Private Polynomial Research Articles

A constant round quantum secure protocol for oblivious polynomial evaluation

Oblivious polynomial evaluation (OPE) is a secure two-party cryptographic protocol that enables a receiver to obliviously retrieve polynomial value on its private input for a private polynomial of a sender. This paper deals with the problem of OPE in the quantum domain. Herein, we propose the construction of the first quantum secure OPE, namely QuOPE. Our design is a constant round protocol that only requires quantum communication of one quantum superposition state that achieves all the stipulated security requirements of an OPE. In particular, privacy and security of the sender and receiver’s private data are ensured. In addition, we extend the idea of QuOPE to quantum rational polynomial evaluation (QuORPE). Furthermore, we explore the possibility of utilizing QuOPE as a building block in the automated medical diagnosis platform.

Private Polynomial Function Computation for Noncolluding Coded Databases

We consider the problem of private polynomial computation (PPC) from a distributed storage system (DSS). In such setting a user wishes to compute a multivariate polynomial of degree at most <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$g$ </tex-math></inline-formula> over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f$ </tex-math></inline-formula> variables (or messages) stored in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> noncolluding coded databases, i.e., databases storing data encoded with an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$[n,k]$ </tex-math></inline-formula> linear storage code, while revealing no information about the desired polynomial evaluation to the databases. For a DSS setup where data is stored using linear storage codes, we derive an outer bound on the PPC rate, which is defined as the ratio of the (minimum) desired amount of information and the total amount of downloaded information, and construct two novel PPC schemes. In the first scheme, we consider Reed-Solomon coded databases with Lagrange encoding, which leverages ideas from recently proposed star-product private information retrieval and Lagrange coded computation. The second scheme considers the special case of coded databases with systematic Lagrange encoding. Both schemes yield improved rates, while asymptotically, as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f\rightarrow \infty $ </tex-math></inline-formula> , the systematic scheme gives a significantly better computation retrieval rate compared to all known schemes up to some storage code rate that depends on the maximum degree of the candidate polynomials.

Private Secure Coded Computation

We present private secure coded computation that ensures both the master’s privacy and data security against the workers, where the master aims to compute a function of its private data and a specific data in a library exclusively owned by the external workers. For privacy, the master should conceal the index of the specific data in the library against the workers. For security, the master should encrypt its private data. As an achievable scheme, we propose private secure polynomial codes and compare the proposed scheme with private polynomial codes in private coded computation and the optimal scheme of robust private information retrieval (RPIR).

Cryptanalysis of NTRU where the private polynomial has one or more consecutive zero coefficients

Let h and (f, g) be the public and private keys of NTRU respectively. We present a new lattice to attack NTRU when the private polynomial g has one or more consecutive coefficients which are equal zeros, i.e., zero patterns. The target vector of the new lattice contains the private polynomial f, and the ratio between the size of the target vector and the expected size of a shortest non-zero vector in the new lattice is smaller than the ratio declared in the previous results. The zero patterns are not necessarily of the same or long lengths. Experimental results show that our attack succeeded to cryptanalyze NTRU faster than the previous attacks.