In this paper, we investigate how the use of a channel with perfect authenticity but no privacy can be used to repair the defects of a channel with imperfect privacy but no authenticity. More precisely, let us assume that Alice and Bob wish to agree on a secret random bit string, and have at their disposal an imperfect private channel and a perfect public channel. The private channel is imperfect in various ways: transmission errors can occur, and partial information can leak to an eavesdropper, Eve, who also has the power to suppress, inject, and modify transmissions arbitrarily. On the other hand, the public channel transmits information accurately, and these transmissions cannot be modified or suppressed by Eve, but their entire contents becomes known to her. We consider the situation in which a random bit string x has already been transmitted from Alice to Bob over the private channel, and we describe interactive public channel protocols that allow them, with high probability: (1) to assess the extent to which the private channel transmission has been corrupted by tampering and channel noise; and (2) if this corruption is not too severe, to repair Bob’s partial ignorance of the transmitted string and Eve’s partial knowledge of it by distilling from the transmitted and received versions of the string another string, in general shorter than x, upon which Alice and Bob have perfect information, while Eve has nearly no information (or in some cases exactly none), except for its length. These protocols remain secure against unlimited computing power.
In this paper, we investigate how the use of a channel with perfect authenticity but no privacy can be used to repair the defects of a channel with imperfect privacy but no authenticity. More precisely, let us assume that Alice and Bob wish to agree on a secret random bit string, and have at their disposal an imperfect private channel and a perfect public channel. The private channel is imperfect in various ways: transmission errors can occur, and partial information can leak to an eavesdropper, Eve, who also has the power to suppress, inject, and modify transmissions arbitrarily. On the other hand, the public channel transmits information accurately, and these transmissions cannot be modified or suppressed by Eve, but their entire contents becomes known to her. We consider the situation in which a random bit string x has already been transmitted from Alice to Bob over the private channel, and we describe interactive public channel protocols that allow them, with high probability: (1) to assess the extent to which the private channel transmission has been corrupted by tampering and channel noise; and (2) if this corruption is not too severe, to repair Bob’s partial ignorance of the transmitted string and Eve’s partial knowledge of it by distilling from the transmitted and received versions of the string another string, in general shorter than x, upon which Alice and Bob have perfect information, while Eve has nearly no information (or in some cases exactly none), except for its length. These protocols remain secure against unlimited computing power.
Privacy Amplification, introduced by Bennett et. al in [1] and generalized in [2], is the technique of shrinking a partially secret string to a highly secret one by public discussion. More precisely, in the privacy amplification phase, two legitimate parties, Alice and Bob, who share a partially secret string S about an opponent Eve has partial information to extract, and a shorter but highly-secret string S' by communication through an public channel such that Eve’s information about S' is negligible, i.e., ε= ≥ −S' U uH ( / ) log S2 | '| holds with very high probability for some small ε > 0, where the random variable U summarizes Eve’s complete knowledge about S', and u is the particular value known to Eve [2]. Privacy amplification is an important step in both quantum cryptography and key theoretically secret information agreement [3].
Several protocols are presented that allow two parties Alice and Bob not sharing any secret information initially (except possibly a short key to be used for authentication) to generate a long shared secret key such that even an enemy Eve with unlimited computing power is unable to obtain a non-negligible amount of information (in Shannon's sense) about this key. Two different models are considered. In a first model we assume that Alice can send information to Bob over a noisy main channel but that Eve is able to receive the same information over a parallel independent noisy channel from Alice to Eve. In a second, more general model we assume that Alice, Bob and Eve receive the output of a random source (e.g., a satellite broadcasting random bits) over three independent individual channels. The condition that the channels be independent can be replaced by the condition that they be independent only to a known, arbitrarily small degree. We demonstrate that even when Eve's channel is sup...
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
Communications over the public channel should be authenticated to make privacy amplification against active adversaries possible. A new method of constructing authentication codes from error correcting codes is proposed and analyzed, and an arbitrary q ary code( q is a prime power) can be transformed into an authentication code according to the method. When the two communicants share n bit partially secret string S , an authentication scheme can be constructed from a ( 2 m,k,2 m-k+1 ) extended Reed Solomon code, where m=[XXZS-ZSX*3Y0]n/k[XXYS-YSX*3Y0] and 2 mk . As long as the two parties share an authentication key of at least 3 m bit, there exists a strong protocol for privacy amplification which ensures that the probability that the adversary successfully implements active attacks is at most k/2 m , and a highly secret string can be distilled of length about the Renyi entropy of S when given Eve's complete knowledge U=u about S . As n is large enough, more than 3 m bits secret string can always be distilled, thus the protocol is practical.
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.
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.