Short integer solution (SIS) andring-SIS problems are twoaverage-case problems that are used inlattice-based cryptography constructions. Lattice-based cryptography began in 1996 from a seminal work byMiklós Ajtai^[1] who presented a family ofone-way functions based on SIS problem. He showed that it is secure in an average case if theshortest vector problem${\displaystyle {\mathrm{S}\mathrm{V}\mathrm{P}}_{\gamma }}$ (where${\displaystyle \gamma =n^c}$ for some constant${\displaystyle c>0}$) is hard in the worst case.

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.

Afull ranklattice${\displaystyle \mathfrak{L}\subset {\mathbb{R}}^n}$ is a set of integer linear combinations of${\displaystyle n}$ linearly independent vectors${\displaystyle \{b_1,\dots ,b_n\}}$, namedbasis:

${\displaystyle \mathfrak{L}(b_1,\dots ,b_n)=\left\{\sum _{i=1}^n{z}_i{b}_i:z_i\in \mathbb{Z}\right\}=\{B\mathit{z}:\mathit{z}\in {\mathbb{Z}}^n\}}$

where${\displaystyle B\in {\mathbb{R}}^{n\times n}}$ is a matrix having basis vectors in its columns.

Remark: Given${\displaystyle B_1,B_2}$ two bases for lattice${\displaystyle \mathfrak{L}}$, there exist unimodular matrices${\displaystyle U_1}$ such that${\displaystyle B_1=B_2{U}_1^{-1},B_2=B_1{U}_1}$.

Definition: Rotationalshift operator on${\displaystyle {\mathbb{R}}^n(n\ge 2)}$ is denoted by${\displaystyle \mathrm{rot}}$, and is defined as:

${\displaystyle \mathrm{\forall }\mathit{x}=(x_1,\dots ,x_{n-1},x_n)\in {\mathbb{R}}^n:\mathrm{rot}(x_1,\dots ,x_{n-1},x_n)=(x_n,x_1,\dots ,x_{n-1})}$

Micciancio introducedcyclic lattices in his work in generalizing the compactknapsack problem to arbitrary rings.^[2] A cyclic lattice is a lattice that is closed under rotational shift operator. Formally, cyclic lattices are defined as follows:

Definition: A lattice${\displaystyle \mathfrak{L}\subseteq {\mathbb{Z}}^n}$ is cyclic if${\displaystyle \mathrm{\forall }\mathit{x}\in \mathfrak{L}:\mathrm{rot}(\mathit{x})\in \mathfrak{L}}$.

Examples:^[3]

1. ${\displaystyle {\mathbb{Z}}^n}$ itself is a cyclic lattice.
2. Lattices corresponding to any ideal in the quotient polynomial ring${\displaystyle R=\mathbb{Z}[x]/(x^n-1)}$ are cyclic:

consider the quotient polynomial ring${\displaystyle R=\mathbb{Z}[x]/(x^n-1)}$, and let${\displaystyle p(x)}$ be some polynomial in${\displaystyle R}$, i.e.${\displaystyle p(x)=\sum _{i=0}^{n-1}{a}_i{x}^i}$ where${\displaystyle a_i\in \mathbb{Z}}$ for${\displaystyle i=0,\dots ,n-1}$.

Define the embedding coefficient${\displaystyle \mathbb{Z}}$-module isomorphism${\displaystyle \rho }$ as:

Let${\displaystyle I\subset R}$ be an ideal. The lattice corresponding to ideal${\displaystyle I\subset R}$, denoted by${\displaystyle {\mathfrak{L}}_I}$, is a sublattice of${\displaystyle {\mathbb{Z}}^n}$, and is defined as

${\displaystyle {\mathfrak{L}}_I:=\rho (I)=\left\{(a_0,\dots ,a_{n-1})\mid \sum _{i=0}^{n-1}{a}_i{x}^i\in I\right\}\subset {\mathbb{Z}}^n.}$

Theorem:${\displaystyle \mathfrak{L}\subset {\mathbb{Z}}^n}$ is cyclic if and only if${\displaystyle \mathfrak{L}}$ corresponds to some ideal${\displaystyle I}$ in the quotient polynomial ring${\displaystyle R=\mathbb{Z}[x]/(x^n-1)}$.

