# Questions tagged [hardness-assumptions]

Mathematical problems that are thought to be difficult to solve for all cases in polynomial time

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### Why MDDH is DDH when k in Matrix D is 1?

I am reading a paper Multi-authority ABE for Non-monotonic Access Structures, when the author defines the DDH problem in preliminaries. The definition is I find it is a type of Matrix DDH when $k=1$....
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### Questions about SIS hard problem

The definition of $\mathrm{SIS}_{q,n,m,\beta}$ problem is as below. Let $A\in\mathbb{Z}_q^{n\times m}$ be an $n\times m$ matrix with entries in $\mathbb{Z}_q$ that consists of $m$ uniformly random ...
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### One way function bulit by singular matrices

Our one-way function $F:\mathbb{F}_q^{n\times n}\times \mathbb{F}_q^{n\times n}\to\mathbb{F}_q^{n\times n}$ differs from traditional OWFs, which focus on being hard to invert. Instead, ours aims to be ...
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### Is the discrete root considered a hard problem?

I know that in groups from large prime order the discrete log problem is considered hard. For example, it is hard to compute $x$ from $g^x$ and $g$. Does the same holds for the root problem? For ...
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### Cryptographic assumptions with bilinear pairings

I have two statements which I think are correct: $\beta, s$ are unknown. Only $g^\beta$ is known to the prover but not any of $g^{\beta s^i}$ for $i \ge 1$. If the prover is able to find group ...
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### Inquiries on Strong Diffie-Hellman assumption

According to the SDH assumption, it is hard for an adversary to output a pair $(c,g_1^{\frac{1}{c+\alpha}})$ given a $q+2$ tuple ($g_1$,$g_2$,$g_2^{\alpha}$, ...,$g_2^{{\alpha}^q}$) Now, consider $g_1$...
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### Learning the LWE secret with advice

I am trying to argue about the hardness of LWE, but in a setting that is different from the standard one. Consider the task of learning the LWE secret $s$ from noisy samples. The specifications of the ...
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### What does the "static" assumption mean?

We can prove the security of a cryptographic scheme in standard model and non-standard model. Standard models like using some computational assumptions, on the other hand, non-standard models like ...
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### Special indistinguishability problem

I need some help for the following simple game: An adversary is given a multiplicative group $\mathbb{G}$ and the 4-tuple $(g_1, g_2, g_3, g_1^a \cdot g_2^b \cdot g_3^c)$ where $g_1$, $g_2$ and $g_3$ ...
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### (type-3) Variant of the decisional Diffie-Hellman

At a high level, the Uber assumption states that it is not possible to compute (distinguish) linearly independent elements. In the decisional version, the problem is restricted to $G_T$, but it is ...
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### Average- and worst-case complexity

The terms "average-case", "worst-case" hardness are quite confusing. What do they mean when they say certain problems (like lattices) have an average-case to worst-case ...
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### Given two unrelated generators $G_1$ and $G_2$, and a third with $H = G_1 + G_2$. Is it hard to compute $xG_1$ from $xH$?

Given some group in which both discrete logarithms and the computational Diffie-Hellman problem are hard. Furthermore, two random, unrelated group generators $G_1, G_2$, and a third generator defined ...
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### What's wrong with this simple reduction of discrete logarithms to the Diffie-Hellman problem?

This recent paper shows that discrete logarithms are solvable if you have an oracle for the Diffie–Hellman problem. However, to me, it seems there is a much simpler reduction and I wonder where I am ...
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### Is the composite order matrix-DDH secure?

I recently read a paper that proposed a matrix-DDH which is a matrix variant of DDH assumption. The brief definition is follows: Let $G$ be a group of prime order $q$. Then, the matrix-DDH says that ...
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### Variant of Decisional Diffie Hellman

Given a cryptographic prime $p$ and a generator $g$ of $\mathbb{F}_p$, the Decisional Diffie Hellman problem asks us to distinguish $(g^a, g^b, g^{ab})$ from $(g^a, g^b, g^z)$ for random $a, b, z$. ...
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### Hardness of a modified version of NTRU

Let the modified NTRU be $h=f/g$ such that $f$ is not necessarily a short polynomial, is the NTRU problem still hard in this case?
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### A random access machine with lots of random data on its tape is a stronger assumption than the existence of OWFs

Suppose we have a random access machine with $(n+1)2^n$ random bits on its tape. This assumption is weaker than assuming the existence of a random oracle, but using this assumption we can construct a ...
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### Two LWE samples with the same secret

I am considering the following problem related to Learning with Errors. Recall that the LWE assumption is that no adversary can distinguish $(A,As+e)$ from $(A,u)$ with non-negligible advantage, where ...
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### When does the need of random data become an assumption?

