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arXiv:1605.04576 · 2017 Computing Conference

Why is deep random suitable for cryptology

Thibault de Valroger

Published Jul 1, 2017• 1 citation
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Abstract

We present a new form of randomness, called "Deep Randomness", generated in such a way that probability distribution of the output signal is made unknowledgeable for an observer. By limiting, thanks to Deep Randomness, the capacity of the opponent observer to perform bayesian inference over public information to estimate private information, we can design protocols, beyond Shannon limit, enabling two legitimate partners, sharing originally no common private information, to exchange secret information with accuracy as close as desired from perfection, and knowledge as close as desired from zero by any unlimitedly powered opponent. We discuss the theoretical foundation of Deep Randomness, which lies on Prior Probability theory, introduced and developed by authors like Laplace, Cox, Carnap, Jefferys and Jaynes; and we introduce computational method to generate such Deep Randomness.

DOI
10.1109/sai.2017.8252187

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We present a new form of randomness, called "Deep Randomness", generated in such a way that probability distribution of the output signal is made unknowledgeable for an observer. By limiting, thanks to Deep Randomness, the capacity of the opponent observer to perform bayesian inference over public information to estimate private information, we can design protocols, beyond Shannon limit, enabling two legitimate partners, sharing originally no common private information, to exchange secret information with accuracy as close as desired from perfection, and knowledge as close as desired from zero by any unlimitedly powered opponent. We discuss the theoretical foundation of Deep Randomness, which lies on Prior Probability theory, introduced and developed by authors like Laplace, Cox, Carnap, Jefferys and Jaynes; and we introduce computational method to generate such Deep Randomness.
We present a new idea to design perfectly secure information exchange protocol, based on so called Deep Randomness, which means randomness relying on hidden probability distribution. Such idea drives us to introduce a new axiom in probability theory, thanks to which we can design a protocol, beyond Shannon limit, enabling two legitimate partners, sharing originally no common private information, to exchange secret information with accuracy as close as desired from perfection, and knowledge as close as desired from zero by any unlimitedly powered opponent.
We have formerly introduced Deep Random Secrecy, a new cryptologic technique capable to ensure secrecy as close as desired from perfection against unlimited passive eavesdropping opponents. We have also formerly introduced an extended protocol, based on Deep Random Secrecy, capable to resist to unlimited active MITM. The main limitation of those protocols, in their initial presented version, is the important quantity of information that needs to be exchanged between the legitimate partners to distill secure digits. We have defined and shown existence of an absolute constant, called Cryptologic Limit, which represents the upper-bound of Secrecy rate that can be reached by Deep Random Secrecy protocols. At last, we have already presented practical algorithms to generate Deep Randomness from classical computing resources. This article is presenting an optimization technique, based on recombination and reuse of random bits; this technique enables to dramatically increase the bandwidth performance of formerly introduced protocols, without jeopardizing the entropy of secret information. That optimization enables to envision an implementation of Deep Random Secrecy at very reasonable cost. The article also summarizes former results in the perspective of a comprehensive implementation.
We have introduced in former work the concept of Deep Randomness and its interest to design Unconditionally Secure communication protocols. We have in particular given an example of such protocol and introduced how to design a Deep Random Generator associated to that protocol. Deep Randomness is a form of randomness in which, at each draw of random variable, not only the result is unpredictable bu also the distribution is unknown to any observer. In this article, we remind formal definition of Deep Randomness, and we expose two practical algorithmic methods to implement a Deep Random Generator within a classical computing resource. We also discuss their performances and their parameters.
In the bounded-storage model for information-theoretically secure encryption and key-agreement one can prove the security of a cipher based on the sole assumption that the adversary's storage capacity is bounded, say by s bits, even if her computational power is unlimited. Assume that a random t-bit string R is either publicly available (e.g. the signal of a deep space radio source) or broadcast by one of the legitimate parties. If s < t, the adversary can store only partial information about R. The legitimate sender Alice and receiver Bob, sharing a short secret key K initially, can therefore potentially generate a very long n-bit one-time pad X with n jKj about which the adversary has essentially no information, thus at rst glance apparently contradicting Shannon's bound on the key size of a perfect cipher.
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.
This monograph describes principles of information theoretic secrecy generation by legitimate parties with public discussion in the presence of an eavesdropper. The parties are guaranteed secrecy in the form of independence from the eavesdropper’s observation of the communication. Part I develops basic technical tools for secrecy generation, many of which are potentially of independent interest beyond secrecy settings. Various information theoretic and cryptographic notions of secrecy are compared. Emphasis is placed on central themes of interactive communication and common randomness as well as on core methods of balanced coloring and leftover hash for extracting secret uniform randomness. Achievability and converse results are shown to emerge from “single shot” incarnations that serve to explain essential structure. Part II applies the methods of Part I to secrecy generation in two settings: a multiterminal source model and a multiterminal channel model, in both of which the legitimate parties are afforded privileged access to correlated observations of which the eavesdropper has only partial knowledge. Characterizations of secret key capacity bring out inherent connections to the data compression concept of omniscience and, for a specialized source model, to a combinatorial problem of maximal spanning tree packing in a multigraph. Interactive common information is seen to govern the minimum rate of communication needed to achieve secret key capacity in the two-terminal source model. Furthermore, necessary and sufficient conditions are analyzed for the secure computation of a given function in the multiterminal source model. Based largely on known recent results, this self-contained monograph also includes new formulations with associated new proofs. Supplementing each chapter in Part II are descriptions of several open problems.