Questions tagged [sis]
For questions involving/related to the Short Integer Solutions(SIS) problem.
26
questions
1
vote
0answers
22 views
polynomial time reduction from SIS to decisional-LWE?
Is the claim "If there is an efficient algorithm that solves SIS, then there is an efficient algorithm that solves decisional LWE" is sufficient?
or,
Is the claim above is equivalent to the ...
1
vote
1answer
76 views
Frobenius inner product polynomial rings
I'm trying to implement the zero-knowledge proof presented in this paper. The proof has a rejection step (page 14), which can be computed as follows:
Where B and Z are in $R^{m \times n}$ for some ...
1
vote
1answer
62 views
SIS vs LWE Problem
The Ajtai one way function is defined by
$$f_A(x)= Ax \; mod\; q $$ where the x $\in \{0,1\}^m$ and A $\in \mathbb{Z_q}^{n \times m}$. $f_A(x)$ is one way function ( Ajtai 96)
While the Regev One way ...
3
votes
1answer
107 views
SIS without the modulus
Consider the following modification to the Short Integer Solution (SIS) problem:
Let $n$ be an integer and $\alpha=\alpha(n),\beta=\beta(n),m=m(n)>\Omega(n\log \alpha)$ be functions of $n$. Sample ...
2
votes
1answer
164 views
The equivalence of SIS and ISIS(Inhomogeneous SIS)
I would like to know whether these two problems are equivalent or not, namely:
$SIS_\alpha$: Given $A \in \mathbb{Z}_q^{n\times m}$ find $ e \in \mathbb{Z}_q^{m}$ such that $ Ae = 0$ and and $\|e\| \...
1
vote
1answer
92 views
Solving modular matrix equations via Gaussian elimination or System of linear equations (SIS assumption?)
Suppose $S \in \mathbb{Z}_q^{m \times m}$, and the norm of $S$ is less than an upper-bound $\beta$.
Additionally, $A_1, \cdots, A_k, C_1, \cdots, C_k \in \mathbb{Z}_q^{m \times n}$.
Here, $k \geq m>...
1
vote
1answer
39 views
Reduction of decison SIS
In Lyu12, Lemma 3.6 is as follows.
Lemma 3.6 For any non-negative integer $\alpha$ such that $gcd(2\alpha+1, q)=1$, there is a polynomial time reduction from the $SIS_{q, n, m, d}$ decsion ...
2
votes
1answer
60 views
average-case SIS and average-case BDD
In lattice based cryptography, we say the average-case SIS (short integer solution) problem because it is such kind problem
" $A \stackrel{\$}{\leftarrow} \mathbb{Z}^{n\times m}_{...
7
votes
1answer
183 views
Is LPN not as important as LWE and SVP?
I've been learning about lattice cryptography and have noticed that most resources such as this survey by Chris Peikart, the Winter School on Lattice Cryptography etc don't include material on LPN, ...
5
votes
0answers
135 views
Parameters for high density SIS
I am considering the SIS problem of finding $x\in \mathbb{Z}^m$ such that for random $A\in\mathbb{Z}_q^{n\times m}$, $Ax=0$ and $\lVert x\rVert < \beta$ for some $p$-norm and bound $\beta < q$. ...
6
votes
1answer
883 views
When does the SIS (Short Integer Solution) Lattice-problem start becoming easy (According to the parameters size)?
SIS (Short Integer Solution) Problem : Given $m$ uniformly random vectors $a \in Z_q^n$, grouped as the columns of a matrix $A \in Z_q^{n.m}$, find a nonzero integer vector $z \in Z^m$ with $||z|| \...
2
votes
0answers
68 views
Estimating the Security of SIS-Based Signature, by verifiying a subset of coordinates?
As I understood, the GPV signature scheme works as follows:
KeyGen($1^n$) : Generate a Lattice with public $A \in Z_q^{n.m}$ and a secret trapdoor $t$.
Sign $m$: compute $\vec y = H(m) \in Z_q^n$ ...
2
votes
1answer
119 views
Why does the following SIS-based decision language not make sense?
I'm currently reading about important lattices problems and noticed that while CVP, SVP, and LWE have decisional versions, SIS does not. I read in the question Relation between decisional SIS and ...
1
vote
0answers
68 views
How does the polynomial module impact the security of ring/lattices-based SIS problem?
