Randomized algorithms and protocols assume the availability of a perfect source of randomness. In real life, however, perfect randomness is rare and is almost never guaranteed. The gap between these two facts motivated much of the work on randomness and derandomization in theoretical computer science. In this work, we define a new type of randomized algorithms (and protocols), that we call doubly-randomized algorithms (protocols). Such algorithms have access to two separate (read-once) random strings. The first string is trusted to be perfectly random, but its length is bounded by some parameter k = k ( n ) (where n is the length of the input). We think of k as relatively small, say sub-linear or poly-logarithmic in n . The second string is of unbounded length and is assumed to be random, but its randomness is not trusted. The output of the algorithm is either an output in the set of possible outputs of the problem, or a special symbol, interpreted as do not know and denoted by ⊥ . On every input for the algorithm, the output of the algorithm must satisfy the following two requirements: We discuss relations of this new definition to several previously studied notions in randomness and derandomization. For example, when considering polynomial-time algorithms, if k is we get the complexity class ZPP , while if k we get the complexity class BPP , and a general k , the viewed an interactive with a probabilistic polynomial-time and a probabilistic polynomial-time verifier, random number of trusted random bits. We also give query complexity separations that show cases where the untrusted random string is provenly helpful. Specifically, we show that there are promise problems that can be solved by doubly-randomized protocols with only one query and just a logarithmic number of trusted random bits, whereas any randomized protocol requires either a linear number of random bits or an exponential number of queries, and any zero-error randomized protocol requires a polynomial number of queries.
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