Short integer solution problem


Short integer solution and ring-SIS problems are two average-case problems that are used in lattice-based cryptography constructions. Lattice-based cryptography began in 1996 from a seminal work by Ajtai 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 is hard in a worst-case scenario.
Average case problems are the problems that are hard to be solved for some randomly selected instances. For cryptography applications, worst case complexity is not sufficient, and we need to guarantee cryptographic construction are hard based on average case complexity.

Lattices

A full rank lattice is a set of integer linear combinations of linearly independent vectors, named basis:
where is a matrix having basis vectors in its columns.
Remark: Given two bases for lattice, there exist unimodular matrices such that.

Ideal lattice

Definition: Rotational shift operator on is denoted by, and is defined as:

Cyclic lattices

Micciancio introduced cyclic lattices in his work in generalizing the compact knapsack problem to arbitrary rings. A cyclic lattice is a lattice that is closed under rotational shift operator. Formally, cyclic lattices are defined as follows:
Definition: A lattice is cyclic if.
Examples:
  1. itself is a cyclic lattice.
  2. Lattices corresponding to any ideal in the quotient polynomial ring are cyclic:
consider the quotient polynomial ring, and let be some polynomial in, i.e. where for.
Define the embedding coefficient -module isomorphism as:
Let be an ideal. The lattice corresponding to ideal, denoted by, is a sublattice of, and is defined as
Theorem: is cyclic if and only if corresponds to some ideal in the quotient polynomial ring.
proof:
We have:
Let be an arbitrary element in. Then, define. But since is an ideal, we have. Then,. But,. Hence, is cyclic.
Let be a cyclic lattice. Hence.
Define the set of polynomials :
  1. Since a lattice and hence an additive subgroup of, is an additive subgroup of.
  2. Since is cyclic,.
Hence, is an ideal, and consequently,.

Ideal lattices Craig Gentry. http://portal.acm.org/citation.cfm?id=1536414.1536440 [Fully Homomorphic Encryption Using Ideal Lattices]. In ''the 41st ACM Symposium on Theory of Computing (STOC)'', 2009.http://web.cse.ohio-state.edu/~lai/5359-aut13/05.Gentry-FHE-concrete-scheme.pdf

Let be a monic polynomial of degree. For cryptographic applications, is usually selected to be irreducible. The ideal generated by is:
The quotient polynomial ring partitions into equivalence classes of polynomials of degree at most :
where addition and multiplication are reduced modulo.
Consider the embedding coefficient -module isomorphism. Then, each ideal in defines a sublattice of called ideal lattice.
Definition:, the lattice corresponding to an ideal, is called ideal lattice. More precisely, consider a quotient polynomial ring, where is the ideal generated by a degree polynomial. , is a sublattice of, and is defined as:
Remark:
  1. It turns out that Lattice problem| for even small is typically easy on ideal lattices. The intuition is that the algebraic symmetries causes the minimum distance of an ideal to lie within a narrow, easily computable range.
  2. By exploiting the provided algebraic symmetries in ideal lattices, one can convert a short nonzero vector into linearly independent ones of the same length. Therefore, on ideal lattices, Shortest vector problem| and Lattice problem| are equivalent with a small loss. Furthermore, even for quantum algorithms, Shortest vector problem| and Lattice problem| are very hard in the worst-case scenario.

    Short integer solution problem

SIS and Ring-SIS are two average case problems that are used in lattice-based cryptography constructions. Lattice-based cryptography began in 1996 from a seminal work by Ajtai who presented a family of one-way functions based on SIS problem. He showed that it is secure in an average case if is hard in a worst-case scenario.

SIS''n'',''m'',''q'',''β''

Let be an matrix with entries in that consists of uniformly random vectors :. Find a nonzero vector such that:
A solution to SIS without the required constraint on the length of the solution is easy to compute by using Gaussian elimination technique. We also require, otherwise is a trivial solution.
In order to guarantee has non-trivial, short solution, we require:
Theorem: For any, any , and any sufficiently large , solving with nonnegligible probability is at least as hard as solving the and for some with a high probability in the worst-case scenario.

Ring-SIS

Ring-SIS problem, a compact ring-based analogue of SIS problem, was studied in.
They consider quotient polynomial ring with and, respectively, and extend the definition of norm on vectors in to vectors in as follows:
Given a vector where are some polynomial in. Consider the embedding coefficient -module isomorphism :
Let. Define norm as:
Alternatively, a better notion for norm is achieved by exploiting the canonical embedding. The canonical embedding is defined as:
where is the complex root of for.

R-SIS''m'',''q'',''β''

Given the quotient polynomial ring, define
. Select independent uniformly random elements. Define vector. Find a nonzero vector such that:
Recall that to guarantee existence of a solution to SIS problem, we require. However, Ring-SIS problem provide us with more compactness and efficacy: to guarantee existence of a solution to Ring-SIS problem, we require.
Definition: The nega-circulant matrix of is defined as:
When the quotient polynomial ring is for, the ring multiplication can be efficiently computed by first forming, the nega-circulant matrix of, and then multiplying with, the embedding coefficient vector of .
Moreover, R-SIS problem is a special case of SIS problem where the matrix in the SIS problem is restricted to negacirculant blocks:.