Suppose an encrypion scheme uses a random string to encrypt and decrypt, which is publicly available, such as an IV or nonce. In all the cases I am aware of, the existence of say an IV is not "...
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### Proving stategies for computational properties

As far as I understand, a property is computational if it holds in a computationally-bounded context, so for ANY computationally-bounded involved entity (even if an unbounded one could discover the ...
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### Models and assumptions in the post quantum world

I'm currently trying to get an overview of post-quantum cryptography. Now I'm struggling with correlations and adjustments of the PQ-world and the Modern-world of cryptography. My Questions: Can you ...
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### What kind of assumptions usually go into the design of block ciphers?

What are some standard assumptions made in showing the security of a block cipher? For example, is it commonly assumed that $P\not=NP$? To this end, are there any block ciphers whose security does ...
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### Why is the discrete logarithm problem hard?

Why is the discrete logarithm problem assumed to be hard? Someone else asked the same question but the answers only explain that exponentiation is in $O(\log(n))$ while the fastest known algorithms to ...
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### Breaking CDH also breaks DHI

I am trying to show that by breaking the Computational Diffie-Hellmann (CDH) assumption one also breaks the Diffie-Hellmann inverse assumption. Unfortunately, I am a bit stuck and do not know where to ...
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### Definition of Multilin DDH

I am on the abbreviation mutlin. DDH, which probably stands for mutliniear Decision Diffie Hellmann. I am currently looking for a definition for this term, but unfortunately cannot find a source. Can ...
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### Definition of the strong Diffie Hellman problem

I am looking for the definition of the strong Diffie Hellmann problem. However, I can only find definitions for the $\ell$ or $q$-strong Diffie Hellmann. Is it possible that the strong Diffie Hellman ...
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### LWE with the matrix A repeated

Consider the following version of Learning With Errors. You are either given $(A, As_1 + e_1, As_2 + e_2, \ldots, As_k + e_k)$ or $(A, u_1, u_2, \ldots, u_k)$, where $A$ is an $m \times n$ matrix ...
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### Can Shor's algorithm factor over finite fields/rings/groups?

Shor's algorithm can (efficiently) solve equations of the form: $$n = pq$$ and $$n = x^{2} + y^{2}$$ This question is simple: Can Shor's algorithm solve these equations in polynomial time when they ...
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### Can Shor's algorithm factor over the gaussian integers?

This is related to this question about solving the following expression: $$x^{2} + y^{2}$$ This can be factored over the gaussian integers as $$(x + iy)(x - iy)$$ If one could factor a sum of two ...
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### Why RLWE is lighter than LWE and why we can pick $a_i$ as a permutation of $a_1$ in RLWE but not LWE?

In LWE, we have $$<a_1,s> + e + \mu_1\in \mathbb{Z}_q$$ for a secret key $s\in \{0,1\}^n$ and $a_1\in \mathbb{Z}_q^n$ This is an encryption of a number $\mu_1$. If we want to encrypt $n$ ...
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### What does hard instance mean in cryptography?

I'm learning cryptography recently. I read that for game-based formal security analysis, it is important to embed the hard instance during reduction. Does "hard instance" mean hard-to-solve ...
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### The existence of OWFs vs $\mathbf{EXP} \neq \mathbf{BPP}$

In CRYPTO 2021, Liu and Pass published a paper with title "On the Possibility of Basing Cryptography on $\mathbf{EXP} \neq \mathbf{BPP}$. One of the main results of this work can be interpreted ...
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### LWE and extended trapdoor claw free functions

Let $q \geq 2$ be a prime integer. Consider two functions, given by: $$f(b, x) = Ax + b \cdot u + e~~~(\text{mod}~q),$$ $$g(b, x) = Ax + b \cdot (As + e') + e~~~(\text{mod}~q),$$ where we have: \begin{...
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### Hardness of a variant of the CDH problem

Given $g$, a generator of a multiplicative group (over some finite field or elliptic curve), and the group elements $\left( g^x, g^a, g^b, g^c, g^{x(a+b)}, g^{x(b+c)} \right)$, is possible to ...
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### Does breaking CDH also break DLP? [duplicate]

Does breaking the computational Diffie-Hellman problem in a group also always break discrete logarithms in that group?
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### The security of DDH with multiple instances?

Let $G$ be a finite group of prime order $p$, and $g$ a generator of $G$. The standard DDH is hard to distinguish two distributions  \{ (g, g^a, g^b, g^{ab}) : a, b \leftarrow \mathbb{Z}_p\} \text{ ...
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### Decisional Diffie-Hellman Assumption over Group of Quadratic Residue

Consider the Decision Diffie-Hellman (DDH) over $QR_n$ (the quadratic residue group over $n=pq$ where $p$ and $q$ are safe primes).According to Boneh's paper, DDH should be hard over $QR_n$ (https://...
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