All Questions
Tagged with cryptography nt.number-theory
26 questions with no upvoted or accepted answers
5
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317
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Elliptic curve sequences needed for universal forgery
Elliptic Curve Digital Signature Algorithm (ECDSA) admits universal forgery (UF) if the Attacker can solve the equation
$$z=\frac{f_{k-1}(x,y)f_{k+1}(x,y)}{f_{k}(x,y)^2},$$
where $k$ is unknown, $f_{k}...
- 12.3k
4
votes
0
answers
169
views
Square hidden number problem
Suppose I have a mystery number $m$ modulo $p$ that I wish to find. I know the value of $m+x_i^2$ where $x_i$ is randomly chosen modulo $p$ for some large number of different $x_i$, $N$ many, $N \gg \...
- 591
4
votes
0
answers
214
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factorising an integer with certain bound on the factors
Can we count the no. of $x$ where $ p^{\alpha -1} < x < p^{\alpha}$ , $gcd(x, 2p)=1$ and if $d |x$ and $d < p ^{\beta}$ for some $1< \beta<\alpha-1$ then $ \frac {x} {d} > p^{\alpha -...
- 897
3
votes
0
answers
85
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When is the number-theoretic transform of small vectors again small?
I am currently working on an idea in the context of lattice-based cryptography, but the problem that I am currently stuck on seems to have almost nothing to do with lattices anymore.
In particular, my ...
- 131
3
votes
0
answers
215
views
Why we are interested in p>3 Schoof's algorithm
In the Schoof's algorithm we are particularly interested in $char(K)>3$, where $K$ is the field. I know Schoof's algorithm is mostly used over large prime fields. Also, when we are transforming ...
- 149
2
votes
0
answers
110
views
How to apply Pohlig Hellman using a very limited set of auxiliary inputs in that case?
So I was reading about Talotti, Paier, and Miculan - ECC’s Achilles’ Heel: Unveiling Weak Keys in Standardized Curves. The underlying idea is to lift the discrete logarithm problem to $\mathrm{prime}−...
- 87
2
votes
0
answers
184
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Will Coppersmith's method work for this bivariate modular polynomial shape?
I have a bivariate modular polynomial of shape
$$f(x,y)=x^2y-g(x)\equiv 0\bmod q$$
where
$q=(2p-1)(2p+1)$ is a product of two primes $2p-1$ and $2p+1$,
$g(x)\in\mathbb Z[x]$ is of degree four and
$f(...
- 13.9k
2
votes
0
answers
87
views
Complexity of finding solutions of trapdoored polynomial?
Related to this question Cryptography signature scheme based on hardness of finding points on varieties.
Working over $K=\mathbb{Q}[x_1,...,x_n,y_1,...y_m]$.
By abuse of notation, for polynomial $f$, ...
- 25.4k
2
votes
0
answers
96
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Cryptography signature scheme based on hardness of finding points on varieties?
Related to this question Complexity of finding solutions of trapdoored polynomial.
I am trying to build signature scheme based on hardness
of finding points on varieties.
Let $K$ be field and $M=K[x_1,...
- 25.4k
2
votes
0
answers
123
views
On choosing the correct square root of $g^{4n}$ modulo primes
Let $p$ be prime congruent to $3$ modulo $4$.
The discrete logarithm problem asks: given $g,a,p$
such that $g^x \equiv a \pmod{p}$, find $x$.
Assume $g$ is of maximal multiplicative order.
In an ...
- 25.4k
2
votes
0
answers
121
views
How to choose a prime p s.t. n-th cyclotomic polynomial splits into as much as possible irreducible polynomials while p is almost constant size?
The reason I ask this question is that cyclotomic polynomial is critical to the construction of lattice-based cryptography. In most of the existing lattice-based cryptographic schemes, $n$ is usually ...
- 33
2
votes
0
answers
88
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Example of action of an infinitely countable group that has important ergodic/statistical property?
I work in probability and I am looking for an important example of action of an amenable countable group in other areas of math for which the (pointwise) ergodic theorem is actually quite important. ...
- 21
2
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0
answers
145
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Showing that a crypto hash function is not permutation, possibly conditionally?
Let $f$ be some crypto hash function, say MD5 with output $n$ bits. Restrict the input to $n$ bits.
