In this article, we discuss a naive method of randomness reduction for cryptographic schemes, which replaces the required perfect randomness with output distribution of a computationally secure pseudorandom generator (PRG). We propose novel ideas and techniques for evaluating the indistinguishability between the random and pseudorandom cases, even against an adversary with computationally unbounded attack algorithm. Hence the PRG-based randomness reduction can be effective even for informationtheoretically secure cryptographic schemes, especially when the amount of information received by the adversary is small. In comparison to a preceding result of Dubrov and Ishai (STOC 2006), our result removes the requirement of generalized notion of “nb-PRGs ” and is effective for more general kinds of protocols. We give some numerical examples to show the effectiveness of our result in practical situations, and we also propose a further idea for improving the effect of the PRG-based randomness reduction.
In this article, we discuss a naive method of randomness reduction for cryptographic schemes, which replaces the required perfect randomness with output distribution of a computationally secure pseudorandom generator (PRG). We propose novel ideas and techniques for evaluating the indistinguishability between the random and pseudorandom cases, even against an adversary with computationally unbounded attack algorithm. Hence the PRG-based randomness reduction can be effective even for informationtheoretically secure cryptographic schemes, especially when the amount of information received by the adversary is small. In comparison to a preceding result of Dubrov and Ishai (STOC 2006), our result removes the requirement of generalized notion of “nb-PRGs ” and is effective for more general kinds of protocols. We give some numerical examples to show the effectiveness of our result in practical situations, and we also propose a further idea for improving the effect of the PRG-based randomness reduction.
In this paper, we discuss a naive method of randomness reduction for cryptographic schemes, which replaces the required perfect randomness with output distribution of a computationally secure pseudorandom generator (PRG). We propose novel ideas and techniques for evaluating the indistinguishability between the random and pseudorandom cases, even against an adversary with computationally unbounded attack algorithm. Hence, the PRG-based randomness reduction can be effective even for information-theoretically secure cryptographic schemes, especially when the amount of information received by the adversary is small. In comparison to a preceding result of Dubrov and Ishai (STOC 2006), our result removes the requirement of generalized notion of “nb-PRGs” and is effective for more general kinds of protocols. We give some numerical examples to show the effectiveness of our result in practical situations, and we also propose a further idea for improving the effect of the PRG-based randomness reduction.
Randomness plays a central role in computer science, in particular cryptography. Almost all cryptographic primitives depend crucially on randomness because randomness and unpredictability in secret keys provides the means for security. Usually one assumes that perfect randomness, a sequence of independently and uniformly distributed bits, are accessible to algorithms. This is a strong assumption. Physical sources of randomness are neither uniformly random, nor produce necessarily independent bits. Therefore, the aim of this thesis is to start from a realistic model of randomness, investigate notions of secrecy and their randomness requirements and finally find practical methods for generation of randomness that matches the requirements of cryptographic primitives. We consider a model of random source where the source output follows one distribution from a set of possible distributions, each with the property that the maximum probability of symbols is bounded and can not be arbitrarily close to 1. This model does not assume independence or uniformity of the output symbols, and is considered to be a realistic model of randomness. From this point, the thesis can be divided into two main parts: In the first part, considering various notions of information theoretic secrecy, a fundamental problem is to find the properties of randomness needed to achieve security in these notions. Traditional cryptographic protocols simply assume perfect randomness and build on this assumption. We explore the results that show secrecy can not be based on imperfect randomness that is not uniform or independent. Thus a line of work attempts to relax notions of secrecy in such a way that they can be constructed with non-perfect sources, and possibly require smaller key sizes. Yet they should match real life applications. An important work in this context is entropic security where the key size could be smaller than the message depending on message distribution. Inspired by this, we propose two relaxed notions of secrecy that are motivated by practical applications. In the first notion, motivated by an application in biometrics authentication, we propose guessing secrecy where the probability of guessing the message for a computational unbounded adversary with the best strategy remains the same when a ciphertext is given or when it is not. We compare the randomness requirements of guessing secrecy with stronger notions and show that in some cases such as key length, the requirements are the same. For key distribution however, we found a family of distributions that provide guessing secrecy but not perfect secrecy. In the second notion, we investigate randomness requirements of multiple message encryption. Considering a natural extension of secrecy definition to multiple messages, we show that independent keys are needed to encrypt each message. We then propose a relaxed notion in which the security of last message is more important than past messages, although the leakage of past messages is bounded using entropic security. By this assumption, we achieve smaller key length compared to indistinguishability, and comparable to key length for entropic security. This notion has applications such as location privacy. In the second part of the thesis, since secrecy crucially depends on perfect randomness, we investigate how perfect randomness can be practically generated, specifically from human game-play. Unlike many random number generators that assume independent or uniform random bits in the random source, we base our work on the realistic model of randomness. Our main observation is that human game-play has an element of randomness coming from the errors in their game-play which is the main entertaining factor of the game. We also observed that this game-play can distinguish among a group of people if the right features are collected from the game-play. We incrementally changed our game design until the distinguishability among a small population is maximized, and then run the experiments required to show the viability of this approach over a larger population. This approach can also provide a hard to delegate authentication property where a human could not emulate the behavior of another human even given statistical information about their game-play.
