We propose BQTRU, a non-commutative NTRU-like cryptosystem over quaternion algebras. This cryptosystem uses bivariate polynomials as the underling ring. The multiplication operation in our cryptosystem can be performed with high speed using quaternions algebras over finite rings. As a consequence, the key generation and encryption process of our cryptosystem is faster than NTRU in comparable parameters. Typically using Strassen's method, the key generation and encryption process is approximately $16/7$ times faster than NTRU for an equivalent parameter set. Moreover, the BQTRU lattice has a hybrid structure that makes inefficient standard lattice attacks on the private key. This entails a higher computational complexity for attackers providing the opportunity of having smaller key sizes. Consequently, in this sense, BQTRU is more resistant than NTRU against known attacks at an equivalent parameter set. Moreover, message protection is feasible through larger polynomials and this allows us to obtain the same security level as other NTRU-like cryptosystems but using lower dimensions.
本文发展了（Z_2）^n-交换代数（n in N）上的线性代数理论，其中包括著名的超线性代数作为特例（n=1）.这种分次交换代数的例子是Clifford代数，特别是四元数代数H。遵循上同调的方法，我们引入类似的概念的痕迹和行列式。我们的构造在经典的交换情况下，通过可逆矩阵对上外幂的作用，简化为行列式的无坐标描述，在超交换情况下，它与Berezinian的著名上同调解释一致。