Cryptographer told me it is open problem if such restricted collision
exists, i.e. $f(x)=f(y),x \ne ...
- 25.4k
1
vote
0
answers
69
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Is it in theory possible to perform general Miller’s algorithm inversion as used with the optimal ate pairing with large trace in subexponential time?
Let’s I have the following :
2 curves $G_1$ defined on $F_p$ and $G_2$ being the $G_1$ curve’s twist defined on $F_p^2$ both having the same prime order ; a large trace ; and $F_p^{12}$ as their ...
- 87
1
vote
0
answers
69
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If we allow DH operations in addition to exponentiation and multiplication can we get a lower bound for discrete logarithm?
In https://crypto.stackexchange.com/questions/72969/proof-dlog-is-hard-in-generic-group-model/ it is shown if we allow only exponentiation and multiplication we can get an exponential complexity lower ...
- 13.9k
1
vote
0
answers
102
views
Question on definition of inverse number theoretic transformation
In the paper Porkodi and Arumuganathan - Public key cryptosystem based on number theoretic transforms I found the following statement on the second page regarding the Inverse Number Theoretic ...
- 11
1
vote
0
answers
98
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Hardness of solving $0=\sum_{i=1}^k \operatorname{linear}_i(x_1,\ldots,x_n)^D$ over the rationals
This is related to cryptography and this question
and another question.
In short, we are asking about decomposing multivariate polynomial
as sum of perfect powers of linear polynomials.
Working over $\...
- 25.4k
1
vote
0
answers
133
views
The security of one-time digital signatures from a solution to a diophantine equations
I wonder how well arbitrary Diophantine equations can be used to make one time digital signature schemes.
For our one-time digital signature scheme, the public key is a collection of polynomials $f_1(...
1
vote
0
answers
212
views
Generate algorithmically an elliptic curve with its exact class group structure?
Is it possible to generate an elliptic curve $E$ (randomly), together with knowing its class group $\mathrm{Cl}(\mathcal{O})$ structure? where $\mathcal{O}$ is its endomorphism rings $\mathsf{End}(E)$ ...
- 181
1
vote
0
answers
276
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Method of Coppersmith optimal for multivariate?
It is shown that Coppersmith method yields optimal integer root extraction for univariate polynomials in https://arxiv.org/abs/1605.08065 and a follow up work attempts this for bivariate polynomials ...
- 13.9k
1
vote
0
answers
343
views
Diophantine approximation
Say absolute values of $a,b,c$ is $O(log^{k}{n})$ for some positive constant $k$.
Given positive integer $n$ that is reasonably large, we cannot always find integers $a,b,c$ such that $|a{b^{c}} - n|$ ...
- 800
0
votes
0
answers
141
views
State of the art on attempts to solve the elliptic curve discrete logarithm problem through transfering the problem to a weaker curve
Let an elliptic curve $E$, and 2 points on such curve $P$ and $O$ the methods I’m talking about consist in creating a weaker elliptic curve $F$ and mapping $P$ and $O$ to $F$ while successfully ...
- 87
0
votes
0
answers
122
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Is it in theory possible to create a subexponential algorithm for solving discrete logarithms in multiplicative subgroups or within an Integer range?
As far I understand, when it comes to finite fields, Pollard rho and Pollard’s lambda are still the best algorithm for solving discrete logarithms in a multiplicative subgroup/suborder…
Index calculus ...
- 87
0
votes
0
answers
103
views
On MSB and LSB of Diffie Hellman
Given generator $g$ of multiplicative cyclic group modulo $p$ a prime and two elements $h_1$ and $h_2$ such that there are $x_1$ and $x_2$ respectively satisfying $g^{x_i}=h_i\bmod p$ at every $i\in\{...
- 13.9k
0
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0
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135
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Can factorization of very large numbers be aided by associating them with a series (described below) of quadratic polynomials?
My name is J. Calvin Smith. I graduated in 1979 with a Bachelor of Arts in Mathematics from Georgia College in Milledgeville, Georgia. My Federal career (1979-2012) in the US Department of Defense led ...
0
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0
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562
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Reducing two variable linear Diophantine equation to modular inversion
I'm in the field of secure multiparty computation using Homomrphic encryption or secret sharing. I want to implement a secure protocol to compute the GCD of two encrypted numbers.
To calculate the ...
- 193