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Revision #1 to TR20-089 | 16th October 2020 19:34

#### Lower Bounds on the Time/Memory Tradeoff of Function Inversion

Revision #1
Authors: Dror Chawin, Iftach Haitner, Noam Mazor
Accepted on: 16th October 2020 19:34
Keywords:

Abstract:

We study time/memory tradeoffs of function inversion: an algorithm, i.e., an inverter, equipped with an $s$-bit advice on a randomly chosen function $f\colon [n] \mapsto [n]$ and using $q$ oracle queries to $f$, tries to invert a randomly chosen output $y$ of $f$, i.e., to find $x\in f^{-1}(y)$. Much progress was done regarding adaptive function inversion - the inverter is allowed to make adaptive oracle queries. Hellman [IEEE transactions on Information Theory '80] presented an adaptive inverter that inverts with high probability a random $f$. Fiat and Naor [SICOMP '00] proved that for any $s,q$ with $s^3 q = n^3$ (ignoring low-order terms), an $s$-advice, $q$-query variant of Hellman's algorithm inverts a constant fraction of the image points of any function. Yao [STOC '90] proved a lower bound of $sq\ge n$ for this problem. Closing the gap between the above lower and upper bounds is a long-standing open question.

Very little is known for the non-adaptive variant of the question - the inverter chooses its queries in advance. The only known upper bounds, i.e., inverters, are the trivial ones (with $s+q= n$), and the only lower bound is the above bound of Yao. In a recent work, Corrigan-Gibbs and Kogan [TCC '19] partially justified the difficulty of finding lower bounds on non-adaptive inverters, showing that a lower bound on the time/memory tradeoff of non-adaptive inverters implies a lower bound on low-depth Boolean circuits. Bounds that, for a strong enough choice of parameters, are notoriously hard to prove.

We make progress on the above intriguing question, both for the adaptive and the non-adaptive case, proving the following lower bounds on restricted families of inverters:

- Linear-advice (adaptive inverter): If the advice string is a linear function of $f$ (e.g., $A\times f$, for some matrix $A$, viewing $f$ as a vector in $[n]^n$), then $s+q \in \Omega(n)$. The bound generalizes to the case where the advice string of $f_1 + f_2$, i.e., the coordinate-wise addition of the truth tables of $f_1$ and $f_2$, can be computed from the description of $f_1$ and $f_2$ by a low communication protocol.

- Affine non-adaptive decoders: If the non-adaptive inverter has an affine decoder - it outputs a linear function, determined by the advice string and the element to invert, of the query answers - then $s \in \Omega(n)$ (regardless of $q$).

- Affine non-adaptive decision trees: If the non-adaptive inversion algorithm is a $d$-depth affine decision tree - it outputs the evaluation of a decision tree whose nodes compute a linear function of the answers to the queries - and $q 0$, then $s\in \Omega(n/d \log n)$.

Changes to previous version:

Minor corrections, generalized linear advice theorem, responded to reviewer feedback.

### Paper:

TR20-089 | 8th June 2020 19:14

#### Lower Bounds on the Time/Memory Tradeoff of Function Inversion

TR20-089
Authors: Dror Chawin, Iftach Haitner, Noam Mazor
Publication: 9th June 2020 16:19
Keywords:

Abstract:

We study time/memory tradeoffs of function inversion: an algorithm, i.e., an inverter, equipped with an $s$-bit advice for a randomly chosen function $f\colon [n] \mapsto [n]$ and using $q$ oracle queries to $f$, tries to invert a randomly chosen output $y$ of $f$ (i.e., to find $x$ such that $f(x)=y$). Much progress was done regarding adaptive function inversion - the inverter is allowed to make adaptive oracle queries. Hellman [IEEE transactions on Information Theory '80] presented an adaptive inverter that inverts with high probability a random $f$. Fiat and Naor [SICOMP '00] proved that for any $s,q$ with $s^2 q = n^2$ (ignoring low-order terms), an $s$-advice, $q$-query variant of Hellman's algorithm inverts a constant fraction of the image points of any function. Yao [STOC '90] proved a lower bound of $sq\ge n$ for this problem. Closing the gap between the above lower and upper bounds is a long-standing open question.

Very little is known for the non-adaptive variant of the question - the inverter chooses its queries in advance. The only known upper bounds, i.e., inverters, are the trivial ones (with $s+q= n$), and the only lower bound is the above bound of Yao. In a recent work, Corrigan-Gibbs and Kogan [TCC '19] partially justified the difficulty of finding lower bounds on non-adaptive inverters, showing that a lower bound on the time/memory tradeoff of non-adaptive inverters implies a lower bound on low-depth Boolean circuits. Bounds that for a strong enough choice of parameters, are notoriously hard to prove.

We make progress on the above intriguing question, both for the adaptive and the non-adaptive case, proving the following lower bounds on restricted families of inverters:

- Linear-advice (adaptive inverter): If the advice string is a linear function of $f$ (e.g., $A\times f$, viewing $f$ as a vector in $[n]^n$), then $s+q \in \Omega(n)$.

- Affine non-adaptive decoders: If the non-adaptive inverter has an affine decoder - it outputs a linear function, determined by the advice string and the element to invert, of the query answers - then $s \in \Omega(n)$ (regardless of $q$).

- Affine non-adaptive decision trees: If the non-adaptive inversion algorithm is a $d$-depth affine decision tree - it outputs the evaluation of a decision tree whose nodes compute a linear function of the answers to the queries - and $q \le cn$ for some universal $c>0$, then $s\in \Omega(n/d \log n)$.

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