Abstract
As a category of algorithms to solve the shortest lattice vector problem, sieve algorithms have drawn more and more attention due to the prominent performance in recent years. Enumeration algorithms used to perform better in practice even though sieve algorithms are asymptotically faster. Combined with techniques like locality-sensitive hashing and rank reduction, sieve algorithms now are capable of competing with enumeration algorithms. In this work, we study sieve algorithms in solving the shortest vector problem on lattices by categorizing various sieve algorithms and elaborating on ideas and techniques used to improve sieve algorithms. In addition, we present several prospective directions worth future research.
Highlights
Originated in the 19th century, lattice theory has been thoroughly studied by famous mathematicians like Gauss, Minkowski and Hermite et al At first, lattice theory was introduced to solve geometric problems such as sphere packing and sphere covering
For the first time the application of lattice theory in cryptography was as a cryptanalysis tool for that the security of many non-lattice-based cryptosystems can be reduced to the hardness of hard lattice problems
In 1996, Ajtai [1] proposed a method of constructing a random class of lattices, on which finding a short vector is as hard as several worst-case hard lattice problems
Summary
Originated in the 19th century, lattice theory has been thoroughly studied by famous mathematicians like Gauss, Minkowski and Hermite et al At first, lattice theory was introduced to solve geometric problems such as sphere packing and sphere covering. The birth and development of quantum computing have brought challenges to the security of conventional public key cryptosystems such as RSA [69], ElGamal [29] and ECC [42], [63] since that solving the hard mathematical problems they rely on become feasible with quantum algorithms [74], [75] and large-scale quantum computers Under this circumstance, the issue of information security in the upcoming quantum era is attracting more and more attention and cryptographers seek to design ‘‘post-quantum’’ cryptosystems resistant to quantum attacks. Z. Sun et al.: Review of Sieve Algorithms in Solving the Shortest Lattice Vector Problem post-quantum cryptography standardization while 5 out of the 7 cryptographic schemes are based on hard lattice problems.
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