Abstract
Goldreich (ECCC 2000) suggested a simple construction of a candidate one-way function f: {0,1}n → {0,1}m where each bit of output is a fixed predicate P of a constant number d of (random) input bits. We investigate the security of this construction in the regime m = Dn, where D(d) is a sufficiently large constant. We prove that for any predicate P that correlates with either one or two of its variables, f can be inverted with high probability.
We also prove an amplification claim regarding Goldreich’s construction. Suppose we are given an assignment x′ ∈ {0,1}n that has correlation ε > 0 with the hidden assignment x ∈ {0,1}n. Then, given access to x′, it is possible to invert f on x with high probability, provided D = D(d, ε) is sufficiently large.
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Bogdanov, A., Qiao, Y. (2009). On the Security of Goldreich’s One-Way Function. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2009 2009. Lecture Notes in Computer Science, vol 5687. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03685-9_30
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DOI: https://doi.org/10.1007/978-3-642-03685-9_30
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