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We introduce the notion of pseudo-mixing time of a graph, defined as the number of steps in a random walk that suffices for generating a vertex that looks random to any polynomial-time observer. Here, in addition to the tested vertex, the observer is also provided with oracle access to the incidence function of the graph.

Assuming the existence of one-way functions, we show that the pseudo-mixing time of a graph can be much smaller than its mixing time. Specifically, we present bounded-degree N-vertex Cayley graphs that have pseudo-mixing time t for any . Furthermore, the vertices of these graphs can be represented by string of length , and the incidence function of these graphs can be computed by Boolean circuits of size .