Questions tagged [hardness-assumptions]

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

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26
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2answers
3k views

What are standard cryptographic assumptions?

I am struggling to understand what is meant by "standard cryptographic assumption". The Wikipedia artice on the Goldwasser–Micali system (GM) reads "GM has the distinction of being the first ...
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1answer
1k views

Can you give me a summary of cryptographic hardness assumptions?

Until recently, I had a link to a website which summarizes up-to-date cryptographic hardness assumptions. But, unfortunately I cannot find it. The webpage is categorized well problems such as, DL ...
13
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4answers
8k views

What is the relation between Discrete Log, Computational Diffie-Hellman and Decisional Diffie-Hellman?

How are the three problems Discrete Logarithm, Computational Diffie-Hellman and Decisional Diffie-Hellman related? From my understanding, since the Discrete Log (DL) Problem is considered hard, then ...
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915 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 ...
9
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1answer
388 views

Nash cryptosystem

In 1955, Nash proposed a cryptosystem in a declassified handwritten letters sent to the National Security Agency. The letters also include a conjecture which is equivalent to the famous $P \ne NP$ ...
9
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1answer
504 views

About random self-reducibility of DDH

In Section 8 of this, Lindell presents a construction of an oblivious transfer protocol which is secure in the malicious model under the following variant of the DDH assumption (page 53): [F]or ...
9
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1answer
126 views

Sum of two squares problem

I would like to know if there is any existing research on the following problem: $$\text{For }a, b \in \mathbb Z \text{, given }n = a^2 + b^2, \text{output }a, b$$. Searching for "sum of squares", "...
8
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2answers
201 views

How hard is Self-DLOG?

This question while asking for something different brings up an intriguing problem: If you can find $x$ such that $x\equiv R^x\pmod p$ then you can break DSA. Now I thought that one might be able to ...
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2answers
936 views

What is a q-type assumption?

I've seen the term "$q$-type assumption" used in a few papers without a definition. A Google search doesn't seem to come up with anything useful either (except the same papers without a definition). ...
6
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1answer
147 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|| \...
6
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1answer
309 views

Is $a b + e \mod P$ hard to invert?

Given $a, p$, $c$ where $c = a b + e \mod p$, can $e$ be recovered? Additional details: $p$ is prime $a, p$ are re-used across many instances of the problem $b, e$ (and by extension $c$) are always ...
4
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1answer
71 views

Global minimum based cryptography?

When using the Back propagation Algorithm for Machine Learning, it is often said finding the global minimum of a cost function over $\mathbb{R}^n$ is very hard, and as $n$ increases it gets even more ...
4
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2answers
98 views

Hardness of LPN problem with small secret

The Learning Parity with Noise (LPN) assumption states that, for a fixed secret $s$ chosen uniformly from $\{0,1\}^n$, then the distribution that outputs $(a,a\cdot s+e)$, where $a$ is uniform in $\{0,...
4
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1answer
163 views

Are there post-quantum cryptosystems with a gap between classical and quantum security?

Is there a gap between classical attacks and quantum attacks against some post-quantum security assumptions? (I'm particularly interested in asymmetric cryptography.) I understand that there is no ...
4
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2answers
305 views

About converting Leveled FHE to Pure FHE and the Circular Security Assumption

As it was pointed out in one answer of this question, the BGV paper has a footnote that says One can obtain a "pure" FHE scheme (with a constant-size public key) from these leveled FHE schemes by ...
4
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1answer
220 views

Is there a group where CDH is easy but DLog is hard?

The question is quite simple: Is there a group where solving the CDH problem can be shown to be easy but solving the discrete logarithm problem is assumed to be hard? Refresher on the problems: CDH:...
4
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2answers
112 views

What is the computational cost of guessing symmetric keys for two ciphers that decrypt to the same input?

A message m is encrypted with two symmetric keys, resulting in two ciphers c, c1 = encrypt(m, key1) c2 = encrypt(m, key2) What is the computational cost of guessing symmetric keys for both ciphers ...
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2answers
192 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
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1answer
115 views

Is this secure?

I have an encryption scheme where the secret keys are $(\alpha_1, \alpha_2)$ and $(y_1, y_2)$. $(y_1', y_2')$ are public information. The ciphertext is: \begin{align*} \mathit{ct}_1 &= r_1 \...
3
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1answer
243 views

How to prove hardness of approximate-GCD problem?

