We present a polynomial time algorithm for breaking NTRU encryption schemes with multiple keys. Our algorithm takes advantage of the specific sampling regime used in NTRU encryption, which samples secret polynomials with a fixed number of coefficients of 1 and \(-1\). By constructing an equation system on the secret keys, we are able to recover the unique secret key when n multiple keys sharing a common denominator are given for an extension degree n. This result shows that NTRU encryption schemes with multiple keys can be solved in polynomial time in n.

中文翻译：

---

破解多密钥 NTRU 加密的多项式时间算法

我们提出了一种多项式时间算法，用于破解具有多个密钥的 NTRU 加密方案。我们的算法利用了 NTRU 加密中使用的特定采样机制，它使用固定数量的系数 1 和\(-1\) 对秘密多项式进行采样。通过在密钥上构建一个方程组，当给定扩展度n的n个共享公分母的多个密钥时，我们能够恢复唯一的密钥。该结果表明，具有多个密钥的 NTRU 加密方案可以在n 的多项式时间内求解。