This paper explores what kinds of information two parties must communicate in order to correct errors which occur in a shared secret string W. Any bits they communicate must leak a significant amount of information about W --- that is, from the adversary's point of view, the entropy of W will drop significantly. Nevertheless, we construct schemes with which Alice and Bob can prevent an adversary from learning any useful information about W. Specifically, if the entropy of W is sufficiently high, then there is no function f(W) which the adversary can learn from the error-correction information with significant probability.This leads to several new results: (a) the design of noise-tolerant "perfectly one-way" hash functions in the sense of Canetti et al. [7], which in turn leads to obfuscation of proximity queries for high entropy secrets W; (b) private fuzzy extractors [11], which allow one to extract uniformly random bits from noisy and nonuniform data W, while also insuring that no sensitive information about W is leaked; and (c) noise tolerance and stateless key re-use in the Bounded Storage Model, resolving the main open problem of Ding [10].The heart of our constructions is the design of strong randomness extractors with the property that the source W can be recovered from the extracted randomness and any string W' which is close to W.
This paper explores what kinds of information two parties must communicate in order to correct errors which occur in a shared secret string W. Any bits they communicate must leak a significant amount of information about W --- that is, from the adversary's point of view, the entropy of W will drop significantly. Nevertheless, we construct schemes with which Alice and Bob can prevent an adversary from learning any useful information about W. Specifically, if the entropy of W is sufficiently high, then there is no function f(W) which the adversary can learn from the error-correction information with significant probability.This leads to several new results: (a) the design of noise-tolerant "perfectly one-way" hash functions in the sense of Canetti et al. [7], which in turn leads to obfuscation of proximity queries for high entropy secrets W; (b) private fuzzy extractors [11], which allow one to extract uniformly random bits from noisy and nonuniform data W, while also insuring that no sensitive information about W is leaked; and (c) noise tolerance and stateless key re-use in the Bounded Storage Model, resolving the main open problem of Ding [10].The heart of our constructions is the design of strong randomness extractors with the property that the source W can be recovered from the extracted randomness and any string W' which is close to W.
We study the question of basing symmetric key cryptography on weak secrets. In this setting, Alice and Bob share an n-bit secret W, which might not be uniformly random, but the adversary has at least k bits of uncertainty about it (formalized using conditional min-entropy). Since standard symmetrickey primitives require uniformly random secret keys, we would like to construct an authenticated key agreement protocol in which Alice and Bob use W to agree on a nearly uniform key R, by communicating over a public channel controlled by an active adversary Eve. We study this question in the information theoretic setting where the attacker is computationally unbounded. We show that single-round (i.e. one message) protocols do not work when k ≤ n 2, and require poor parameters even when n 2 < k ≪ n. On the other hand, for arbitrary values of k, we design a communication efficient two-round (challenge-response) protocol extracting nearly k random bits. This dramatically improves the previous construction of Renner and Wolf [RW03], which requires Θ(λ + log(n)) rounds where λ is the security parameter. Our solution takes a new approach by studying and constructing “non-malleable” seeded randomness extractors — if an attacker sees a random seed X and comes up with an arbitrarily related seed X ′, then we bound the relationship between R = Ext(W;X) and R ′ = Ext(W;X ′). We also extend our two-round key agreement protocol to the “fuzzy ” setting, where Alice and Bob share “close ” (but not equal) secrets WA and WB, and to the Bounded Retrieval Model (BRM) where the size of the secret W is huge.
This is the third part of a three-part paper on se- cret-key agreement secure against active adversaries. Here, we con- sider the special case where the legitimate partners already share a mutual string which might, however, be partially known to the adversary. The problem of generating a key in this case has been well studied in the passive-adversary model—for instance, in the context of quantum key agreement—under the name of privacy amplification. We consider the same problem with respect to an ac- tive adversary and propose two protocols, one based on universal hashing and one based on extractors, allowing for privacy ampli- fication secure against an adversary whose knowledge about the initial partially string is limited to one third of the length of this string. Our results are based on novel techniques for au- thentication secure even against adversaries knowing a substantial amount of the secret key. information about . From Eve's point of view, the length of can in this case be roughly equal to the Renyi entropy of . In this paper, we investigate the same problem under the as- sumption that the communication between Alice and Bob is not authenticated. Note that, in contrast to the model where many independent repetitions of the involved random variables are given (15), the same piece of information must be used here both for authentication and as the input for privacy amplifica- tion. Two problems that arise in this context are authentication with an only partially key, and the fact that this authenti- cation leaks information about , hence potentially also about , to the adversary. We show that for our purpose, a new, inter- active, authentication method is better than one-way authentica- tion by strongly universal hashing, and that so-called extractors, requiring fewer random bits (i.e., shorter messages to be com- municated), are a better technique for privacy amplification than
Consider two parties, Alice and Bob, who hold private inputs x and y, and wish to compute a function f (x, y) privately in the information theoretic sense; that is, each party should learn nothing beyond f (x, y). However, the communication channel available to them is noisy. This means that the channel can introduce errors in the transmission between the two parties. Moreover, the channel is adversarial in the sense that it knows the protocol that Alice and Bob are running, and maliciously introduces errors to disrupt the communication, subject to some bound on the total number of errors. A fundamental question in this setting is to design a protocol that remains private in the presence of large number of errors. If Alice and Bob are only interested in computing f (x, y) correctly, and not privately, then quite robust protocols are known that can tolerate a constant fraction of errors. However, none of these solutions is applicable in the setting of privacy, as they inherently leak information about the parties' inputs. This leads to the question whether we can simultaneously achieve privacy and error-resilience against a constant fraction of errors. We show that privacy and errorresilience are contradictory goals. In particular, we show that for every constant c > 0, there exists a function f which is privately computable in the error-less setting, but for which no private and correct protocol is resilient against a c-fraction of errors.