proof:${\displaystyle \Leftarrow )}$ We have:

${\displaystyle \mathfrak{L}={\mathfrak{L}}_I:=\rho (I)=\left\{(a_0,\dots ,a_{n-1})\mid \sum _{i=0}^{n-1}{a}_i{x}^i\in I\right\}}$

Let${\displaystyle (a_0,\dots ,a_{n-1})}$ be an arbitrary element in${\displaystyle \mathfrak{L}}$. Then, define${\displaystyle p(x)=\sum _{i=0}^{n-1}{a}_i{x}^i\in I}$. But since${\displaystyle I}$ is an ideal, we have${\displaystyle xp(x)\in I}$. Then,${\displaystyle \rho (xp(x))\in {\mathfrak{L}}_I}$. But,${\displaystyle \rho (xp(x))=\mathrm{rot}(a_0,\dots ,a_{n-1})\in {\mathfrak{L}}_I}$. Hence,${\displaystyle \mathfrak{L}}$ is cyclic.

${\displaystyle \Rightarrow )}$

Let${\displaystyle \mathfrak{L}\subset {\mathbb{Z}}^n}$ be a cyclic lattice. Hence${\displaystyle \mathrm{\forall }(a_0,\dots ,a_{n-1})\in \mathfrak{L}:\mathrm{rot}(a_0,\dots ,a_{n-1})\in \mathfrak{L}}$.

Define the set of polynomials${\displaystyle I:=\left\{\sum _{i=0}^{n-1}{a}_i{x}^i\mid (a_0,\dots ,a_{n-1})\in \mathfrak{L}\right\}}$:

1. Since${\displaystyle \mathfrak{L}}$ a lattice and hence an additive subgroup of${\displaystyle {\mathbb{Z}}^n}$,${\displaystyle I\subset R}$ is an additive subgroup of${\displaystyle R}$.
2. Since${\displaystyle \mathfrak{L}}$ is cyclic,${\displaystyle \mathrm{\forall }p(x)\in I:xp(x)\in I}$.

Hence,${\displaystyle I\subset R}$ is an ideal, and consequently,${\displaystyle \mathfrak{L}={\mathfrak{L}}_I}$.

Source:^[4]

Let${\displaystyle f(x)\in \mathbb{Z}[x]}$ be amonic polynomial of degree${\displaystyle n}$. For cryptographic applications,${\displaystyle f(x)}$ is usually selected to be irreducible. The ideal generated by${\displaystyle f(x)}$ is:

${\displaystyle (f(x)):=f(x)\cdot \mathbb{Z}[x]=\{f(x)g(x):\mathrm{\forall }g(x)\in \mathbb{Z}[x]\}.}$

The quotient polynomial ring${\displaystyle R=\mathbb{Z}[x]/(f(x))}$ partitions${\displaystyle \mathbb{Z}[x]}$ into equivalence classes of polynomials of degree at most${\displaystyle n-1}$:

${\displaystyle R=\mathbb{Z}[x]/(f(x))=\left\{\sum _{i=0}^{n-1}{a}_i{x}^i:a_i\in \mathbb{Z}\right\}}$

where addition and multiplication are reduced modulo${\displaystyle f(x)}$.

Consider the embedding coefficient${\displaystyle \mathbb{Z}}$-module isomorphism${\displaystyle \rho }$. Then, each ideal in${\displaystyle R}$ defines a sublattice of${\displaystyle {\mathbb{Z}}^n}$ calledideal lattice.

Definition:${\displaystyle {\mathfrak{L}}_I}$, the lattice corresponding to an ideal${\displaystyle I}$, is called ideal lattice. More precisely, consider a quotient polynomial ring${\displaystyle R=\mathbb{Z}[x]/(p(x))}$, where${\displaystyle (p(x))}$ is the ideal generated by a degree${\displaystyle n}$ polynomial${\displaystyle p(x)\in \mathbb{Z}[x]}$.${\displaystyle {\mathfrak{L}}_I}$, is a sublattice of${\displaystyle {\mathbb{Z}}^n}$, and is defined as:

