International Workshop on Public Key Cryptography

PKC 2011: Public Key Cryptography – PKC 2011 pp 387-402

Round-Efficient Sub-linear Zero-Knowledge Arguments for Linear Algebra

  • Jae Hong Seo
Conference paper

DOI: 10.1007/978-3-642-19379-8_24

Volume 6571 of the book series Lecture Notes in Computer Science (LNCS)
Cite this paper as:
Seo J.H. (2011) Round-Efficient Sub-linear Zero-Knowledge Arguments for Linear Algebra. In: Catalano D., Fazio N., Gennaro R., Nicolosi A. (eds) Public Key Cryptography – PKC 2011. PKC 2011. Lecture Notes in Computer Science, vol 6571. Springer, Berlin, Heidelberg


The round complexity of interactive zero-knowledge arguments is an important measure along with communication and computational complexities. In the case of zero-knowledge arguments for linear algebraic relations over finite fields, Groth proposed (at CRYPTO 2009) an elegant methodology that achieves sub-linear communication overheads and low computational complexity. He obtained zero-knowledge arguments of sub-linear size for linear algebra using reductions from linear algebraic relations to equations of the form z = x*′y, where x, \(\mathbf{y}\in\mathbb{F}_p^n\) are committed vectors, \(z\in\mathbb{F}_p\) is a committed element, and \(*':\mathbb{F}_p^n\times\mathbb{F}_p^n\rightarrow\mathbb{F}_p\) is a bilinear map. These reductions impose additional rounds on zero-knowledge arguments of sub-linear size. We focus on minimizing such additional rounds, and we reduce the rounds of sub-linear zero-knowledge arguments for linear algebraic relations as compared with Groth’s zero-knowledge arguments for the same relations. To reduce round complexity, we propose a general transformation from a t-round zero-knowledge argument, satisfying mild conditions, to a (t − 2)-round zero-knowledge argument; this transformation is of independent interest.


Round-efficient zero-knowledge argumentssub-linear zero-knowledge argumentslinear algebra
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Copyright information

© International Association for Cryptologic Research 2011

Authors and Affiliations

  • Jae Hong Seo
    • 1
  1. 1.Department of Mathematical Sciences and ISaC-RIMSeoul National UniversitySeoulKorea