模部分幂理论的统一
Unification modulo a partial theory of exponentiation
摘要：
模幂运算是现代密码学中一种常见的数学运算。这一点，以及在基和指数级别上的模乘（到不同的模），在大量的密钥协商协议中起着重要的作用。在我们的早期工作中，我们给出了许多可判定性以及不可判定性的结果，涉及到模幂运算的各种性质。在这里，我们考虑一个部分子理论只集中在指数和乘法运算符。证明了两个主要结果。第一个结果是肯定的，即，上述理论的统一问题（其中没有额外的性质是假设的乘法运算符）是可判定的。第二个结果是否定的：如果我们假设这两个乘法算子属于两个不同的交换群，那么统一问题就变得不可判定。
英文摘要：
Modular exponentiation is a common mathematical operation in modern cryptography. This, along with modular multiplication at the base and exponent levels (to different moduli) plays an important role in a large number of key agreement protocols. In our earlier work, we gave many decidability as well as undecidability results for multiple equational theories, involving various properties of modular exponentiation. Here, we consider a partial subtheory focussing only on exponentiation and multiplication operators. Two main results are proved. The first result is positive, namely, that the unification problem for the above theory (in which no additional property is assumed of the multiplication operators) is decidable. The second result is negative: if we assume that the two multiplication operators belong to two different abelian groups, then the unification problem becomes undecidable.
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