By Raphael Pass, Abhi Shelat
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Extra resources for A Course in Cryptography
5. 1). 5 Collections of One-Way Functions In the last two sections, we have come to suitable definitions for strong and weak one-way functions. These two definitions are concise and elegant, and can nonetheless be used to construct generic schemes and protocols. However, the definitions are more suited for research in complexity-theoretic aspects of cryptography. Practical considerations motivate us to introduce a more flexible definition that combines the practicality of a weak OWF with the security of a strong OWF.
3. e. ” 4. Our experience indicates that “natural” functions that are not known to be computable in polynomial-time require much more time to compute, so the separation we propose seems well-founded. Note, our treatment of computation is an asymptotic one. In practice, concrete running time needs to be considered carefully, as do other hidden factors such as the size of the description of A. Thus, when porting theory to practice, one needs to set parameters carefully. 1 Some computationally “hard” problems Many commonly encountered functions are computable by efficient algorithms.
By these facts and unique factorization, we can write X= ∏ p νp ( X ) > 2 x p<2x 1 32 chapter 2. computational hardness where the product is over primes p less than 2x and νp ( X ) denotes the integral power of p in the factorization of X. Taking logs on both sides, we have ∑ νp ( X ) log p > x p<2x We now employ the following claim proven below. 4 log 2x log p > νp ( X ) Substituting this claim, we have ∑ p<2x log 2x log p log p = log 2x ∑ 1 >x p<2x Notice that the second sum is precisely π (2x ); thus π (2x ) > x 1 2x = · log 2x 2 log 2x which establishes the theorem for even values.