|
Indistinguishability Obfuscation
In cryptography, indistinguishability obfuscation (abbreviated IO or iO) is a type of software obfuscation with the defining property that obfuscating any two programs that compute the same mathematical function results in programs that cannot be distinguished from each other. Informally, such obfuscation hides the implementation of a program while still allowing users to run it. Formally, iO satisfies the property that obfuscations of two circuits of the same size which implement the same function are computationally indistinguishable. Indistinguishability obfuscation has several interesting theoretical properties. Firstly, iO is the "best-possible" obfuscation (in the sense that any secret about a program that can be hidden by any obfuscator at all can also be hidden by iO). Secondly, iO can be used to construct nearly the entire gamut of cryptographic primitives, including both mundane ones such as public-key cryptography and more exotic ones such as deniable encryption and f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cryptography
Cryptography, or cryptology (from "hidden, secret"; and ''graphein'', "to write", or ''-logy, -logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of Adversary (cryptography), adversarial behavior. More generally, cryptography is about constructing and analyzing Communication protocol, protocols that prevent third parties or the public from reading private messages. Modern cryptography exists at the intersection of the disciplines of mathematics, computer science, information security, electrical engineering, digital signal processing, physics, and others. Core concepts related to information security (confidentiality, data confidentiality, data integrity, authentication, and non-repudiation) are also central to cryptography. Practical applications of cryptography include electronic commerce, Smart card#EMV, chip-based payment cards, digital currencies, password, computer passwords, and military communications. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probabilistic Polynomial-time
In complexity theory, PP, or PPT is the class of decision problems solvable by a probabilistic Turing machine in polynomial time, with an error probability of less than 1/2 for all instances. The abbreviation PP refers to probabilistic polynomial time. The complexity class was defined by Gill in 1977. If a decision problem is in PP, then there is an algorithm running in polynomial time that is allowed to make random decisions, such that it returns the correct answer with chance higher than 1/2. In more practical terms, it is the class of problems that can be solved to any fixed degree of accuracy by running a randomized, polynomial-time algorithm a sufficient (but bounded) number of times. Turing machines that are polynomially-bound and probabilistic are characterized as PPT, which stands for probabilistic polynomial-time machines. This characterization of Turing machines does not require a bounded error probability. Hence, PP is the complexity class containing all problems sol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NC (complexity)
In computational complexity theory, the class NC (for "Nick's Class") is the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors. In other words, a problem with input size ''n'' is in NC if there exist constants ''c'' and ''k'' such that it can be solved in time using parallel processors. Stephen Cook coined the name "Nick's class" after Nick Pippenger, who had done extensive research on circuits with polylogarithmic depth and polynomial size.Arora & Barak (2009) p.120 As in the case of circuit complexity theory, usually the class has an extra constraint that the circuit family must be ''uniform'' ( see below). Just as the class P can be thought of as the tractable problems ( Cobham's thesis), so NC can be thought of as the problems that can be efficiently solved on a parallel computer.Arora & Barak (2009) p.118 NC is a subset of P because polylogarithmic parallel computations can be simulated by polynomi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudorandom Generator
In theoretical computer science and cryptography, a pseudorandom generator (PRG) for a class of statistical tests is a deterministic procedure that maps a random seed to a longer pseudorandom string such that no statistical test in the class can distinguish between the output of the generator and the uniform distribution. The random seed itself is typically a short binary string drawn from the uniform distribution. Many different classes of statistical tests have been considered in the literature, among them the class of all Boolean circuits of a given size. It is not known whether good pseudorandom generators for this class exist, but it is known that their existence is in a certain sense equivalent to (unproven) circuit lower bounds in computational complexity theory. Hence the construction of pseudorandom generators for the class of Boolean circuits of a given size rests on currently unproven hardness assumptions. Definition Let \mathcal A = \ be a class of functions. Thes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parity Learning
Parity learning is a problem in machine learning. An algorithm that solves this problem must find a function ''ƒ'', given some samples (''x'', ''ƒ''(''x'')) and the assurance that ''ƒ'' computes the parity of bits at some fixed locations. The samples are generated using some distribution over the input. The problem is easy to solve using Gaussian elimination provided that a sufficient number of samples (from a distribution which is not too skewed) are provided to the algorithm. Noisy version ("Learning Parity with Noise") In Learning Parity with Noise (LPN), the samples may contain some error. Instead of samples (''x'', ''ƒ''(''x'')), the algorithm is provided with (''x'', ''y''), where for random boolean b \in \ y = \begin f(x), & \textb \\ 1-f(x), & \text \end The noisy version of the parity learning problem is conjectured to be hard and is widely used in cryptography. See also * Learning with errors In cryptography, learning with errors (LWE) is a ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Learning With Errors
In cryptography, learning with errors (LWE) is a mathematical problem that is widely used to create secure encryption algorithms. It is based on the idea of representing secret information as a set of equations with errors. In other words, LWE is a way to hide the value of a secret by introducing noise to it. In more technical terms, it refers to the computational problem of inferring a linear n-ary function f over a finite Ring (mathematics), ring from given samples y_i = f(\mathbf_i) some of which may be erroneous. The LWE problem is conjectured to be hard to solve, and thus to be useful in cryptography. More precisely, the LWE problem is defined as follows. Let \mathbb_q denote the ring of integers Modular arithmetic, modulo q and let \mathbb_q^n denote the set of n-Vector (mathematics and physics), vectors over \mathbb_q . There exists a certain unknown linear function f:\mathbb_q^n \rightarrow \mathbb_q, and the input to the LWE problem is a sample of pairs (\mathbf,y), wher ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
