Minimal Linear Codes with Few Weights and Their Secret Sharing

Year 2019, Volume: 8 Issue: 4, 77 - 87, 01.12.2019

Sıhem Mesnager Ahmet Sınak , Oguz Yayla

Abstract

Minimal linear codes with few weights have significant applications in secure two-party computation and secret sharing schemes. In this paper, we construct two-weight and three-weight minimal linear codes by using weakly regular plateaued functions in the well-known construction method based on the second generic construction. We also give punctured codes and subcodes for some constructed minimal codes. We finally obtain secret sharing schemes with high democracy from the dual codes of our minimal codes.

Keywords

Minimal linear code, weakly regular plateaued function, secret sharing scheme

References

• [1] A. Ashikhmin and A. Barg. Minimal vectors in linear codes. IEEE Transactions on Information Theory, 44(5):2010–2017, (1998).
• [2] G. R. Blakley et al. Safeguarding cryptographic keys. In Proceedings of the national computer conference, volume 48, pages 313–317, 1979.
• [3] A. Calderbank and J. Goethals. Three-weight codes and association schemes. Philips J. Res, 39(4-5):143–152, 1984.
• [4] R. Calderbank and W. Kantor. The geometry of two-weight codes. Bulletin of the London Mathematical Society, 18(2):97– 122, 1986.
• [5] C. Ding. Linear codes from some 2-designs. IEEE Transactions on information theory, 61(6):3265–3275, 2015.
• [6] C. Ding. A construction of binary linear codes from boolean functions. Discrete mathematics, 339(9):2288–2303, 2016.
• [7] K. Ding and C. Ding. A class of two-weight and threeweight codes and their applications in secret sharing. IEEE Transactions on Information Theory, 61(11):5835–5842, 2015.
• [8] O. Goldreich, S. Micali, and A. Wigderson. How to play any mental game. In Proceedings of the nineteenth annual ACM symposium on Theory of computing, pages 218–229. ACM, 1987.
• [9] J. L. Massey. Minimal codewords and secret sharing. In Proceedings of the 6th joint Swedish-Russian international workshop on information theory, pages 276–279, 1993.
• [10] R. J. McEliece and D. V. Sarwate. On sharing secrets and reedsolomon codes. Communications of the ACM, 24(9):583–584, 1981.
• [11] S. Mesnager, F. Ozbudak, and A. Sınak. A new class of three- ¨ weight linear codes from weakly regular plateaued functions. In Proceedings of the Tenth International Workshop on Coding and Cryptography (WCC) 2017.
• [12] S. Mesnager, F. Ozbudak, and A. Sınak. Linear codes from ¨ weakly regular plateaued functions and their secret sharing schemes. Designs, Codes and Cryptography, 87(2-3):463–480, 2019.
• [13] S. Mesnager, A. Sınak, and O. Yayla. Three-weight minimal linear codes and their applications. In Proceedings of the Second International Workshop on Cryptography and its Applications (2IWCA’19), pages 216-220, 19-20 June 2019, Oran, Algeria.
• [14] S. Mesnager and A. Sınak. Several classes of minimal linear codes with few weights from weakly regular plateaued functions. IEEE Transaction on Information Theory, DOI: 10.1109/TIT.2019.2956130, 2019.
• [15] J. Pieprzyk and X. Zhang. Ideal secret sharing schemes from mds codes. In Proc. 5th Int. Conf. Information Security and Cryptology (ICISC 2002), pages 269–279, 2002.
• [16] A. Renvall and C. Ding. The access structure of some secretsharing schemes. In Information Security and Privacy, pages 67–78. Springer, 1996.
• [17] B. Schoenmakers. A simple publicly verifiable secret sharing scheme and its application to electronic voting. In Annual International Cryptology Conference, pages 148–164. Springer, 1999.
• [18] A. Shamir. How to share a secret. Communications of the ACM, 22(11):612–613, 1979.
• [19] A. Sınak. Three-weight and four-weight minimal linear codes and their secret sharing schemes. Submitted to the international journal, 2019.
• [20] C. Tang, N. Li, Y. Qi, Z. Zhou, and T. Helleseth. Linear codes with two or three weights from weakly regular bent functions. IEEE Transactions on Information Theory, 62(3):1166–1176, 2016.
• [21] C. Tang, C. Xiang, and K. Feng. Linear codes with few weights from inhomogeneous quadratic functions. Designs, Codes and Cryptography, 83(3):691–714, 2017.
• [22] C.-N. Yang and J.-B. Lai. Protecting data privacy and security for cloud computing based on secret sharing. In 2013 International Symposium on Biometrics and Security Technologies, pages 259–266. IEEE, 2013.
• [23] Y. Zheng and X.-M. Zhang. Plateaued functions. In ICICS, volume 99, pages 284–300. Springer, 1999.
• [24] Z. Zhou, N. Li, C. Fan, and T. Helleseth. Linear codes with two or three weights from quadratic bent functions. Designs, Codes and Cryptography, 81(2):283–295, 2016.
• [25] G. Zyskind, O. Nathan, et al. Decentralizing privacy: Using blockchain to protect personal data. In 2015 IEEE Security and Privacy Workshops, pages 180–184. IEEE, 2015.