${\displaystyle {\mathfrak{L}}_I:=\rho (I)=\left\{(a_0,\dots ,a_{n-1})\mid \sum _{i=0}^{n-1}{a}_i{x}^i\in I\right\}\subset {\mathbb{Z}}^n.}$

Remark:^[5]

1. It turns out that${\displaystyle {\text{GapSVP}}_{\gamma }}$ for even small${\displaystyle \gamma =\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}(\mathrm{n})}$ 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${\displaystyle n}$ linearly independent ones of (nearly) the same length. Therefore, on ideal lattices,${\displaystyle {\mathrm{S}\mathrm{V}\mathrm{P}}_{\gamma }}$ and${\displaystyle {\mathrm{S}\mathrm{I}\mathrm{V}\mathrm{P}}_{\gamma }}$ are equivalent with a small loss.^[6] Furthermore, even forquantum algorithms,${\displaystyle {\mathrm{S}\mathrm{V}\mathrm{P}}_{\gamma }}$ and${\displaystyle {\mathrm{S}\mathrm{I}\mathrm{V}\mathrm{P}}_{\gamma }}$ are believed to be very hard in the worst-case scenario.

The Short Integer Solution (SIS) problem is anaverage case problem that is used in lattice-based cryptography constructions. Lattice-based cryptography began in 1996 from a seminal work by Ajtai^[1] who presented a family of one-way functions based on the SIS problem. He showed that it is secure in an average case if${\displaystyle {\mathrm{S}\mathrm{V}\mathrm{P}}_{\gamma }}$ (where${\displaystyle \gamma =n^c}$ for some constant${\displaystyle c>0}$) is hard in a worst-case scenario. Along with applications in classical cryptography, the SIS problem and its variants are utilized in several post-quantum security schemes includingCRYSTALS-Dilithium andFalcon.^[7][8]

Let${\displaystyle A\in {\mathbb{Z}}_q^{n\times m}}$ be an${\displaystyle n\times m}$ matrix with entries in${\displaystyle {\mathbb{Z}}_q}$ that consists of${\displaystyle m}$ uniformly random vectors${\displaystyle {\mathit{a}}_{\mathit{i}}\in {\mathbb{Z}}_q^n}$:${\displaystyle A=[{\mathit{a}}_{\mathbf{1}}|\cdots |{\mathit{a}}_{\mathit{m}}]}$. Find a nonzero vector${\displaystyle \mathit{x}\in {\mathbb{Z}}^m}$ such that for some norm${\displaystyle \Vert \cdot \Vert }$:

• ,
• ${\displaystyle f_A(\mathit{x}):=A\mathit{x}=\mathbf{0}\in {\mathbb{Z}}_q^n}$.

A solution to SIS without the required constraint on the length of the solution (${\displaystyle \Vert \mathit{x}\Vert \le \beta }$) is easy to compute by usingGaussian elimination technique. We also require, otherwise${\displaystyle \mathit{x}=(q,0,\dots ,0)\in {\mathbb{Z}}^m}$ is a trivial solution.

In order to guarantee${\displaystyle f_A(\mathit{x})}$ has non-trivial, short solution, we require:

• ${\displaystyle \beta \ge \sqrt{n\log q}}$, and
• ${\displaystyle m\ge n\log q}$

Theorem: For any${\displaystyle m=\mathrm{poly}(n)}$, any${\displaystyle \beta >0}$, and any sufficiently large${\displaystyle q\ge \beta n^c}$ (for any constant${\displaystyle c>0}$), solving${\displaystyle {\mathrm{SIS}}_{n,m,q,\beta }}$ with nonnegligible probability is at least as hard as solving the${\displaystyle {\mathrm{GapSVP}}_{\gamma }}$ and${\displaystyle {\mathrm{SIVP}}_{\gamma }}$ for some${\displaystyle \gamma =\beta \cdot O(\sqrt{n})}$ with a high probability in the worst-case scenario.

