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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Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of splitting fields, and to the understanding of a linear operator.
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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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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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Homomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures. The word homomorphism comes from the ancient Greek language: ¿¿¿¿ (homos) meaning "same" and ¿¿¿¿¿ (morphe) meaning "shape". Isomorphisms, automorphisms, and endomorphisms are all types of homomorphism.
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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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