We suggest the usage of algebraic subsets instead of subgroups in public-key cryptography. In particular, we present the subset version of two protocols introduced by Shpilrain and Ushakov with some examples in ascending HNN-extensions of free-abelian groups and discuss their resistance to length and distance based attacks. We also introduce several new group theoretic problems arising from this work.

翻译：我们建议在公钥密码学中使用代数子集替代子群。特别地，我们提出了Shpilrain与Ushakov所引入的两个协议的子集版本，并以自由阿贝尔群的升序HNN扩张为例进行说明，同时讨论了它们对基于长度和距离攻击的抵抗能力。此外，本文还介绍了由此工作衍生出的若干新的群论问题。