In this survey, we present several computational analogues of entropy and illustrate how they are useful for constructing cryptographic primitives. Specifically, we focus on constructing pseudorandom generators and statistically hiding commitments from arbitrary one-way functions, and demonstrate that:
- The security properties of these (and other) cryptographic primitives can be understood in terms of various computational analogues of entropy, and in particular how these computational measures of entropy can be very different from real, information-theoretic entropy.
- It can be shown that every one-way function directly exhibits some gaps between real entropy and the various computational entropies.
- Thus we can construct the desired cryptographic primitives by amplifying and manipulating the entropy gaps in a one-way function, through forms of repetition and hashing.
The constructions we present (which are from the past decade) are much simpler and more efficient than the original ones, and are based entirely on natural manipulations of new notions of computational entropy. The two constructions are "dual" to each other, whereby the construction of pseudorandom generators relies on a form of computational entropy ("pseudoentropy") being larger than the real entropy, while the construction of statistically hiding commitments relies on a form of computational entropy ("accessible entropy") being smaller than the real entropy. Beyond that difference, the two constructions share a common structure, using a very similar sequence of manipulations of real and computational entropy. As a warmup, we also "deconstruct" the classic construction of pseudorandom generators from one-way permutations using the modern language of computational entropy.
This survey is written in honor of Shafi Goldwasser and Silvio Micali.