Selected schemes for the purpose of key exchange, also called key establishment, key encapsulation and key encapsulation mechanism (KEM): Selected lattice-based cryptographic schemes for the purpose of hashing: Selected schemes for the purpose of homomorphic encryption: In 1999, Nguyen published a critical flaw in the scheme's design. GGH encryption scheme, which is based in the closest vector problem (CVP).Selected schemes for the purpose of encryption: This section presents selected lattice-based schemes, grouped by primitive. Many (though not all) lattice-based cryptographic constructions are known to be secure if SVP is in fact hard in this regime. In linear algebra, a lattice L ⊂ R n, and even with a quantum computer. For example, in 2009, Craig Gentry introduced the first fully homomorphic encryption scheme, which was based on a lattice problem. Much more work has been devoted to constructing additional cryptographic primitives based on LWE and related problems. Since then, much follow-up work has focused on improving Regev's security proof and improving the efficiency of the original scheme. The first lattice-based public-key encryption scheme whose security was proven under worst-case hardness assumptions was introduced by Oded Regev in 2005, together with the Learning with Errors problem (LWE). However, their scheme is not known to be at least as hard as solving a worst-case lattice problem. Silverman introduced a lattice-based public-key encryption scheme, known as NTRU. In 1998, Jeffrey Hoffstein, Jill Pipher, and Joseph H. She then showed a cryptographic hash function whose security is equivalent to the computational hardness of SIS. In 1996, Miklós Ajtai introduced the first lattice-based cryptographic construction whose security could be based on the hardness of well-studied lattice problems, and Cynthia Dwork showed that a certain average-case lattice problem, known as Short Integer Solutions (SIS), is at least as hard to solve as a worst-case lattice problem. Furthermore, many lattice-based constructions are considered to be secure under the assumption that certain well-studied computational lattice problems cannot be solved efficiently. Unlike more widely used and known public-key schemes such as the RSA, Diffie-Hellman or elliptic-curve cryptosystems - which could, theoretically, be defeated using Shor's algorithm on a quantum computer - some lattice-based constructions appear to be resistant to attack by both classical and quantum computers. Lattice-based constructions are currently important candidates for post-quantum cryptography. Lattice-based cryptography is the generic term for constructions of cryptographic primitives that involve lattices, either in the construction itself or in the security proof. IV, Sect.Constructions of cryptographic primitives that involve lattices Kurosh, "Lectures on general algebra", Chelsea (1963) pp. Grätzer, "Lattice theory", Freeman (1971)Ī.G. Beran, "Orthomodular lattices", Reidel (1985) pp. The principal ideal in $L$ determined by $\alpha$ is also denoted (more accurately) by $[\alpha)$.Ī partially ordered set is a complete lattice if and only if it has a zero and every ideal in $L$ is principal. The right principal ideal $R(\alpha)$ contains all the elementsĪnd the two-sided principal ideal $J(\alpha)$ contains all elements of the form The left principal ideal $L(\alpha)$ of a ring $A$ contains, in addition to the element $\alpha$ itself, also all the elements
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