Questions tagged [hardness-assumptions]
Mathematical problems that are thought to be difficult to solve for all cases in polynomial time
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Proof by reduction definition in “Serious Cryptography”: Cipher reduced to hardness problem or other way around?
In Serious Cryptography by Jean-Philippe Aumasson on p. 46, paragraph "Provable Security", it says:
Provable security is about proving that breaking your crypto scheme is at least as hard as ...
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2answers
44 views
Protocol for proof of knowledge of $l$-th root
Assume we have Group G in which the adaptive root assumption holds.
This assumption states that if we choose an element $w$ and after that, if we receive a prime value $l$ it is hard to find the $u$ ...
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1answer
63 views
(How) Is DDH generally broken in groups of composite order?
In a somewhat recent lecture a claim was made that I couldn't back up myself but it got me curious whether it actually holds:
If the order of the group is not prime, then the DDH assumption does ...
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1answer
62 views
Is it possible to depict the bitcoin mining problem as a TSP-metric problem?
I wonder if someone can represent the blockchain encryption problem (used to mine bitcoins) as a TSP-metric problem (where TSP = Traveling-Salesman Problem). Any approach or intuition of the idea to ...
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1answer
75 views
Can quantum algorithms solve the approximate GCD hard problem efficiently?
Some cryptographic schemes are based on the hardness of this problem. The answer to this question determines if those schemes are quantum resistant or not. There are a number similar questions but ...
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1answer
221 views
Is this problem on $\mathbb{Z}_p$ really hard?
I just want to know if there's something obvious that renders this hard problem useless. Not a full cryptoanalisys. Any hint on whatever is welcomed.
We will work with the Ring $\mathbb{Z}_{p}$, $p$ ...
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0answers
97 views
Define a Decryption algorithm on a given group-based Cramer Shoup lite scheme
I am currently working on public key encryption schemes and I want some help to figure out how decryption algorithms work. Suppose we have a public key $pk = (G,p,g,e)$ with $e \in Z^*_p$ . (where $G$ ...
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27 views
If they exist a relation between decisional Diffie-Hellman assumption and composite decisional residuosity assumption
From the cryptographic hardness assumptions, we have DDH and CDR assumptions.
It is known that the composite decisional residuosity assumption is related to a factoring problem, while the DDH is ...
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1answer
121 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
23 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 ...
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2answers
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What are standard cryptographic assumptions?
I am struggling to understand what is meant by "standard cryptographic assumption".
The Wikipedia artice on the GoldwasserāMicali system (GM) reads "GM has the distinction of being the first ...
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4answers
328 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 ...
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1answer
175 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
74 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
76 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=XYXā1,B=XZXā1,Y and Z are public key.And X is secret permutation. then encryption is C=...
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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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127 views
Is Permutation conjugate problem hard?
Let $x$,$y$,$z$ be permutations. Then public key is $z=xyx^{ā1}$ 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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259 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
158 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
69 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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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)?
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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
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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 ...
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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
289 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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121 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....
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1answer
78 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
212 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 ...
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230 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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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
74 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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136 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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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
108 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-...
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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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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
114 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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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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693 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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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
529 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 ...
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1answer
255 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 ...
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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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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 ...
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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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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
248 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 ...
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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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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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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-...
