Interactive proof systemIn computational complexity theory, an interactive proof system is an abstract machine that models computation as the exchange of messages between two parties: a prover and a verifier. The parties interact by exchanging messages in order to ascertain whether a given string belongs to a language or not. The prover possesses unlimited computational resources but cannot be trusted, while the verifier has bounded computation power but is assumed to be always honest.
Payment systemA payment system is any system used to settle financial transactions through the transfer of monetary value. This includes the institutions, payment instruments such as payment cards, people, rules, procedures, standards, and technologies that make its exchange possible. A common type of payment system, called an operational network, links bank accounts and provides for monetary exchange using bank deposits. Some payment systems also include credit mechanisms, which are essentially a different aspect of payment.
E-commerce payment systemAn e-commerce payment system (or an electronic payment system) facilitates the acceptance of electronic payment for offline transfer, also known as a subcomponent of electronic data interchange (EDI), e-commerce payment systems have become increasingly popular due to the widespread use of the internet-based shopping and banking. Credit cards remain the most common forms of payment for e-commerce transactions. As of 2008, in North America, almost 90% of online retail transactions were made with this payment type.
Zero-knowledge proofIn cryptography, a zero-knowledge proof or zero-knowledge protocol is a method by which one party (the prover) can prove to another party (the verifier) that a given statement is true, while avoiding conveying to the verifier any information beyond the mere fact of the statement's truth. The intuition underlying zero-knowledge proofs is that it is trivial to prove the possession of certain information by simply revealing it; the challenge is to prove this possession without revealing the information, or any aspect of it whatsoever.
Fiber product of schemesIn mathematics, specifically in algebraic geometry, the fiber product of schemes is a fundamental construction. It has many interpretations and special cases. For example, the fiber product describes how an algebraic variety over one field determines a variety over a bigger field, or the pullback of a family of varieties, or a fiber of a family of varieties. Base change is a closely related notion. The of schemes is a broad setting for algebraic geometry.
Scheme (mathematics)In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers). Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise "Éléments de géométrie algébrique"; one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne).
Group schemeIn mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not necessarily connected, smooth, or defined over a field. This extra generality allows one to study richer infinitesimal structures, and this can help one to understand and answer questions of arithmetic significance.
PaymentA payment is the voluntary tender of money or its equivalent or of things of value by one party (such as a person or company) to another in exchange for goods or services provided by them, or to fulfill a legal obligation/philanthropy desire. The party making the payment is commonly called the payer, while the payee is the party receiving the payment. Payments can be effected in a number of ways, for example: the use of money, cheque, or debit, credit, or bank transfers, whether through mobile payment or otherwise the transfer of anything of value, such as stock, or using barter, or personal data, the exchange of one good or service for another.
Mobile paymentA mobile payment, also referred to as mobile money, mobile money transfer and mobile wallet, is any of various payment processing services operated under financial regulations and performed from or via a mobile device, as the cardinal class of digital wallet. Instead of paying with cash, cheque, or credit cards, a consumer can use a payment app on a mobile device to pay for a wide range of services and digital or hard goods.
Formal schemeIn mathematics, specifically in algebraic geometry, a formal scheme is a type of space which includes data about its surroundings. Unlike an ordinary scheme, a formal scheme includes infinitesimal data that, in effect, points in a direction off of the scheme. For this reason, formal schemes frequently appear in topics such as deformation theory. But the concept is also used to prove a theorem such as the theorem on formal functions, which is used to deduce theorems of interest for usual schemes.
