Key-agreement protocols whose security is proven in the random oracle model are an important alternative to protocols based on public-key cryptography. In the random oracle model, the parties and the eavesdropper have access to a shared random function (an "oracle"), but the parties are limited in the number of queries they can make to the oracle. The random oracle serves as an abstraction for black-box access to a symmetric cryptographic primitive, such as a collision resistant hash. Unfortunately, as shown by Impagliazzo and Rudich [STOC '89] and Barak and Mahmoody [Crypto '09], such protocols can only guarantee limited secrecy: the key of any l-query protocol can be revealed by an O(l^2)-query adversary. This quadratic gap between the query complexity of the honest parties and the eavesdropper matches the gap obtained by the Merkle's Puzzles protocol of Merkle [CACM '78]. In this work we tackle a new aspect of key-agreement protocols in the random oracle model: their communication complexity. In Merkle's Puzzles, to obtain secrecy against an eavesdropper that makes roughly l^2 queries, the honest parties need to exchange Omega(l) bits. We show that for protocols with certain natural properties, ones that Merkle's Puzzle has, such high communication is unavoidable. Specifically, this is the case if the honest parties' queries are uniformly random, or alternatively if the protocol uses non-adaptive queries and has only two rounds. Our proof for the first setting uses a novel reduction from the set-disjointness problem in two-party communication complexity. For the second setting we prove the lower bound directly, using information-theoretic arguments. Understanding the communication complexity of protocols whose security is proven (in the random-oracle model) is an important question in the study of practical protocols. Our results and proof techniques are a first step in this direction.
Key-agreement protocols whose security is proven in the random oracle model are an important alternative to protocols based on public-key cryptography. In the random oracle model, the parties and the eavesdropper have access to a shared random function (an "oracle"), but the parties are limited in the number of queries they can make to the oracle. The random oracle serves as an abstraction for black-box access to a symmetric cryptographic primitive, such as a collision resistant hash. Unfortunately, as shown by Impagliazzo and Rudich [STOC '89] and Barak and Mahmoody [Crypto '09], such protocols can only guarantee limited secrecy: the key of any l-query protocol can be revealed by an O(l^2)-query adversary. This quadratic gap between the query complexity of the honest parties and the eavesdropper matches the gap obtained by the Merkle's Puzzles protocol of Merkle [CACM '78]. In this work we tackle a new aspect of key-agreement protocols in the random oracle model: their communication complexity. In Merkle's Puzzles, to obtain secrecy against an eavesdropper that makes roughly l^2 queries, the honest parties need to exchange Omega(l) bits. We show that for protocols with certain natural properties, ones that Merkle's Puzzle has, such high communication is unavoidable. Specifically, this is the case if the honest parties' queries are uniformly random, or alternatively if the protocol uses non-adaptive queries and has only two rounds. Our proof for the first setting uses a novel reduction from the set-disjointness problem in two-party communication complexity. For the second setting we prove the lower bound directly, using information-theoretic arguments. Understanding the communication complexity of protocols whose security is proven (in the random-oracle model) is an important question in the study of practical protocols. Our results and proof techniques are a first step in this direction.
