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**31 - 33**of**33**### Bitwise Quantum Min-Entropy Sampling and New Lower Bounds for Random Access Codes

, 2011

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### Average-case Complexity, and Error Correcting Codes. Direct Product Theorems are more formal statements with

, 2009

"... the following general intuition: “if there is a problem which is hard to solve on the average, then solving multiple instances of the problem becomes even harder.” Such theorems are useful in the following settings: (i) Cryptography: Much of Cryptography is based on existence of problems which are h ..."

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the following general intuition: “if there is a problem which is hard to solve on the average, then solving multiple instances of the problem becomes even harder.” Such theorems are useful in the following settings: (i) Cryptography: Much of Cryptography is based on existence of problems which are hard to solve on average. Direct Product Theorems provide a consistent way to amplify security properties of certain cryptographic protocols. For instance, [IJK07] uses such theorems to show amplification of the gap between the success of a human user and a computer in solving a CAPTCHA2 test in order to distinguish between the two. (ii) Derandomization: A series of results (e.g. [NW94, IW97]) show a very interesting Hardness-vs-Randomness tradeoff. These results show the following sequence of implications: if there is a function which is hard in the worst-case, then there exists a function which is mildly hard-on-average. A Direct Product construction is then used to amplify the average-case hardness of such functions. The harder function is then used to construct a pseudorandom generator. This generator, which can extend a random seed by an exponential amount, is finally used to derandomize probabilistic computation. In short, a non-trivial circuit lower bound implies that randomness does not help as long as efficient computation is concerned.