The fundamental security of cryptographic protocols hinges on two intertwined pillars: secrecy and randomness. In classical cryptography, the relationship between these concepts is well-understood, often formalized through information theory and computational complexity. However, the advent of quantum computing poses a significant threat to many classical cryptographic primitives, necessitating a transition to post-quantum cryptography (PQC). This paper explores the critical notion of the secrecy-randomness equivalence within the context of post-quantum protocols. We delve into how the definitions and relationships between perfect secrecy, computational secrecy, true randomness, and pseudorandomness are impacted by quantum adversaries and quantum information theory. Specifically, we investigate whether the classical equivalence – where sufficient randomness can guarantee secrecy and robust secrecy often relies on high-quality randomness – holds or requires re-evaluation in a post-quantum landscape. We analyze the implications for the design and security proofs of various PQC families, including lattice-based, code-based, and hash-based schemes, considering the role of quantum random number generators (QRNGs) and the potential for quantum-enhanced attacks on randomness. Our findings suggest that while the core principles remain, the practical realization and theoretical guarantees of this equivalence become more nuanced, demanding careful consideration of quantum-resistant randomness sources and revised security models to ensure the long-term integrity of post-quantum cryptographic systems.
Cryptographic protocols are commonly designed and their security proven under the assumption that the protocol parties have access to perfect (uniform) randomness. Physical randomness sources deployed in practical implementations of these protocols often fall short in meeting this assumption, but instead provide only a steady stream of bits with certain high entropy. Trying to ground cryptographic protocols on such imperfect, weaker sources of randomness has thus far mostly given rise to a multitude of impossibility results, including the impossibility to construct provably secure encryption, commitments, secret sharing, and zero-knowledge proofs based solely on a weak source. More generally, indistinguishability-based properties break down for such weak sources. In this paper, we show that the loss of security induced by using a weak source can be meaningfully quantified if the source is bounded, e.g., for the well-studied Santha-Vazirani (SV) sources. The quantification relies on a novel relaxation of indistinguishability by a quantitative parameter. We call the resulting notion dierential indistinguishability in order to reflect its structural similarity to dierential privacy. More concretely, we prove that indistinguishability with uniform randomness implies dierential indistinguishability with weak randomness. We show that if the amount of weak randomness is limited (e.g., by using it only to seed a PRG), all cryptographic primitives and protocols still achieve dierential indistinguishability.
Cryptographic protocols, though used in a wide range of everyday tasks, are typically built for certain unrealistic ideal conditions in which the user is assumed to work. For example, one such condition that is implicitly assumed is that the user has perfect randomness that she can use as her cryptographic secret key. Another commonly made assumption is that the user runs her cryptographic tasks (such as encryption) on a device that is perfectly secure; that is, its internals are not tampered with or even observed by an adversary.
In this thesis, we focus on designing cryptographic techniques that are secure under less-than-ideal conditions. In order to remove the assumption that users share perfectly random secret keys, we study the problem of privacy amplification: key agreement between two parties who both know a weak secret w, such as a password. The goal of the protocol is to convert this non-uniform secret w into a uniformly distributed string R that is fully secret from the adversary. R may then be used as a key for running symmetric cryptographic protocols (such as encryption or authentication). Because we make no computational assumptions, the entropy in R can come only from w. Thus such a protocol must minimize the entropy loss during its execution, so that R is as long as possible. In this thesis, we present the first protocol for information-theoretic key agreement that has entropy loss linear in the security parameter. The result is optimal up to constant factors. We also extend our results to the case where the users share correlated secrets w and w' . This setting occurs, for example, when the secrets are biometric readings taken at different points of time and is more generally known as information-reconciliation.
The question of how to construct optimally efficient secure protocols is a central question in cryptography and in the computer security world at large. We focus in this work on several facets of this question, with a particular view towards the role of randomness in secure computation.
The use of random inputs is ubiquitous in cryptographic primitives. However, the ability to consistently draw from large random sources may be a barrier in practice. We examine the existence of optimally strong pseudorandom sources with particularly efficient implementations; namely, we construct exponentially hard pseudorandom generators that can be computed by circuits that have size linear in the generator output.
Conversely, we also examine efficient protocols that rely only minimally on their random inputs. In the resettable security model, parties may be forced to use the same random input across polynomially many interactions with other parties. We examine this security model in the case of zero knowledge proofs, which are a primitive frequently required in secure computation protocols when one party must prove that they have executed the protocol correctly to another party without revealing secret inputs and compromising security.
Finally, we examine the secure and efficient implementation of a specific functionality, including its various required cryptographic primitives (such as zero knowledge arguments of knowledge). In particular, we construct a protocol that securely realizes pattern matching, including single character wildcards and substring matching.