Consider the following SIS problem: for a function $f_A(s)$=$As$, where $A$ is a fixed, randomly-chosen matrix in $(R_q)^{r \times n}$=$\left(\mathbb{Z}_q[X]/(X^N+1)\right)^{r \times n}$ and $q$ a ...
2
votes
1answer
117 views
Can I connect the hardness of a linear short integer solution problem to that of SIS problem?
As we know, SIS problem is defined as: for a function $f_A(s)$=$As$, where $A$ is a fixed, randomly-chosen matrix in $\mathbb{Z}_q^{r \times n}$, it is hard to find elements $s \in \mathbb{Z}_q^{n}$ ...
10
votes
2answers
1k views
Concrete evidence for the asymptotics of $\lambda_1(\Lambda^\perp(A))$?
A recent eprint paper claims to bound $\lambda_1(\Lambda^\perp(\mathbf{A}))$ for $\mathbf{A}\in\mathbb{Z}^{n\times m}$, a uniformly random matrix, by $O(1)$, specifically by $4$. This has applications ...
3
votes
1answer
53 views
Solutions to $\gamma \equiv \sum_{i=1}^m \xi_i\cdot x_i\bmod p$ with $|x_i| < \ell$
Are there any clear conditions on $p,\ell$ and $m$ under which the equation $\gamma \equiv \sum_{i=1}^m \xi_i\cdot x_i\bmod p$ has at most one solution with $|x_i|<\ell$ with high probability over ...
3
votes
2answers
291 views
How is the matrix A related to the lattice space L in SIS?
Is the matrix $A= (b_1|,...,|b_m)$ where B=$(b_1,...,b_m)$ is the basis of the lattice space, $L$(B)? Not sure if the answer is trivial however I'm having trouble seeing how SIS is a lattice hard ...
3
votes
0answers
61 views
Size of $q$ in reductions from lattice problems to R-SIS
The Short integer solution problem is parameterized by four values:
$n$, the dimension of the vectors that must be added
$m$, the number of samples (dimension of the solution)
$\beta$, upper-bound ...
5
votes
1answer
431 views
Hardness of Short Interger Solution in Lattices
Short Integer Solution ($SIS_{n,m,q,\beta}$) is defined as:
Given a matrix $A \in \mathbb{Z}_{q}^{n \times m}$, find a non-zero vector $x \in \mathbb{Z}^{m}$ such that $A \cdot x = 0\mod q$ and $||x|| ...
2
votes
0answers
157 views
Hardness of $SIS$ and its reduction to an NP-complete problem
Short Integer Solution ($ SIS_\gamma^{(q,n,m,\beta)}$): Given a matrix $A\in Z_{q}^{n×m}$, find $x \in Z^m $, such that $Ax=0\mod q$ and $||x|| \le \beta$
Is $SIS\in NP$ ?
If $SIS \in NP$, then it ...
3
votes
0answers
319 views
Short integer solution lattice problem with q=2
For large values of $q$, we know that there are worst-case lattice problems which reduce to the average-case short integer solution (SIS) problem. Does this means that for $q=2$, the SIS problem is ...
4
votes
1answer
213 views
Is there any reduction from Short Integer Solution to $\textrm{SIVP}_\gamma$
Short Integer Solution (SIS) is proved to be hard by reducing $\textrm{SIVP}_\gamma$ to SIS, i.e., if we solve SIS, then we can solve $\textrm{SIVP}_\gamma$.
Is there any way to reduce an instance of ...
7
votes
1answer
586 views
Relation between decisional SIS and leftover hash lemma in lattices
The semantic security of Regev's cryptosystem [Reg05] is based on the LWE assumption and leftover hash lemma. This lemma implies that because $m \approx (n+1)\log q$ is large enough, so for uniform $A\...
4
votes
1answer
850 views
How to estimate the hardness of SIS instances?
The Short Integer Solution (SIS) problem is to find, given a matrix $A \in \mathbb{F}_q^{n \times m}$ with uniformly random coefficients, a vector $\mathbf{x} \in \mathbb{Z}^m \backslash \{\mathbf{0}\}...
5
votes
2answers
302 views
ZK Proof for SIS
Let $A x = 0 \bmod q$ with $\Vert x \Vert < \beta$ as part of a lattice SIS problem. Does there exist an efficient zero knowledge proof of knowledge for such a solution?
My idea is to use it for ...