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Dagstuhl Seminar Proceedings

Public-Key Cryptosystems from the Worst-Case Shortest Vector Problem

Chris Peikert

Published Jan 1, 2009• 192 citations
Publisher

Abstract

We construct public-key cryptosystems that are secure assuming the *worst-case* hardness of approximating the shortest vector problem on lattices. Prior cryptosystems with worst-case connections (e.g., the Ajtai-Dwork system) were based either on a *special case* of the shortest vector problem, or on the conjectured hardness of lattice problems for *quantum* algorithms. Our main technical innovation is a reduction from certain variants of the shortest vector problem to corresponding versions of the "learning with errors" (LWE) problem; previously, only a quantum reduction of this kind was known. In addition, we construct new cryptosystems based on LWE, including a very natural chosen ciphertext-secure system that has a much simpler description and tighter underlying worst-case approximation factor than prior constructions. (Duration: 30 minutes, on or before Wednesday.)

DOI
10.4230/dagsemproc.08491.4

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•Jan 1, 2009•0 citations
We construct public-key cryptosystems that are secure assuming the worst-case hardness of approximating the shortest vector problem on lattices. Prior cryptosystems with worst-case connections (e.g., the Ajtai-Dwork system) were based either on a special case of the shortest vector problem, or on the conjectured hardness of lattice problems for quantum algorithms. Our main technical innovation is a reduction from certain variants of the shortest vector problem to corresponding versions of the "learning with errors" (LWE) problem; previously, only a quantum reduction of this kind was known. In addition, we construct new cryptosystems based on LWE, including a very natural chosen ciphertext-secure system that has a much simpler description and tighter underlying worst-case approximation factor than prior constructions. (Duration: 30 minutes, on or before Wednesday.)
We construct public-key cryptosystems that are secure assuming theworst-case hardness of approximating the minimum distance on n-dimensional lattices to within small Poly(n) factors. Prior cryptosystems with worst-case connections were based either on the shortest vector problem for a special class of lattices (Ajtai and Dwork, STOC 1997; Regev, J. ACM 2004), or on the conjectured hardness of lattice problems for quantum algorithms (Regev, STOC 2005). Our main technical innovation is a reduction from variants of the shortest vector problem to corresponding versions of the "learning with errors" (LWE) problem; previously, only a quantum reduction of this kind was known. As an additional contribution, we construct a natural chosen ciphertext-secure cryptosystem having a much simpler description and tighter underlying worst-case approximation factor than prior schemes.
Our main result is a reduction from worst-case lattice problems such as SVP and SIVP to a certain learning problem. This learning problem is a natural extension of the 'learning from parity with error' problem to higher moduli. It can also be viewed as the problem of decoding from a random linear code. This, we believe, gives a strong indication that these problems are hard. Our reduction, however, is quantum. Hence, an efficient solution to the learning problem implies a quantum algorithm for SVP and SIVP. A main open question is whether this reduction can be made classical.Using the main result, we obtain a public-key cryptosystem whose hardness is based on the worst-case quantum hardness of SVP and SIVP. Previous lattice-based public-key cryptosystems such as the one by Ajtai and Dwork were only based on unique-SVP, a special case of SVP. The new cryptosystem is much more efficient than previous cryptosystems: the public key is of size Õ(n2) and encrypting a message increases its size by Õ(n)(in previous cryptosystems these values are Õ(n4) and Õ(n2), respectively). In fact, under the assumption that all parties share a random bit string of length Õ(n2), the size of the public key can be reduced to Õ(n).
Our main result is a reduction from worst-case lattice problems such as GapSVP and SIVP to a certain learning problem. This learning problem is a natural extension of the “learning from parity with error” problem to higher moduli. It can also be viewed as the problem of decoding from a random linear code. This, we believe, gives a strong indication that these problems are hard. Our reduction, however, is quantum. Hence, an efficient solution to the learning problem implies a quantum algorithm for GapSVP and SIVP. A main open question is whether this reduction can be made classical (i.e., nonquantum). We also present a (classical) public-key cryptosystem whose security is based on the hardness of the learning problem. By the main result, its security is also based on the worst-case quantum hardness of GapSVP and SIVP. The new cryptosystem is much more efficient than previous lattice-based cryptosystems: the public key is of size Õ( n 2 ) and encrypting a message increases its size by a factor of Õ( n ) (in previous cryptosystems these values are Õ( n 4 ) and Õ( n 2 ), respectively). In fact, under the assumption that all parties share a random bit string of length Õ( n 2 ), the size of the public key can be reduced to Õ( n ).
We introduce the use of Fourier analysis on lattices as an integral part of a lattice-based construction. The tools we develop provide an elegant description of certain Gaussian distributions around lattice points. Our results include two cryptographic constructions that are based on the worst-case hardness of the unique shortest vector problem. The main result is a new public key cryptosystem whose security guarantee is considerably stronger than previous results ( O ( n 1.5 ) instead of O ( n 7 )). This provides the first alternative to Ajtai and Dwork's original 1996 cryptosystem. Our second result is a family of collision resistant hash functions with an improved security guarantee in terms of the unique shortest vector problem. Surprisingly, both results are derived from one theorem that presents two indistinguishable distributions on the segment [0, 1). It seems that this theorem can have further applications; as an example, we use it to solve an open problem in quantum computation related to the dihedral hidden subgroup problem.
One attractive feature of lattice-based cryptosystems is the existence of security reductions relating the difficulty of breaking the cryptosystem to the difficulty of solving variants of the shortest vector problem (Regev, STOC 2005; Peikert, ePrint 2008). As there are no known polynomial-time algorithms which solve these lattice problems, this implies the asymptotic security of the cryptosystem. However, current lattice-based cryptosystems using the learning with errors (LWE) problem select parameters for which the reduction to the underlying lattice problem gives no meaningful assurance of concrete security. We analyze the runtime of the algorithm constructed in the reductions and select parameters for a cryptosystem under which the reductions give 128-bit security. While the resulting LWE-based cryptosystem is somewhat cumbersome, requiring a dimension of n = 1460, this is less than 2 times the dimension in the recently proposed Frodo cryptosystem (Bos et al., ACM CCS 2016), and could be implemented without catastrophic damage to communication times. We also investigate the runtime necessary for a reduction to give meaningful security assurances for current cryptosystems.
We describe public key encryption schemes with security provably based on the worst case hardness of the approximate Shortest Vector Problem in some structured lattices, called ideal lattices. Under the assumption that the latter is exponentially hard to solve even with a quantum computer, we achieve CPA-security against subexponential at- tacks, with (quasi-)optimal asymptotic performance: if n is the security parameter, both keys are of bit-length O(n) and the amortized costs of both encryption and decryption are O(1) per message bit. Our construc- tion adapts the trapdoor one-way function of Gentry et al. (STOC'08), based on the Learning With Errors problem, to structured lattices. Our main technical tools are an adaptation of Ajtai's trapdoor key genera- tion algorithm (ICALP'99) and a re-interpretation of Regev's quantum reduction between the Bounded Distance Decoding problem and sam- pling short lattice vectors.