XDH Assumption
The external Diffie–Hellman (XDH) assumption is a computational hardness assumption used in elliptic curve cryptography. The XDH assumption holds if there exist certain subgroups of elliptic curves which have useful properties for cryptography. Specifically, XDH implies the existence of two distinct groups \langle_1, _2\rangle with the following properties: # The discrete logarithm problem (DLP), the computational Diffie–Hellman problem (CDH), and the computational co-Diffie–Hellman problem are all intractable in _1 and _2. # There exists an efficiently computable bilinear map (pairing) e(\cdot, \cdot) : _1 \times _2 \rightarrow _T. # The decisional Diffie–Hellman problem (DDH) is intractable in _1. The above formulation is referred to as asymmetric XDH. A stronger version of the assumption (symmetric XDH, or SXDH) holds if DDH is ''also'' intractable in _2. The XDH assumption is used in some pairing-based cryptographic protocols. In certain ellip ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Hardness Assumption
In computational complexity theory, a computational hardness assumption is the hypothesis that a particular problem cannot be solved efficiently (where ''efficiently'' typically means "in polynomial time"). It is not known how to prove (unconditional) hardness for essentially any useful problem. Instead, computer scientists rely on reductions to formally relate the hardness of a new or complicated problem to a computational hardness assumption about a problem that is better-understood. Computational hardness assumptions are of particular importance in cryptography. A major goal in cryptography is to create cryptographic primitives with provable security. In some cases, cryptographic protocols are found to have information theoretic security; the one-time pad is a common example. However, information theoretic security cannot always be achieved; in such cases, cryptographers fall back to computational security. Roughly speaking, this means that these systems are secure ''assumin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Provable Security
Provable security refers to any type or level of computer security that can be proved. It is used in different ways by different fields. Usually, this refers to mathematical proofs, which are common in cryptography. In such a proof, the capabilities of the attacker are defined by an adversarial model (also referred to as attacker model): the aim of the proof is to show that the attacker must solve the underlying hard problem in order to break the security of the modelled system. Such a proof generally does not consider side-channel attacks or other implementation-specific attacks, because they are usually impossible to model without implementing the system (and thus, the proof only applies to this implementation). Outside of cryptography, the term is often used in conjunction with secure coding and security by design, both of which can rely on proofs to show the security of a particular approach. As with the cryptographic setting, this involves an attacker model and a model o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P = NP
The P versus NP problem is a major unsolved problem in theoretical computer science. Informally, it asks whether every problem whose solution can be quickly verified can also be quickly solved. Here, "quickly" means an algorithm exists that solves the task and runs in polynomial time (as opposed to, say, exponential time), meaning the task completion time is bounded above by a polynomial function on the size of the input to the algorithm. The general class of questions that some algorithm can answer in polynomial time is " P" or "class P". For some questions, there is no known way to find an answer quickly, but if provided with an answer, it can be verified quickly. The class of questions where an answer can be ''verified'' in polynomial time is "NP", standing for "nondeterministic polynomial time".A nondeterministic Turing machine can move to a state that is not determined by the previous state. Such a machine could solve an NP problem in polynomial time by falling into th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open-source Software
Open-source software (OSS) is Software, computer software that is released under a Open-source license, license in which the copyright holder grants users the rights to use, study, change, and Software distribution, distribute the software and its source code to anyone and for any purpose. Open-source software may be developed in a collaborative, public manner. Open-source software is a prominent example of open collaboration, meaning any capable user is able to online collaboration, participate online in development, making the number of possible contributors indefinite. The ability to examine the code facilitates public trust in the software. Open-source software development can bring in diverse perspectives beyond those of a single company. A 2024 estimate of the value of open-source software to firms is $8.8 trillion, as firms would need to spend 3.5 times the amount they currently do without the use of open source software. Open-source code can be used for studying and a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Hierarchy
In computational complexity theory, the polynomial hierarchy (sometimes called the polynomial-time hierarchy) is a hierarchy of complexity classes that generalize the classes NP and co-NP. Each class in the hierarchy is contained within PSPACE. The hierarchy can be defined using oracle machines or alternating Turing machines. It is a resource-bounded counterpart to the arithmetical hierarchy and analytical hierarchy from mathematical logic. The union of the classes in the hierarchy is denoted PH. Classes within the hierarchy have complete problems (with respect to polynomial-time reductions) that ask if quantified Boolean formulae hold, for formulae with restrictions on the quantifier order. It is known that equality between classes on the same level or consecutive levels in the hierarchy would imply a "collapse" of the hierarchy to that level. Definitions There are multiple equivalent definitions of the classes of the polynomial hierarchy. Oracle definition For the oracle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
