Abstract In the ElGamal signature and encryption schemes, an element x of the underlying group for a prime p is also considered as an exponent, for example in gx, where g is a generator of G. This ElGamal map is poorly understood, and one may wonder whether it has some randomness properties. This work presents two pieces of evidence for randomness. Firstly, experiments with small primes suggest that the map behaves like a uniformly random permutation with respect to two properties that we consider. Secondly, the theory of Sidon sets shows that the graph of this map is equidistributed in a suitable sense. It remains an open question to prove more randomness properties, for example, that the ElGamal map is pseudorandom.

中文翻译：

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ElGamal 函数的 Sidon 集和统计

摘要 在 ElGamal 签名和加密方案中，素数 p 的底层群的元素 x 也被视为指数，例如在 gx 中，其中 g 是 G 的生成器。想知道它是否具有一些随机性。这项工作为随机性提供了两个证据。首先，使用小素数的实验表明，对于我们考虑的两个属性，映射的行为类似于均匀随机排列。其次，西顿集理论表明，该图的图在适当意义上是等分布的。证明更多的随机性属性仍然是一个悬而未决的问题，例如 ElGamal 映射是伪随机的。