We prove that the Ring Learning With Errors (RLWE) and the Polynomial Learning With Errors (PLWE) problems over the cyclotomic field \({\mathbb {Q}}(\zeta _n)\) are not equivalent. Precisely, we show that reducing one problem to the other increases the noise by a factor that is more than polynomial in n. We do so by providing a lower bound, holding for infinitely many positive integers n, for the condition number of the Vandermonde matrix of the nth cyclotomic polynomial.

中文翻译：

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分圆场上的 RLWE 和 PLWE 不等价

我们证明了在分圆域\({\mathbb {Q}}(\zeta _n)\)上的带误差环学习 (RLWE) 和带误差的多项式学习 (PLWE) 问题是不等价的。准确地说，我们表明将一个问题减少到另一个问题会使噪声增加的因子大于n中的多项式。我们通过为第n个分圆多项式的 Vandermonde 矩阵的条件数提供一个下界来做到这一点，该下界对于无限多个正整数n保持不变。