Exploring Post-Quantum Cryptography and Constructive Reductions
Delve into the realm of post-quantum cryptography through an insightful journey of constructive reductions, rethinking cryptography assumptions, and the relationship between classical and post-quantum regimes. Discover the challenges, advancements, and goals in the quest for durable cryptographic algorithms that withstand quantum adversaries.
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Constructive Post-quantum Reductions Yael Kalai MSR/MIT Nir Bitansky TAU Zvika Brakerski Weizmann *Some slides inspired by Zvika
Post-quantum Cryptography classical protocol/scheme P2 P1 quantum adversary A A
Rethinking Cryptography Assumptions lattice instead of Factoring Adversary interface quantum vs classical signing oracle? Reductions OWFs still imply all symmetric key crypto?
(Security) Reductions A A breaks P P e.g. PRG R R breaks Q Q e.g. OWF/LWE Breaking P (e.g. PRG) is no easier than breaking Q (e.g. OWF)
Can classical reductions be lifted to post-quantum regime? Most classical reductions treat A as a black box .
Problematic in Interactive Setting A A ? P disturbed R R Q interactive P BB-reducible to LWE, but quantumly broken [BCMVV18]
Our Focus: Non-interactive Primitives/Assumptions A A A A A A A A reduction execution R R
Quantum Auxiliary Input A A A A A A A A ? ? ? ? ? ? ? R R Where does ? come from? - Expensive preprocessing - Intermediate state in a protocol - Friendly advice from aliens Auxiliary input state disturbed
Just Copy the State? A A A A A A A A ? ? ? ? R R Non-constructive Quantum cloning impossible!
Goal: Goal: Constructive Reductions A A R R breaks Q A A ? breaks P ? Win-Win: broken scheme explicit algorithmic advance Targeted also classically (uniform reductions) [Bellare, Rogaway] Goal II: Goal II: Durability, new algorithm should work forever.
Results Lifting large class of classical reductions Any R from breaking Q to breaking P such that: - R is black box - R is non-adaptive - P is a decision assumption (e.g. PRG) or has few solutions (e.g Injective OWF) Resulting post-quantum reduction is constructive and durable. Intermediate models of stateful adversaries, also of interest classically Negative result Restriction on P is somewhat inherent.
Bridge Between Stateful and Stateless Adversaries A A A A A A A A ? ? ? ? ? ? One-shot ? Gap A A A A A A A A Stateless classical reduction applicable
Bridge 1: One-Shot to Persistent A A A A A A A A ? ? ? ? ? ? One-shot ? quantum rewinding [Chiesa-Ma-Spooner-Zhandry] Gap 1 restriction to decision etc. A A A A A A A A ? ? ? ? Persistent keeps solving, without losing steam
Isnt Persistent Enough? A A A A A A A A ? ? ? ? Persistent keeps solving, without losing steam Solvable set drifts reduction queries may be correlated (e.g., Goldreich-Levin)
Bridge 2: Persistent to Memoryless A A Persistent A A A A A A ? ? ? ? simulation argument restriction to non-adaptive Gap 2 main tech contribution A A A A A A A A ? ? ? ? Memoryless (& persistent) keeps clock, strategy fixed ahead
Simulating Memoryless Behavior Idea: adversary state poly-bounded, limited memory of past queries To make ?-th query ??: plant real ??in random location A A A A ? ??? ??? ???, , ??? ??marginal of ?-th query A A ?? A A ? ? ? ??? #state-qubits ?-close to execution with predetermined oracles for ? ?2 (quantum mutual information argument)
Bridge 3: Memoryless to Stateless A A Memoryless A A A A A A ? ? ? ? keeps clock, strategy fixed ahead simulation argument restriction to non-adaptive Gap 3 (shuffle queries ) A A A A A A A A Stateless classical reduction applicable
Bridge Between Stateful and Stateless Adversaries A A A A A A A A ? ? ? ? ? ? One-shot ? Persistent main tech contribution Gap Memoryless also in cosmic [CFP22] A A A A A A A A Stateless classical reduction applicable
Food for Thought What about adaptive reductions? (PRGs from OWFs [HILL]) Thanks!
A Counterexample for Search Assumptions Non-interactive problems P,Q with classical reduction, but no constructive post-quantum reduction P: Given vk for digital signature scheme, and a random message m, output s which is a valid signature for m. Q: Given vk for digital signature scheme, and random messages (m1, m2), output (s1, s2) which are valid signature for (m1, m2). Classically: P-Solver Q-Solver Quantumly: tokenized signature schemes [BS18,CLLZ21] allow to generate a quantum state that can be used to generate exactly one valid signature.
