A Zero-Knowledge Identification Scheme Based on Matrix Power Problem

This paper shows that in some case the matrix power function f(x)=CxDx is injective, where C and D are the the companion matrices of primitive polynomials. And a new zero-knowledge identification scheme based on the matrix power problem is proposed. It is perfect zero-knowledge for honest verifiers. Due to its simplicity, low-memory and low-computation costs, the proposed scheme is suitable for using in computationally limited devices for identification such as smart cards.

[1]  Vladimir Shpilrain.  Cryptanalysis of Stickel's Key Exchange Scheme , 2008, CSR.

[2]  Ciaran Mullan.  Cryptanalysing variants of Stickel's key agreement scheme , 2011, J. Math. Cryptol..

[3]  Jiang Zhan-hua.  Design and implement on the HFEM public key scheme , 2011 .

[4]  Keisuke Tanaka,et al.  Concurrently Secure Identification Schemes Based on the Worst-Case Hardness of Lattice Problems , 2008, ASIACRYPT.

[5]  Rudolf Lide,et al.  Finite fields , 1983 .

[7]  Huawei Huang,et al.  Companion Matrix and Recognition of Primitive Polynomial , 2014 .

[8]  K. Conrad,et al.  Finite Fields , 2018, Series and Products in the Development of Mathematics.

[9]  Richard Lindner,et al.  Improved Zero-Knowledge Identification with Lattices , 2012 .

[10]  Zhao Hong-Wei,et al.  Construct Public Key Encryption Scheme Using Ergodic Matrices over GF(2) , 2007, TAMC.

[11]  M. Singh.  Public key cryptography with matrices , 2004, Proceedings from the Fifth Annual IEEE SMC Information Assurance Workshop, 2004..

[12]  Amit Sahai,et al.  Honest-verifier statistical zero-knowledge equals general statistical zero-knowledge , 1998, STOC '98.

[13]  Zhao Hong-wei.  Public Key Encryption Scheme Based on the Ergodic Matrices , 2010 .

[14]  Eberhard Stickel,et al.  A New Method for Exchanging Secret Keys , 2005, Third International Conference on Information Technology and Applications (ICITA'05).

[15]  Zhao Shi.  On the Properties of the Ergodic Matrix over Finite Field , 2012 .

[16]  Xiaoyi Zhou,et al.  Ergodic Matrix and Hybrid-key Based Image Cryptosystem , 2011 .

[17]  José-Francisco Vicent,et al.  A matricial public key cryptosystem with digital signature , 2008 .

[18]  Ilene H. Morgan,et al.  Primitive normal polynomials over finite fields , 1994 .