The SIS problem solved over an ideal ring is also called the Ring-SIS or R-SIS problem.^[2][9] This problem considers a quotient polynomial ring${\displaystyle R_q={\mathbb{Z}}_q[x]/(f(x))}$ with${\displaystyle f(x)=x^n-1}$ for some integer${\displaystyle n}$ and with some norm${\displaystyle \Vert \cdot \Vert }$. Of particular interest are cases where there exists integer${\displaystyle k}$ such that${\displaystyle n=2^k}$ as this restricts the quotient to cyclotomic polynomials.^[10]

We then define the problem as follows:

Select${\displaystyle m}$ independent uniformly random elements${\displaystyle a_i\in R_q}$. Define vector${\displaystyle \overrightarrow{\mathit{a}}:=(a_1,\dots ,a_m)\in R_q^m}$. Find a nonzero vector${\displaystyle \overrightarrow{\mathit{z}}:=(z_1,\dots ,z_m)\in R^m}$ such that:

• ,
• ${\displaystyle f_{\overrightarrow{\mathit{a}}}(\overrightarrow{\mathit{z}}):={\overrightarrow{\mathit{a}}}^T.\overrightarrow{\mathit{z}}=\sum _{i=1}^m{a}_i.z_i=0\in R_q}$.

Recall that to guarantee existence of a solution to SIS problem, we require${\displaystyle m\approx n\log q}$. However, Ring-SIS problem provide us with more compactness and efficacy: to guarantee existence of a solution to Ring-SIS problem, we require${\displaystyle m\approx \log q}$ .

Definition: Thenega-circulant matrix of${\displaystyle b}$ is defined as:

When the quotientpolynomial ring is${\displaystyle R=\mathbb{Z}[x]/(x^n+1)}$ for${\displaystyle n=2^k}$, the ring multiplication${\displaystyle a_i.p_i}$ can be efficiently computed by first forming${\displaystyle \mathrm{rot}(a_i)}$, the nega-circulant matrix of${\displaystyle a_i}$, and then multiplying${\displaystyle \mathrm{rot}(a_i)}$ with${\displaystyle \rho (p_i(x))\in Z^n}$, the embedding coefficient vector of${\displaystyle p_i}$ (or, alternatively with${\displaystyle \sigma (p_i(x))\in Z^n}$, the canonical coefficient vector.

Moreover, R-SIS problem is a special case of SIS problem where the matrix${\displaystyle A}$ in the SIS problem is restricted to negacirculant blocks:${\displaystyle A=[\mathrm{rot}(a_1)|\cdots |\mathrm{rot}(a_m)]}$.^[10]

The SIS problem solved over a module lattice is also called the Module-SIS or M-SIS problem. Like R-SIS, this problem considers the quotient polynomial ring${\displaystyle R=\mathbb{Z}[x]/(f(x))}$ and${\displaystyle R_q={\mathbb{Z}}_q[x]/(f(x))}$ for${\displaystyle f(x)=x^n-1}$ with a special interest in cases where${\displaystyle n}$ is a power of 2. Then, let${\displaystyle M}$ be a module of rank${\displaystyle d}$ such that${\displaystyle M\subseteq R^d}$ and let${\displaystyle \Vert \cdot \Vert }$ be an arbitrary norm over${\displaystyle R_q^m}$.

We then define the problem as follows:

Select${\displaystyle m}$ independent uniformly random elements${\displaystyle a_i\in R_q^d}$. Define vector${\displaystyle \overrightarrow{\mathit{a}}:=(a_1,\dots ,a_m)\in R_q^{d\times m}}$. Find a nonzero vector${\displaystyle \overrightarrow{\mathit{z}}:=(z_1,\dots ,z_m)\in R^m}$ such that:

• ,
• ${\displaystyle f_{\overrightarrow{\mathit{a}}}(\overrightarrow{\mathit{z}}):={\overrightarrow{\mathit{a}}}^T.\overrightarrow{\mathit{z}}=\sum _{i=1}^m{a}_i.z_i=0\in R_q^d}$.

While M-SIS is a less compact variant of SIS than R-SIS, the M-SIS problem is asymptotically at least as hard as R-SIS and therefore gives a tighter bound on the hardness assumption of SIS. This makes assuming the hardness of M-SIS a safer, but less efficient underlying assumption when compared to R-SIS.^[10]

• Lattice-based cryptography
• Homomorphic encryption
• Ring learning with errors key exchange
• Post-quantum cryptography
• Lattice problem
• TFNP § PPP

• Number theory
• Lattice-based cryptography
• Post-quantum cryptography
• Computational problems
• Computational hardness assumptions

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