I am trying to prove the security of my system using the hardness assumption of the approximate-GCD problem using contradiction, i.e. If the attacker is able to break in our scheme, then attacker ...
3
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1answer
60 views

General factoring and one-way functions

Let a function $f$ be one-way, if there exists a probabilistic polynomial time algorithm to find the preimage of $y = f(x)$ for uniformly chosen $x$ with non-negligible probability. Define the ...
3
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1answer
332 views

Strong LRSW assumption for type 3 pairing groups

Why does the "strong LRSW" assumption by Ateniese et al. [Untraceable RFID Tags via Insubvertible Encryption, CCS'05] hold ONLY for type 3 pairings and NOT for symmetric pairings? Whereas, the LRSW ...
3
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1answer
217 views

How hard is the following q-Strong Diffie-Hellman problem?

From On the $q$-Strong Diffie-Hellman Problem, the following problem is well-known to be hard. For an randomly chosen element $x \in \mathbb{Z}_p$ and a random generator $g \in \mathbb{G}$, the ...
3
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1answer
467 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}\}...
3
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1answer
126 views

Is there any IND-CPA secure stream cipher with a “standard” hardness assumption?

I've read our recent question: "One-time pad using RSA and Diffie-Hellman functions" which asks about the security of a particular way to convert RSA and discrete exponentiation into a stream cipher. ...
3
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1answer
58 views

Is a composition of computational hardness problems still hard?

It is well known that both $g^x$ and $x^2$ are computational hardness problems in certain rings. But I wonder if the composition of them is still hard? Namely, given $(g, g^x, x^2)$ in a ring $Z_n$ ...
3
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1answer
204 views

Does the following Diffie-Hellman problem hold in bilinear groups $G\times G \rightarrow G_T$

For every PPT distinguisher A there exists a negligible function $neg(·)$ such that for all $\lambda$ $|\Pr[A(1^\lambda, g,g_1^a,g_1^b,g_1^{ab}) = 1] - \Pr[A(1^\lambda, g,g_1^a,g_1^b,g_1^z) = 1]| \...
3
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1answer
226 views

Generalization of the DL-assumption in bilinear group pair

When thinking about a pairing-based cryptographic scheme, I encountered the following problem. Let $e \colon G_1, G_2 \to G_T$ be a Type 3 pairing. Then: Given $P, zP \in G_1$ and $Q, zQ \in G_2$, ...
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74 views

Assuming the difference between success probabilities to be positive

Context: I am currently studying Theorem 3.1 in the paper Number-Theoretic Constructions of Efficient Pseudo-Random Functions by Naor and Reingold. The theorem basically states the randomized self-...
3
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Is the ring learning with errors problem still hard if the errors are drawn from some subspace?

Let $R=\mathbb{Z}_p[x]/x^n+1$ be the ring used in normal RLWE, which is linear space over $\mathbb{Z}_p$ with dimension of $n$, let $S$ be a linear subspace of $R$ which described by linear ...
2
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4answers
318 views

Academic breach revealed too late

Do you know if is has already happened (since 1980), that someone (academic or not) has "broken" (even in a weak sense) some cryptographic assumption, but has chosen to not first publish and directly ...
2
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2answers
199 views

More general - what is the hard problem of recovering r from r*p mod q?

I would like to know the cryptographic hard problem that is most closely tied to recovering integer $r$ from the modular product $r\times p\mod q$. (This is a simplification of an earlier post that ...
2
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2answers
712 views

Is computing roots moduli a composite $N$ a hard problem without knowing the factorization of $N$?

Suppose that we are given $\mathbb{Z}_{N}$ and an element $x^u \in \mathbb{Z}_{N}$ with $u \in (0,l]$ where $l$ is the bit-size of $N$. Is it difficult to recover $x$ by knowing $u$ without knowing ...
2
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1answer
71 views

Does DDH hold in $Z_N^*$ where N is product of 2 large primes

I know that DDH does not hold in $Z_p^*$ because given $g, g^a, g^b, x$, we can compute legendre symbols of $g^a, g^b$ and compare it with legendre symbol of $x$. The same attack doesn't work when we ...
2
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1answer
812 views

Is factorization modulo a product of primes an NP-hard problem?

