At Crypto 2011, some of us had proposed a family of cryptographic protocols for key establishment capable of protecting quantum and classical legitimate parties unconditionally against a quantum eavesdropper in the query complexity model. Unfortunately, our security proofs were unsatisfactory from a cryptographically meaningful perspective because they were sound only in a worst-case scenario. Here, we extend our results and prove that for any \eps > 0, there is a classical protocol that allows the legitimate parties to establish a common key after O(N) expected queries to a random oracle, yet any quantum eavesdropper will have a vanishing probability of learning their key after O(N^(1.5-\eps)) queries to the same oracle. The vanishing probability applies to a typical run of the protocol. If we allow the legitimate parties to use a quantum computer as well, their advantage over the quantum eavesdropper becomes arbitrarily close to the quadratic advantage that classical legitimate parties enjoyed over classical eavesdroppers in the seminal 1974 work of Ralph Merkle. Along the way, we develop new tools to give lower bounds on the number of quantum queries required to distinguish two probability distributions. This method in itself could have multiple applications in cryptography. We use it here to study average-case quantum query complexity, for which we develop a new composition theorem of independent interest.
At Crypto 2011, some of us had proposed a family of cryptographic protocols for key establishment capable of protecting quantum and classical legitimate parties unconditionally against a quantum eavesdropper in the query complexity model. Unfortunately, our security proofs were unsatisfactory from a cryptographically meaningful perspective because they were sound only in a worst-case scenario. Here, we extend our results and prove that for any \eps > 0, there is a classical protocol that allows the legitimate parties to establish a common key after O(N) expected queries to a random oracle, yet any quantum eavesdropper will have a vanishing probability of learning their key after O(N^(1.5-\eps)) queries to the same oracle. The vanishing probability applies to a typical run of the protocol. If we allow the legitimate parties to use a quantum computer as well, their advantage over the quantum eavesdropper becomes arbitrarily close to the quadratic advantage that classical legitimate parties enjoyed over classical eavesdroppers in the seminal 1974 work of Ralph Merkle. Along the way, we develop new tools to give lower bounds on the number of quantum queries required to distinguish two probability distributions. This method in itself could have multiple applications in cryptography. We use it here to study average-case quantum query complexity, for which we develop a new composition theorem of independent interest.
In 1974, Ralph Merkle proposed the first unclassified protocol for secure communications over insecure channels. When legitimate communicating parties are willing to spend an amount of computational effort proportional to some parameter N, an eavesdropper cannot break into their communication without spending a time proportional to N2 , which is quadratically more than the legitimate effort. In a quantum world, however, Merkle’s protocol is immediately broken by Grover’s algorithm, but it is easily repaired if we are satisfied with a quantum protocol against which a quantum adversary needs to spend a time proportional to N3/2 in order to break it. Can we do better? We give two new key establishment protocols in the spirit of Merkle’s. The first one, which requires the legitimate parties to have access to a quantum computer, resists any quantum adversary who is not willing to make an effort at least proportional to N5/3 , except with vanishing probability. Our second protocol is purely classical, yet it requires any quantum adversary to work asymptotically harder than the legitimate parties, again except with vanishing probability. In either case, security is proved for a typical run of the protocols: the probabilities are taken over the random (or quantum) choices made by the legitimate participants in order to establish their key as well as over the random (or quantum) choices made by the adversary who is trying to be privy to it.
Key agreement is a cryptographic scenario between two legitimate parties, who need to establish a common secret key over a public authenticated channel, and an eavesdropper who intercepts all their messages in order to learn the secret. We consider query complexity in which we count only the number of evaluations (queries) of a given black-box function, and classical communication channels. Ralph Merkle provided the first unclassified scheme for secure communications over insecure channels. When legitimate parties are willing to ask O(N) queries for some parameter N, any classical eavesdropper needs Omega(N^2) queries before being able to learn their secret, which is is optimal. However, a quantum eavesdropper can break this scheme in O(N) queries. Furthermore, it was conjectured that any scheme, in which legitimate parties are classical, could be broken in O(N) quantum queries. In this thesis, we introduce protocols à la Merkle that fall into two categories. When legitimate parties are restricted to use classical computers, we offer the first secure classical scheme. It requires Omega(N^{13/12}) queries of a quantum eavesdropper to learn the secret. We give another protocol having security of Omega(N^{7/6}) queries. Furthermore, for any k>= 2, we introduce a classical protocol in which legitimate parties establish a secret in O(N) queries while the optimal quantum eavesdropping strategy requires Theta(N^{1/2+k/{k+1}}) queries, approaching Theta(N^{3/2}) when k increases. When legitimate parties are provided with quantum computers, we present two quantum protocols improving on the best known scheme before this work. Furthermore, for any k>= 2, we give a quantum protocol in which legitimate parties establish a secret in O(N) queries while the optimal quantum eavesdropping strategy requires Theta(N^{1+{k}/{k+1}})} queries, approaching Theta(N^{2}) when k increases.
