Concepts inFully homomorphic encryption using ideal lattices
Ideal lattice cryptography
Ideal lattices are a special class of lattices and a generalization of cyclic lattices. Ideal lattices naturally occur in many parts of number theory, but also in other areas. In particular, they have a significant place in cryptography. Micciancio defined a generalization of cyclic lattices as ideal lattices. They can be used in cryptosystems to decrease by a square root the number of parameters necessary to describe a lattice, making them more efficient.
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Homomorphic encryption
Homomorphic encryption is a form of encryption which allows specific types of computations to be carried out on ciphertext and obtain an encrypted result which is the ciphertext of the result of operations performed on the plaintext. For instance, one person could add two encrypted numbers and then another person could decrypt the result, without either of them being able to find the value of the individual numbers. Homomorphic encryption schemes are malleable by design.
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Lattice-based cryptography
Lattice-based cryptography is the generic term for asymmetric cryptographic primitives based on lattices.
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Lattice (group)
In mathematics, especially in geometry and group theory, a lattice in R is a discrete subgroup of R which spans the real vector space R. Every lattice in R can be generated from a basis for the vector space by forming all linear combinations with integer coefficients. A lattice may be viewed as a regular tiling of a space by a primitive cell. Lattices have many significant applications in pure mathematics, particularly in connection to Lie algebras, number theory and group theory.
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Cryptography
Cryptography is the practice and study of techniques for secure communication in the presence of third parties. More generally, it is about constructing and analyzing protocols that overcome the influence of adversaries and which are related to various aspects in information security such as data confidentiality, data integrity, and authentication. Modern cryptography intersects the disciplines of mathematics, computer science, and electrical engineering.
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Cryptosystem
There are two different meanings of the word cryptosystem. One is used by the cryptographic community, while the other is the meaning understood by the public.
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Public-key cryptography
Public-key cryptography refers to a cryptographic system requiring two separate keys, one to lock or encrypt the plaintext, and one to unlock or decrypt the cyphertext. Neither key will do both functions. One of these keys is published or public and the other is kept private. If the lock/encryption key is the one published then the system enables private communication from the public to the unlocking key's owner.
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Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal conceptually generalizes the property of certain subsets of the integers, such as the "even numbers" or "multiples of 3", that the product of any element of the ring with an element of the subset is again in the subset: the product of any integer with an even integer is again an even integer. An ideal is therefore said to absorb the elements of the ring under multiplication.
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