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In computer science, lattice problems are a class of optimization problems related to mathematical objects called lattices.The conjectured intractability of such problems is central to the construction of secure lattice-based cryptosystems: lattice problems are an example of NP-hard problems which have been shown to be average-case hard, providing a test case for the security of cryptographic ...
Lattice-based cryptography began in 1996 from a seminal work by Miklós Ajtai [1] who presented a family of one-way functions based on SIS problem. He showed that it is secure in an average case if the shortest vector problem (where = for some constant >) is hard in a worst-case scenario. Average case problems are the problems that are hard to ...
In 1996, Miklós Ajtai introduced the first lattice-based cryptographic construction whose security could be based on the hardness of well-studied lattice problems, [3] 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. [4]
For quantum computers, Factoring and Discrete Log problems are easy, but lattice problems are conjectured to be hard. [13] This makes some lattice-based cryptosystems candidates for post-quantum cryptography. Some cryptosystems that rely on hardness of lattice problems include: NTRU (both NTRUEncrypt and NTRUSign)
In general terms, ideal lattices are lattices corresponding to ideals in rings of the form [] / for some irreducible polynomial of degree . [1] All of the definitions of ideal lattices from prior work are instances of the following general notion: let be a ring whose additive group is isomorphic to (i.e., it is a free -module of rank), and let be an additive isomorphism mapping to some lattice ...
An early successful application of the LLL algorithm was its use by Andrew Odlyzko and Herman te Riele in disproving Mertens conjecture. [5]The LLL algorithm has found numerous other applications in MIMO detection algorithms [6] and cryptanalysis of public-key encryption schemes: knapsack cryptosystems, RSA with particular settings, NTRUEncrypt, and so forth.
In cryptography, learning with errors (LWE) is a mathematical problem that is widely used to create secure encryption algorithms. [1] It is based on the idea of representing secret information as a set of equations with errors. In other words, LWE is a way to hide the value of a secret by introducing noise to it. [2]
IEEE P1363 is an Institute of Electrical and Electronics Engineers (IEEE) standardization project for public-key cryptography. It includes specifications for: Traditional public-key cryptography (IEEE Std 1363-2000 and 1363a-2004) Lattice-based public-key cryptography (IEEE Std 1363.1-2008) Password-based public-key cryptography (IEEE Std 1363. ...