We prove near-optimal lower bounds for preprocessing attacks on quantum cryptography in the random oracle model. Specifically, we show that a T-query adversary with S qubits of non-uniform advice can recover a random key k from the n-qubit binary phase state ∣ψk⟩∝∑xR(k,x)∣x⟩ with probability at most O(NT2+ST) for N=2n. In contrast, the best known bound for post-quantum one-way functions is O(NT2+ST), with a trivial attack at S=N. This demonstrates a new advantage of quantum cryptography over classical cryptography: n qubits of communication suffice for security against preprocessing attacks with space up to N2 rather than N. Our methodology is simple: express the optimal preprocessing attack as the operator norm of a random matrix, and bound this value in expectation over the random oracle via the trace-moment method. These trace moments have a natural interpretation using compressed oracles [Zhandry, Crypto 2019], which we then analyze. This can be viewed as a simplification and generalization of the approach of Liu [Eurocrypt 2023] for proving the security of post-quantum cryptography against preprocessing attacks. We also prove the following results: (1) We tighten Liu's analysis of post-quantum PRGs in QROM, achieving a distinguishing advantage bound of O(NT2+NST). (2) For unitary synthesis, we extend the one-query lower bound of Lombardi-Ma-Wright [STOC 2024] to hold against adversaries that can make one arbitrary function query along with polynomially many (adaptive) queries to the random oracle, either before or after the function query. This also interprets the original LMW24 result in terms of compressed oracles. (3) Finally, we prove a tight O(NS) bound for the pseudorandomness of random binary phase states against space S distinguishers.
Starting in 1974, Ralph Merkle proposed the first unclassified systems for secure communications over insecure channels. When legitimate communicating parties are willing to spend an amount of computational effort proportional to some parameter N, an eavesdropper cannot break into their communication without spending a time in the order ofN2, which is quadratically more than the legitimate effort. We investigate quantum analogues to this technique. First, we show that Merkle's systems are completely insecure if the legitimate parties are classical but the eavesdropper uses quantum computation. Then, we describe simple modifications on Merkle's proposals, in which the legitimate parties still use classical communication but benefit from local quantum computation to agree on a common key. We show that the optimal quantum eavesdropping strategy against our protocols requires a time in the order o/7V3/2. We conjecture these Quantum Merkle Puzzles to be optimal in the classical communication model, in which case quantum mechanics does more harm than good for the purpose of secure communications over insecure classical channels. This is in sharp contrast with Quantum Key Distribution, which ensures unconditionally secure communications over quantum channels.
We construct a quantum public key encryption scheme from oneway functions in the random oracle model. While our construction only has indistinguishability against restricted adversaries, who perform individual attacks and can only make classical queries to the random oracle, it possesses two practical advantages that were not achieved in previous works. First, legitimate participants can discard noisy quantum public keys. Second, the scheme only requires products of single-qubit states, and it does not require quantum computing or quantum memory for legitimate participants. Although the security does not hold against adversaries capable of large-scale quantum computation, we believe that our protocol is well-suited for the near-term stage of the quantum era.
We make progress towards the impossibility of quantum-computation, classical-communication (QCCC) key agreement by giving the first unconditional polynomial-query attacks that tolerate imperfect completeness in the following settings. First, we give a quantum attack on protocols with arbitrarily many rounds in which both parties' oracle queries and communication are classical before the final round, while local computation may be quantum throughout, and the final round may involve quantum oracle queries and one quantum message. Second, we give a classical attack on constant-round QCCC protocols in which Alice has classical oracle access and Bob may make quantum queries throughout. In both settings, the attacker is computationally unbounded and makes poly(λ) queries to recover the key with nonnegligible probability whenever each party's total honest query bound is at most poly(λ) and the valid agreement probability is inverse-polynomial.
As consequences, we rule out query-bounded IND-CPA security for quantum public-key encryption with classical public and secret keys and classical messages of polynomially bounded length in the QROM in either of two settings: (i) key generation has classical oracle access, while encryption, decryption, and the ciphertext may be quantum; or (ii) encryption has classical oracle access and ciphertexts are classical, while key generation and decryption may have quantum oracle access. Both results assume negligible correctness error and polynomial honest query complexity. Combining the constant-round attack with the constructions of Bartusek and Khurana (CRYPTO 2025), we also rule out constant-round oblivious state preparation with polynomial honest query complexity and negligible correctness error in the QROM against computationally unbounded quantum receivers making polynomially many oracle queries.