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Questions tagged [hardness-assumptions]

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

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1answer
106 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^{...
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0answers
21 views

What is the current state of the Conjugacy Search Problem in Matrix and Permutation Groups?

It results that I have come with a mathematical approach to solve the CSP (Conjugacy Search Problem) where the platform group $G$ is either a permutation group or a Matrix Group. From my part I ...
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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4answers
317 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 ...
4
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1answer
161 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 ...
3
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1answer
59 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 ...
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1answer
70 views

Is permutation conjugate search problem many answer? Are there many answer equal? [closed]

If conjugate search problem of permutation is difficult , then there are next cryptosystem will appear. A=XYX−1,B=XZX−1,Y and Z are public key.And X is secret permutation. then encryption is C=...
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2answers
92 views

From the product of two permutation matrices raised to the same power, is it easy to find the power?

Let $A$ and $B$ be two public permutation matrices. If $r$ is a secret power of large number, can we easily find $r$ from $A^rB^r$?
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1answer
114 views

Is Permutation conjugate problem hard?

Let $x$,$y$,$z$ be permutations. Then public key is $z=xyx^{−1}$ and $y$. Is permutation conjugate search problem easy? if yes, how to find $x$ from $z$ and $y$? Let be a is Alice's secret key as ...
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1answer
131 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|| \...
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1answer
96 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 \...
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1answer
64 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 $s \in \mathbb{Z}_q^{n}$ ...
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1answer
49 views

finding sha256 preimage or secp256k1 private key, which is harder?

Is finding a public key that matches hash of pre-commited public key (second layer security) more computationally hard than finding private key to a known public key (attacking secp256k1)?
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,...
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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 ...
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2answers
909 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 ...
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1answer
243 views

KEA assumption details

In order to understand the construction of a zK-SNARK, I have recently been trying to understand the KEA1 assumption in The Knowledge-of-Exponent Assumptions and 3-Round Zero-Knowledge Protocols by ...
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0answers
105 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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1answer
69 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 ...
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2answers
189 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 ...
4
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1answer
217 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:...
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0answers
128 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 $\phi(n)$...
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1answer
70 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 ...
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1answer
125 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", "...
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1answer
72 views

Proving impossibility for a stronger primitive

I have a primitive $A$ which is impossible to prove under some hardness assumption in a black-box way. Now, if another primitive $B$ is stronger than $A$ - in other words $B$ implies $A$ - will it ...
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1answer
95 views

Is DDH hard over this group?

I'm new to DDH. Reading this survey, I noticed that DDH is (believed to be) hard in many groups, but most of them are prime-order groups (the only one that is not is the cyclic subgroup of order $(p-...
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 \...
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1answer
66 views

Diffie Hellman Key Exchange Security wrt the hardness of discrete Logarithm

Hi I am very new to this field and I was wondering how one would prove or disprove the following statements: 1) if the DH key exchange protocol is secure for a finite cyclic group G, then computing ...
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2answers
86 views

Proving security of an encryption scheme

I have an encryption scheme $Enc(pk, m)$ which outputs ciphertext of the form $c_1 = (pk)^r, c_2 = g^r \cdot m$; where $pk$ is the public key of the form $g^a$, $a$ being the corresponding secret key. ...
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1answer
69 views

Arguing security of an encryption scheme

I have an encryption scheme which masks the message m by multiplying it with a secret random group element i.e $g^r \cdot m$. I should allow decryption and get $g^r$ if I have a valid secret key; ...
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1answer
414 views

Comparing stronger vs. weaker assumptions

If I have to argue that an assumption A is stronger than B, should I argue this way:- An adversary breaking the security of A does not translate to breaking the security of B. But, if an adversary ...
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
449 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 ...
3
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1answer
205 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
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 $Z_n$ ...
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2answers
64 views

Is there any Information on the “Modular Approximate Greatest Divisor” problem?

The Approximate Greatest Common Divisor (AGCD) problem is the difficulty of obtaining the value $\ p\ $ when given samples of $\ pq_i + e_i$ for secret values $p, q_i, e_i$. I would like to know how ...
3
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0answers
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-...
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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1answer
228 views

Difficulty of generating a string “The (md5|sha1) hash of this string is (x)” that hashes to (x) vs. regular hash collision between files?

I know that collisions in md5 and sha1 have been demonstrated in particular pairs of files (by incrementing some none viewable portion of a PDF, etc). Would generating the above string be ...
1
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1answer
76 views

Naor Pinkas OT - senders security; reduction to DDH

This question concerns the security of the sender in the Naor Pinkas OT. The protocol can be found here. We can reduce the security to the DDH assumption. How exactly is this done? Can someone ...
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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 ...
2
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1answer
160 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-...
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0answers
125 views

Random self-reducibility and NP

I was reading the Wikipedia page https://en.m.wikipedia.org/wiki/Random_self-reducibility and it states: "If an NP-complete problem is non-adaptively random self-reducible the polynomial hierarchy ...
2
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1answer
503 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
799 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 ...
1
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1answer
116 views

Extracting $ x $ given $ g^x, g^{x + y}, y $

Given a cyclic group $ G $ of prime order $ q $ with generator $ g $ in which computing the discrete logarithm is hard. Is it (still) hard to find the discrete logarithm $ dlog_g(g^x) = x $ if the ...
3
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1answer
201 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]| \...
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1answer
305 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 ...
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 ...
1
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1answer
79 views

What is the hard problem for this algebraic encryption construct?

I would like to know what cryptographic hard problem this reduces to. Select two large prime numbers $p$ and $q$, and let $N=pq$. Select a random positive integer $r$. Compute the encryption of ...