For example, let, $p$ and $q$ be two large prime numbers. We set $n = p \cdot q$. Now, let $a \cdot b = c \pmod n$. Given $c$ and $n$, is finding the factors $a$ and $b$ computationally difficult? I ...
2
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2answers
1k views

Decisional Diffie-Hellman assumption vs decisional bilinear Diffie-Hellman assumption

For the Decisional Diffie-Hellman (DDH) assumption we know that: Given $g^a$ and $g^b$ for uniformly and independently chosen $a,b \in Z_p$ the value of $g^{ab}$ looks like a random value in group $\...
2
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1answer
101 views

Bit-strength of discrete logarithm for a group of integers modulo a safe prime

Preliminaries Let $p$ be a safe prime number. Let $\mathbb{Z}_p^*$ be the multiplicative group of integers modulo $p$. We have $\mathbb{Z}_p = \{\,a \in \mathbb{Z} \mid 1 \le a \lt p\,\}$ . Let $g \...
2
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1answer
163 views

Universal forgery based on mathematical problem

It is known that DSA admits universal forgery under assumption that the Attacker can solve the equation $x\equiv R^x\pmod p.$ Are there any other protocols admitting universal forgery based on non-...
2
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1answer
225 views

Bilinear map assumpion

Is there an assumption that says from a tag $k\cdot e(g,g_1)^{rx}$ ($k,r$ are secret) it is difficult to forge it with some x': $k\cdot e(g,g_1)^{rx'}$, as long as you cannot solve DL in $\mathbb{G}_1$...
2
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1answer
286 views

What is the notion of an interactive assumption?

In this paper: Sequential Aggregate Signatures with Short Public Keys: Design, Analysis and Implementation Studies the authors sell the paper as the first who propose Aggregate signatures without ...
2
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1answer
243 views

Gap problem for Learning With Errors

Informally, a "Gap problem" arises when solving the computational (or search) version using an oracle for the decisional version. This definition of Gap Problem was introduced by Okamoto and ...
2
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1answer
469 views

How to generate hard subset sum instances

A subset sum problem can be defined as: Given a set of integers $S$ A target integer $x$ Find some subset of elements $s \in S$ such that $\sum_0^{n}s_i = x$ The "density" of a subset sum problem ...
2
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1answer
113 views

Is this problem based on discrete polynomials modulo $(x^3-1)$ strong?

We start working with the Ring $R=\left(\mathbb{Z}/p\mathbb{Z}\right)\left[x\right]/\left(x^{3}-1\right)$, $p$ prime, i.e. degree two polynomials with coefficients modulo $p$ modulo $x^{3}-1$. As $x^{...
2
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1answer
65 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 $s \in \mathbb{Z}_q^{n}$ ...
2
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0answers
41 views

Assumptions underlying the soundness of STARKs

STARKs have recently received quite a lot of attention due to their small proof size and supposedly simple assumptions. The paper introduction itself seems to mainly state that their construction is ...
2
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0answers
106 views

What are the known attacks on $\phi$-hiding assumption? How to chose its parameters?

$\phi$-hiding assumption states the following. Sample 2 random primes $e_0$ and $e_1$ in the range $[5, 2^{\lambda/4}]$. Sample $N = pq$ of length $\lambda$ ($p$ and $q$ are large primes of length $0....
2
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0answers
130 views

Efficient way of knowing large factors of $\phi(n)$ given small prime factors and $n$

Knowing large prime factor$(r > n^{1/4})$ of $\phi(n)$ can easily factorize n and hence learn $\phi(n)$. If we have knowledge on all small prime factors $(2< r_i << n^{1/4})$ of $\phi(n)$...
2
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0answers
39 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 ...
2
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1answer
515 views

q-Strong Bilinear Diffie-Hellman

I'm having trouble finding the first paper that introduced the $q$-Strong Bilinear Diffie-Hellman ($q$-SBDH) assumption which, roughly speaking, is: Let $\mathbb{G},\mathbb{G}_T$ be two groups of ...
2
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1answer
129 views

Security model scalar multiplication in NaCl

I'm having trouble understanding what the “security model” for “scalar multiplication” in NaCl is. Security model crypto_scalarmult is designed to be strong ...