Questions tagged [hardness-assumptions]
Mathematical problems that are thought to be difficult to solve for all cases in polynomial time
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questions
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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
votes
1answer
48 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 ...
6
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1answer
115 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
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1answer
82 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
61 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 $s \in \mathbb{Z}_q^{n}$ ...
0
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1answer
46 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
92 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,...
2
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0answers
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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 ...
10
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2answers
900 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 ...
1
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1answer
223 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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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
65 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 ...
4
votes
1answer
203 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:...
2
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0answers
127 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 $\phi(n)$...
9
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1answer
117 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", "...
1
vote
1answer
71 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 ...
1
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1answer
79 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
votes
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
61 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
79 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
296 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 ...
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0answers
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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
399 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
votes
1answer
191 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
votes
1answer
57 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 $Z_n$ ...
1
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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-...
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2answers
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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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vote
1answer
211 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
vote
1answer
74 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
198 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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0answers
117 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
467 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
734 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
113 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
votes
1answer
194 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
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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
votes
2answers
192 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 ...
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1answer
68 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 ...
2
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2answers
985 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 $\...
9
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1answer
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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 $P \ne NP$ ...
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2answers
284 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 ...
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1answer
70 views
Is DL difficult under the group of Unimodular matrices?
Is discrete logarithm assumed to be computationally hard in a non-abelian group as the subgroup of the general linear group under matrix multiplication formed by the unimodular matrices? The two ...
2
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1answer
123 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 ...
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0answers
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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 ...
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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 ...
3
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1answer
235 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
votes
1answer
121 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. ...
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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 ...
