On the Two-sided Permutation Inversion Problem

Gorjan Alagic; Chen Bai; Alexander Poremba; Kaiyan Shi

doi:10.62056/a0qj89n4e

On the Two-sided Permutation Inversion Problem

Authors

Gorjan Alagic, Chen Bai, Alexander Poremba, Kaiyan Shi

Gorjan Alagic

QuICS, University of Maryland, USA
National Institute of Standards and Technology, USA
galagic at umd dot edu

Chen Bai

QuICS, University of Maryland, USA
Dept. of Electrical and Computer Engineering, University of Maryland, USA
cbai1 at umd dot edu

Alexander Poremba

Computing and Mathematical Sciences, California Institute of Technology, USA
CSAIL and Department of Mathematics, Massachusetts Institute of Technology, USA
poremba at mit dot edu

Kaiyan Shi

QuICS, University of Maryland, USA
Dept. of Computer Science, University of Maryland, USA
kshi12 at umd dot edu

Abstract

In the permutation inversion problem, the task is to find the preimage of some challenge value, given oracle access to the permutation. This fundamental problem in query complexity appears in many contexts, particularly cryptography. In this work, we examine the setting in which the oracle allows for quantum queries to both the forward and the inverse direction of the permutation—except that the challenge value cannot be submitted to the latter. Within that setting, we consider three options for the inversion algorithm: whether it can get quantum advice about the permutation, whether the query algorithm can restrict the distribution with which the challenge input is sampled, and whether it must produce the entire preimage (search) or only the first bit (decision). We prove several theorems connecting the hardness of the resulting variations of the permutation inversion problem and establish lower bounds for them. Our results show that, perhaps surprisingly, the permutation inversion problem does not become significantly easier when the adversary is granted oracle access to the inverse—provided it cannot query the challenge itself.

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DOI: https://doi.org/10.62056/a0qj89n4e

History

Submitted: 2024-01-09
Accepted: 2024-03-05
Published: 2024-04-09

How to cite

Gorjan Alagic, Chen Bai, Alexander Poremba, and Kaiyan Shi, On the Two-sided Permutation Inversion Problem. IACR Communications in Cryptology, vol. 1, no. 1, Apr 09, 2024, doi: 10.62056/a0qj89n4e.

Citations

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This work is licensed under a Creative Commons Attribution (CC BY) license.

@article{CiC-1-1-27,
    author = "Alagic, Gorjan and Bai, Chen and Poremba, Alexander and Shi, Kaiyan",
    journal = "{IACR} {C}ommunications in {C}ryptology",
    publisher = "{I}nternational {A}ssociation for {C}ryptologic {R}esearch",
    title = "On the Two-sided Permutation Inversion Problem",
    volume = "1",
    number = "1",
    date = "2024-04-09",
    year = "2024",
    issn = "3006-5496",
    doi = "10.62056/a0qj89n4e"
}

TY - JOUR
AU - Alagic, Gorjan
AU - Bai, Chen
AU - Poremba, Alexander
AU - Shi, Kaiyan
PY  - 2024
TI  - On the Two-sided Permutation Inversion Problem
JF  - IACR Communications in Cryptology
JA  - CIC
VL  - 1
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DO  - 10.62056/a0qj89n4e
UR  - https://doi.org/10.62056/a0qj89n4e
AB  - <p>  In the permutation inversion problem, the task is to find the preimage of some challenge value, given oracle access to the permutation. This fundamental problem in query complexity appears in many contexts, particularly cryptography. In this work, we examine the setting in which the oracle allows for quantum queries to both the forward and the inverse direction of the permutation—except that the challenge value cannot be submitted to the latter. Within that setting, we consider three options for the inversion algorithm: whether it can get quantum advice about the permutation, whether the query algorithm can restrict the distribution with which the challenge input is sampled, and whether it must produce the entire preimage (search) or only the first bit (decision). We prove several theorems connecting the hardness of the resulting variations of the permutation inversion problem and establish lower bounds for them. Our results show that, perhaps surprisingly, the permutation inversion problem does not become significantly easier when the adversary is granted oracle access to the inverse—provided it cannot query the challenge itself. </p>
ER  -

Gorjan Alagic, Chen Bai, Alexander Poremba, and Kaiyan Shi, On the Two-sided Permutation Inversion Problem. IACR Communications in Cryptology, vol. 1, no. 1, Apr 09, 2024, doi: 10.62056/a0qj89n4e.