In this work we deal with one-round key-agreement protocols, called Merkle’s Puzzles, in the random oracle model, where the players Alice and Bob are allowed to query a random permutation oracle n times. We prove that Eve can always break the protocol by querying the oracle O(n 2 ) times. The long-time unproven optimality of the quadratic bound in the fully general, multi-round scenario has been proven recently by Barak and MahmoodyGhidary. The results in this paper have been found independently of their work. In this work we prove the tight upper-bound on the number of queries needed to break a keyagreement protocol in the random oracle model. The key-agreement protocol called Merkle’s puzzles, developed by Merkle in 1974 a published in 1978 [3] is one of the earliest example of public-key encryption. Following the protocol, two parties can agree on a secret-key by exchanging messages, assuming that they share no secrets beforehand. Informally, Alice creates a message for Bob in the following way - she constructs a large number of puzzles of moderate difficulty, each of them being possible to solve with Bob’s computational resources. All of them are in the form of an encrypted message with an unknown key that is short enough to allow the brute force attack. After receiving the message from Alice, Bob chooses one puzzle uniformly at random and solves it. The solution contains an identifier and a key. Bob encrypts the identifier with the key, and announces it back to Alice. The solution of the puzzle solved by Bob becomes Alice’s and Bob’s secret-key. Since the puzzle’s identifier is sent to Alice as a message encrypted with a key that is unknown to Eve, the eavesdropper’s best strategy to attack the key-agreement protocol is to solve as many puzzles as possible. To achieve constant probability of success, she has to solve a constant fraction of them, which might require much more computational power than Alice and Bob have. In a similar way we construct a key-agreement protocol in the random oracle scenario, where the computational difficulty of key-agreement is expressed by the number of oracle queries that
In the random oracle model, the parties are given oracle access to a random member of a (typically huge) function family, and are assumed to have unbounded computational power (though they can only make a bounded number of oracle queries). This model provides powerful properties that allow proving the security of many protocols, even such that cannot be proved secure in the standard model (under any hardness assumption). The random oracle model is also used to show that a given cryptographic primitive cannot be used in a black-box way to construct another primitive; in their seminal work, Impagliazzo and Rudich [STOC ’89] showed that in the random function model – when the function family is the set of all functions – it is impossible to construct (secure) key-agreement protocols, yielding that key-agreement cannot be black-box reduced to one-way functions. Their work has a long line of followup works (Simon [EC ’98], Gertner et al. [STOC ’00] and Gennaro et al. [SICOMP ’05], to name a few), showing that given oracle access to a certain type of function family (e.g., the family that “implements ” public-key encryption) is not sufficient for building a given cryptographic primitive (e.g., oblivious transfer). Yet, in the more general sense, the following fundamental question remained open: What is the exact power of the random oracle model, and more specifically, of the random function model? We make progress towards answering the above question, showing that any (no private input) semi-honest two-party functionality that can be securely implemented in the random function model, can be securely implemented information theoretically (where parties are assumed to be all powerful, and no oracle is given). We further generalize the above result to function families that provide some natural combinatorial property. To exhibit the power of our result, we use the recent information theoretic impossibility result of McGregor et al. [FOCS ’10], to show the existence of functionalities (e.g., inner product) that cannot be computed both accurately and in a differentially private manner in the random function model; yielding that protocols for computing these functionalities cannot be black-box reduced to the existence of one-way functions.
This paper proves that it is impossible to construct perfectly complete quantum key agreement protocols (QKA) from quantumly secure one-way functions (OWFs) in a black-box manner.
Specifically, consider any protocol in which Alice and Bob exchange only classical messages, make at most qA and qB quantum queries, respectively, to a Boolean-valued random oracle, and agree on a shared key with certainty. This paper shows that there exists an eavesdropper, given the classical messages, that can recover the shared key with certainty using O((qA+qB)5) classical oracle queries. The bound is independent of the number of rounds, transcript length, key length, and oracle-domain size. Previous results only applies to two-round key agreement (Li et al. CRYPTO 26) or relies on unproven conjectures (Austrin et al. CRYPTO 22).
GPT-5.6 Sol Ultra found this proof in a one-shot conversation and drafted a preliminary version of this paper. The authors are fully responsible for the correctness, writing and discussions of this paper.
In this paper we study one-round key-agreement protocols analogous to Merkle's puzzles in the random oracle model. The players Alice and Bob are allowed to query a random permutation oracle n times and upon their queries and communication, they both output the same key with high probability. We prove that Eve can always break such a protocol by querying the oracle O(n2) times. The long-time unproven optimality of the quadratic bound in the fully general, multi-round scenario has been shown recently by Barak and Mahmoody-Ghidary. The results in this paper have been found independently of their work.
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.
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.