 1994:1-12 [5] M Naor, A Shamir Visual cryptography II: Improving the contras via the cover base[c] Lecture Notes in Computer science, SpringerBerlin 1997(1189):197-202 [6] CC Thien, JC Lin Secret image sharing[j] Computer&Graphic,2002,26(5):765-770 [7] Daniel jbernstein "Salsa 20 specification"department of Mathematics, Statistics, and Computer Science (M/C 249) The University of Illinois at Chicago, Date: 20071225 [8] Li Bai A Reliable (k, n) Image Secret Sharing Scheme[C]Dependable, Autonomic and Secure Computing, 2nd IEEE International Symposium on Sept, 2006:31-36 [9] Li Bai A strong ramp secret sharing scheme using matrix projectionworld of Wireless[C], Mobile and Multimedia NetworksInternational Symposium on 26-29 62006, pages:5 pp [10] Hou Zhengfeng, Gao Hanjun A compressible threshold image sharing scheme based on matrix multiplication[j] Geomatics and Information Science of Wuhan University 2008, 33(10):1003-1006 [11] Hou Zhengfeng, Gao Hanjun Multi- secret image sharing based on matrix multiplication,ieee International conference on networks security(2009),vol1,pp184-187 [12] Ran-Zan Wang, Chin-Hui Su Secret image sharing with smaller shadow images [J] Pattern Recognition Letters 27 (2006) 551 555 [13] Chin-Chen Chang, Chia-Chen Lin, Chia- Hsuan Lin, Yi-Hui Chen A novel secret image sharing scheme in color images using small shadow images [J] Information Sciences 178 (2008) 2433 2447 [14] Shi Runhua, Zhong Hong, Huang Liusheng, Luo Yonglong A (t,n) Secret Sharing Scheme for Image Encryption [C] 2008 Congress on Image and Singal Processing 27-30 May 2008:3-6 [15] Chien HY, Jan JK, Tseng YM A practical (t,n) multi-secret sharing scheme[j] IEICE Transactions on Fundamentals 2000, E83- A(12):2762-2765 [16] PANG LJ,WANG YM A new (t,n) multisecret sharing scheme based on Shamir s secret sharing[j] Applied Mathematics andcomputation, 2005,167(2):840-848 [17] Yang CC,Chang TY,Hwang MS A (t,n) multi-secret sharing scheme[j] Applied Mathematics and Computation, 2004, 151(2):483-490 501