Discrete logarithmIn mathematics, for given real numbers a and b, the logarithm logb a is a number x such that bx = a. Analogously, in any group G, powers bk can be defined for all integers k, and the discrete logarithm logb a is an integer k such that bk = a. In number theory, the more commonly used term is index: we can write x = indr a (mod m) (read "the index of a to the base r modulo m") for rx ≡ a (mod m) if r is a primitive root of m and gcd(a,m) = 1. Discrete logarithms are quickly computable in a few special cases.
Timing attackIn cryptography, a timing attack is a side-channel attack in which the attacker attempts to compromise a cryptosystem by analyzing the time taken to execute cryptographic algorithms. Every logical operation in a computer takes time to execute, and the time can differ based on the input; with precise measurements of the time for each operation, an attacker can work backwards to the input. Finding secrets through timing information may be significantly easier than using cryptanalysis of known plaintext, ciphertext pairs.
Chosen-ciphertext attackA chosen-ciphertext attack (CCA) is an attack model for cryptanalysis where the cryptanalyst can gather information by obtaining the decryptions of chosen ciphertexts. From these pieces of information the adversary can attempt to recover the hidden secret key used for decryption. For formal definitions of security against chosen-ciphertext attacks, see for example: Michael Luby and Mihir Bellare et al. A number of otherwise secure schemes can be defeated under chosen-ciphertext attack.
Galois extensionIn mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed by the automorphism group Aut(E/F) is precisely the base field F. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory. A result of Emil Artin allows one to construct Galois extensions as follows: If E is a given field, and G is a finite group of automorphisms of E with fixed field F, then E/F is a Galois extension.
Simple extensionIn field theory, a simple extension is a field extension which is generated by the adjunction of a single element, called a primitive element. Simple extensions are well understood and can be completely classified. The primitive element theorem provides a characterization of the finite simple extensions. A field extension L/K is called a simple extension if there exists an element θ in L with This means that every element of L can be expressed as a rational fraction in θ, with coefficients in K; that is, it is produced from θ and elements of K by the field operations +, −, •, / .
Forward secrecyIn cryptography, forward secrecy (FS), also known as perfect forward secrecy (PFS), is a feature of specific key-agreement protocols that gives assurances that session keys will not be compromised even if long-term secrets used in the session key exchange are compromised. For HTTPS, the long-term secret is typically the private key of the server. Forward secrecy protects past sessions against future compromises of keys or passwords.
HTTPSHypertext Transfer Protocol Secure (HTTPS) is an extension of the Hypertext Transfer Protocol (HTTP). It uses encryption for secure communication over a computer network, and is widely used on the Internet. In HTTPS, the communication protocol is encrypted using Transport Layer Security (TLS) or, formerly, Secure Sockets Layer (SSL). The protocol is therefore also referred to as HTTP over TLS, or HTTP over SSL. The principal motivations for HTTPS are authentication of the accessed website and protection of the privacy and integrity of the exchanged data while it is in transit.
Purely inseparable extensionIn algebra, a purely inseparable extension of fields is an extension k ⊆ K of fields of characteristic p > 0 such that every element of K is a root of an equation of the form xq = a, with q a power of p and a in k. Purely inseparable extensions are sometimes called radicial extensions, which should not be confused with the similar-sounding but more general notion of radical extensions. An algebraic extension is a purely inseparable extension if and only if for every , the minimal polynomial of over F is not a separable polynomial.
Moduli stack of elliptic curvesIn mathematics, the moduli stack of elliptic curves, denoted as or , is an algebraic stack over classifying elliptic curves. Note that it is a special case of the moduli stack of algebraic curves . In particular its points with values in some field correspond to elliptic curves over the field, and more generally morphisms from a scheme to it correspond to elliptic curves over . The construction of this space spans over a century because of the various generalizations of elliptic curves as the field has developed.
Proof theoryProof theory is a major branch of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of a given logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.
