private-key encryption would not be possible, [BM84,GMR88,IL89,ILL89,Rom90]. Given the impact of one-way functions in cryptography and complexity, we be-.
A characterization of one-way functions based on time-bounded Komogorov complexity Lu´ıs Antunes
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Andr´e Souto
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Andreia Teixeira
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{lfa,andresouto,andreiasofia}@dcc.fc.up.pt Universidade do Porto Instituto de Telecomunica¸co ˜es Address: Rua do Campo Alegre, no 1021/1055 4169-007 Porto Portugal
The security of most cryptographic schemes is based implicitly on the security of the cryptographic primitives used, like one-way functions, i.e., functions that are “easy” to compute in polynomial time but are “hard” to invert in probabilistic polynomial time. In fact, the vast majority of the usual primitives implies the existence of one-way functions. The existence of these functions is a strong assumption as it is well known that it implies that P 6= NP, although it is a very important open question to know if this is also a sufficient condition. To emphasize the importance of the existence of one-way functions we observe that if they did not exist then pseudo-random generators, digital signatures, identification schemes and private-key encryption would not be possible, [BM84,GMR88,IL89,ILL89,Rom90]. Given the impact of one-way functions in cryptography and complexity, we believe that it is important to study these functions at the individual level in opposition to its average case behavior. In this work we give a first characterization of one-way functions based on time-bounded Kolmogorov complexity. We hope that this characterization may lead to a finer grained analysis to some cryptographic protocols, such as commitment schemes. Classically there are two types of one-way functions: strong and weak one-way functions. In the case of a strong one-way function, it is required that the inversion happens with low probability and in the weak version the non inversion must happen with non-negligible probability. An intersecting fact about these functions is that not every weak one-way function is a strong one-way function but their existence is equivalent (see [Gol01] for details). Formally: Definition 1 (Weak one-way function). A function f : {0, 1}∗ → {0, 1}∗ is a weak one-way function if it is computable in polynomial time, it is total, one-to-one, ? ?? ???
All the authors are partially supported by CSI 2 (PTDC/EIA- CCO/099951/2008) The author is also supported by the grant SFRH / BD / 28419 / 2006 from FCT The author is also supported by the grant SFRH / BD / 33234 / 2007 from FCT
honest, and there is a polynomial p such that for every probabilistic polynomial time algorithm G and all sufficiently large n’s, P r[G(f (x)) 6= x] >
1 . p(n)
Definition 2 (Strong one-way function). A function f : {0, 1}∗ → {0, 1}∗ is a strong one-way function if it is computable in polynomial time, it is total, one-toone, honest, and for every probabilistic polynomial time algorithm G, every positive polynomial p, and all sufficiently large n’s, P r[G(f (x)) = x]
