# A failure in decryption process for bivariate polynomial reconstruction problem cryptosystem

## TL;DR

This paper identifies a decryption failure in a bivariate polynomial-based cryptosystem when errors exceed a certain threshold.

## Contribution

The paper introduces a new upper bound to prevent decryption failure in bivariate polynomial reconstruction cryptosystems.

## Key findings

- Decryption failure occurs when error weight exceeds the number of monomials in the secret polynomial.
- An upper bound is established to avoid decryption failure in the cryptosystem.

## Abstract

In 1999, the Polynomial Reconstruction Problem (PRP) was put forward as a new hard mathematics problem. A univariate PRP scheme by Augot and Finiasz was introduced at Eurocrypt in 2003, and this cryptosystem was fully cryptanalyzed in 2004. In 2013, a bivariate PRP cryptosystem was developed, which is a modified version of Augot and Finiasz's original work. This study describes a decryption failure that can occur in both cryptosystems. We demonstrate that when the error has a weight greater than the number of monomials in a secret polynomial, p, decryption failure can occur. The result of this study also determines the upper bound that should be applied to avoid decryption failure.