Consider two parties holding correlated random variables W and W′, respectively, that are within distance t of each other in some metric space. These parties wish to agree on a uniformly distributed secret key R by sending a single message over an insecure channel controlled by an all-powerful adversary. We consider both the keyless case, where the parties share no additional secret information, and the keyed case, where the parties share a long-term secret SK that they can use to generate a sequence of session keys {Rj} using multiple pairs {(Wj, W ′ j)}. The former has applications to, e.g., biometric authentication, while the latter arises in, e.g., the bounded storage model with errors. Our results improve upon previous work in several respects: – The best previous solution for the keyless case with no errors (i.e., t = 0) requires the min-entropy of W to exceed 2|W|/3. We show a solution when the min-entropy of W exceeds the minimal threshold |W|/2. – Previous solutions for the keyless case in the presence of errors (i.e., t> 0) required random oracles. We give the first constructions (for certain metrics) in the standard model. – Previous solutions for the keyed case were stateful. We give the first stateless solution.
Motivated by applications in cryptography, we consider a generalization of randomness extraction and the related notion of privacy amplification to the case of two correlated sources. We introduce the notion of correlation extractors, which extract nearly perfect independent instances of a given joint distribution from imperfect, or "leaky," instances of the same distribution. More concretely, suppose that Alice holds a and Bob holds b, where (a, b) are obtained by taking n independent samples from a joint distribution (X, Y) and letting a include all X instances and b include all Y instances. An adversary Eve obtains partial information about (a, b) by choosing a function L with output length t and learning L(a, b). The goal is to design a protocol between Alice and Bob which may use additional fresh randomness, such that for every L as above the following holds. In the end of the interaction, Alice outputs a' and Bob outputs b' such that (a', b') are statistically indistinguishable from m independent instances of (X, Y) even when conditioned on Eve's view, and even when conditioned on the joint view of Eve together with either Alice or Bob. The standard questions of privacy amplification and randomness extraction correspond to the case where X and Y are identical random bits. In this work we address this question for other types of correlations. A central special case is that of OT extractors, which are correlation extractors for the correlation (X, Y) corresponding to the cryptographic primitive of oblivious transfer. Our main result is that for any finite joint distribution (X, Y) there is an explicit correlation extractor which extracts m = ?(n) instances using O(n) bits of communication, even when t = ?(n) bits of information can be leaked to Eve. We present several applications which motivate the concept of correlation extractors and our main result. These include: ? Protecting certain cryptographic protocols against sidechannel attacks. ? A protocol which realizes m instances of oblivious transfer by communicating only O(m) bits. The security of the protocol relies on a number-theoretic intractability assumption. ? A constant-rate unconditionally secure construction of oblivious transfer (for semi-honest parties) from any nontrivial channel. This establishes constant-rate equivalence of any two nontrivial finite channels.
Abstract. Consider two parties Alice and Bob, who hold private inputs x and y, and wish to compute a function f(x, y) privately in the information theoretic sense; that is, each party should learn nothing beyond f(x, y). However, the communication channel available to them is noisy. This means that the channel can introduce errors in the transmission between the two parties. Moreover, the channel is adversarial in the sense that it knows the protocol that Alice and Bob are running, and maliciously introduces errors to disrupt the communication, subject to some bound on the total number of errors. A fundamental question in this setting is to design a protocol that remains private in the presence of large number of errors. If Alice and Bob are only interested in computing f(x, y) correctly, and not privately, then quite robust protocols are known that can tolerate a constant fraction of errors. However, none of these solutions is applicable in the setting of privacy, as they inherently leak information about the parties ’ inputs. This leads to the question whether we can simultaneously achieve privacy and error-resilience against a constant fraction of errors. We show that privacy and error-resilience are contradictory goals. In particular, we show that for every constant c> 0, there exists a function f which is privately computable in the error-less setting, but for which no private and correct protocol is resilient against a c-fraction of errors.