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This paper explores some attacks that someone with a Quantum Computer may be able to perform against NTRUEncrypt, and in particular NTRUEncrypt as implemented by the publicly available library from Security Innovation. We show two ways where the attacker is able to recover plaintext from the ciphertext and public key with less than expected effort.  \section{Introduction}  NTRUEncrypt\cite{Whyte_2005} NTRUEncrypt\cite{Hoffstein_1998}  is a public key encryption system designed by Jeffrey Hoffstein, Jill Pipher and Joseph Silverman. It has several attractive features, one of which is that it is immune to attacks by Shor's algorithm (as it does not rely on a factorization or discrete log hard problem). Hence, it looks to be a logical component as a part of a Quantum-Resistant cryptosystem. NTRU does appear to be immune to Shor's algorithm (which allows the attacker to quickly factor numbers and compute discrete logarithms). However, a Quantum Computer also allows an attacker to run Grover's algorithm\cite{Grover_1996}, which is able to find a $n$ bit solution to a problem in $2^{n/2}$ time. The question we would like to look at is 'how can Grover's algorithm be used to advantage in attacking NTRU?'