TR20-044 Authors: Marshall Ball, Elette Boyle, Akshay Degwekar, Apoorvaa Deshpande, Alon Rosen, Vinod Vaikuntanathan, Prashant Vasudevan

Publication: 8th April 2020 10:58

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Reductions between problems, the mainstay of theoretical computer science, efficiently map an instance of one problem to an instance of another in such a way that solving the latter allows solving the former. The subject of this work is ``lossy'' reductions, where the reduction loses some information about the input instance. We show that such reductions, when they exist, have interesting and powerful consequences for lifting hardness into ``useful'' hardness, namely cryptography.

Our first, conceptual, contribution is a definition of lossy reductions in the language of mutual information. Roughly speaking, our definition says that a reduction $C$ is $t$-lossy if, for any distribution $X$ over its inputs, the mutual information $I(X;C(X)) \leq t$. Our treatment generalizes a variety of seemingly related but distinct notions such as worst-case to average-case reductions, randomized encodings (Ishai and Kushilevitz, FOCS 2000), homomorphic computations (Gentry, STOC 2009), and instance compression (Harnik and Naor, FOCS 2006).

We then proceed to show several consequences of lossy reductions:

1. We say that a language $L$ has an $f$-reduction to a language $L'$ for a Boolean function $f$ if there is a (randomized) polynomial-time algorithm $C$ that takes an $m$-tuple of strings $X = (x_1,\ldots,x_m)$, with each $x_i\in\{0,1\}^n$, and outputs a string $z$ such that with high probability, L'(z) = f(L(x_1),L(x_2),...,L(x_m))

2. Suppose a language $L$ has an $f$-reduction $C$ to $L'$ that is $t$-lossy. Our first result is that one-way functions exist if $L$ is worst-case hard and one of the following conditions holds:

- $f$ is the OR function, $t \leq m/100$, and $L'$ is the same as $L$

- $f$ is the Majority function, and $t \leq m/100$

- $f$ is the OR function, $t \leq O(m\log{n})$, and the reduction has no error

This improves on the implications that follow from combining (Drucker, FOCS 2012) with (Ostrovsky and Wigderson, ISTCS 1993) that result in auxiliary-input one-way functions.

3. Our second result is about the stronger notion of $t$-compressing $f$-reductions -- reductions that only output $t$ bits. We show that if there is an average-case hard language $L$ that has a $t$-compressing Majority reduction to some language for $t=m/100$, then there exist collision-resistant hash functions.

This improves on the result of (Harnik and Naor, STOC 2006), whose starting point is a cryptographic primitive (namely, one-way functions) rather than average-case hardness, and whose assumption is a compressing OR-reduction of SAT (which is now known to be false unless the polynomial hierarchy collapses).

Along the way, we define a non-standard one-sided notion of average-case hardness, which is the notion of hardness used in the second result above, that may be of independent interest.