Geodesic
Infinitely frequently one-sided function based on an assumption on complexity in the mean
Algebra i analiz, Tome 21 (2009) no. 3, pp. 130-144.

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We assume the existence of a function $f$ that is computable in polynomial time but cannot be inverted by a randomized average-case polynomial algorithm. The cryptographic setting is, however, different: even for a weak one-way function, a successful adversary should fail on a polynomial fraction of inputs. Nevertheless, we show how to construct an infinitely-often one-way function based on $f$. 
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E. A. Hirsch; D. M. Itsykson. Infinitely frequently one-sided function based on an assumption on complexity in the mean. Algebra i analiz, Tome 21 (2009) no. 3, pp. 130-144. http://geodesic.mathdoc.fr/item/AA_2009_21_3_a4/

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