Average Case Complete Problems

Levin, Leonid A.

doi:10.1137/0215020

Cited by 337 publications

(176 citation statements)

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“…It may be instructive to relate the notion of weakly-verifiable computational puzzles to the notion of average-case hardness due to Levin [9]. Recall that according to Levin, a distributional problem is a pair (P, D), where P is a (search or decision) problem and D is a distribution on the instances of P. Hence, weaklyverifiable puzzles are a special case of distributional search problems.…”

Section: Discussionmentioning

confidence: 99%

Hardness Amplification of Weakly Verifiable Puzzles

Canetti¹,

Halevi²,

Steiner³

2005

Theory of Cryptography

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Abstract. Is it harder to solve many puzzles than it is to solve just one? This question has different answers, depending on how you define puzzles. For the case of inverting one-way functions it was shown by Yao that solving many independent instances simultaneously is indeed harder than solving a single instance (cf. the transformation from weak to strong one-way functions). The known proofs of that result, however, use in an essential way the fact that for one-way functions, verifying candidate solutions to a given puzzle is easy. We extend this result to the case where solutions are efficiently verifiable only by the party that generated the puzzle. We call such puzzles weakly verifiable. That is, for weakly verifiable puzzles we show that if no efficient algorithm can solve a single puzzle with probability more than ε, then no efficient algorithm can solve n independent puzzles simultaneously with probability more than ε n . We also demonstrate that when the puzzles are not even weakly verifiable, solving many puzzles may be no harder than solving a single one. Hardness amplification of weakly verifiable puzzles turns out to be closely related to the reduction of soundness error under parallel repetition in computationally sound arguments. Indeed, the proof of Bellare, Impagliazzo and Naor that parallel repetition reduces soundness error in threeround argument systems implies a result similar to our first result, albeit with considerably worse parameters. Also, our second result is an adaptation of their proof that parallel repetition of four-round systems may not reduce the soundness error.

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Section: Discussionmentioning

confidence: 99%

Hardness Amplification of Weakly Verifiable Puzzles

Canetti¹,

Halevi²,

Steiner³

2005

Theory of Cryptography

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“…We consider only polynomial-time samplable distributions, i.e., distributions that can be generated in polynomial time from a uniform distribution. The first definition of an average-case polynomial algorithm is due to Levin [6]. According to Levin, an algorithm runs in polynomial time on the average if there exists a positive number such that the expectation of T (x) over all inputs x of length n is O(n), where T (x) is the running time of the algorithm on the input x.…”

Section: §1 Introductionmentioning

confidence: 99%

“…(1) A successful cryptographic adversary may err on a polynomial fraction of inputs [3, Definition 2.2.2], while an average-case polynomial-time algorithm cannot spend exponential time on a polynomial fraction of inputs [2,6]. (2) A (polynomial-time samplable) probability distribution in the cryptographic setting is taken over the inputs of a function, while in the average-case setting, it is usually taken over the outputs (i.e., over the instances of the problem of inverting the function); see, e.g., [4,7].…”

Section: §1 Introductionmentioning

confidence: 99%

An infinitely-often one-way function based on an average-case assumption

Hirsch

Itsykson

2010

St. Petersburg Math. J.

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Abstract. 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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“…However, following Gurevich and Levin [7,9] discussion for syndrome decoding, we believe that both these problems are difficult on average.…”

Section: Security Reductionmentioning

confidence: 99%

A Family of Fast Syndrome Based Cryptographic Hash Functions

Augot

Finiasz

Sendrier

2005

Progress in Cryptology – Mycrypt 2005

110

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Abstract. Recently, some collisions have been exposed for a variety of cryptographic hash functions [20,21] including some of the most widely used today. Many other hash functions using similar constructions can however still be considered secure. Nevertheless, this has drawn attention on the need for new hash function designs. In this article is presented a family of secure hash functions, whose security is directly related to the syndrome decoding problem from the theory of error-correcting codes. Taking into account the analysis by Coron and Joux [4] based on Wagner's generalized birthday algorithm [19] we study the asymptotical security of our functions. We demonstrate that this attack is always exponential in terms of the length of the hash value. We also study the work-factor of this attack, along with other attacks from coding theory, for non asymptotic range, i.e. for practical values. Accordingly, we propose a few sets of parameters giving a good security and either a faster hashing or a shorter description for the function.

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Average Case Complete Problems

Cited by 337 publications

References 1 publication

Hardness Amplification of Weakly Verifiable Puzzles

Hardness Amplification of Weakly Verifiable Puzzles

An infinitely-often one-way function based on an average-case assumption

A Family of Fast Syndrome Based Cryptographic Hash Functions

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