All Questions
Tagged with cryptography nt.number-theory
69 questions
44
votes
1
answer
17k
views
Conjecturally unsafe RSA primes $p=27a^2+27a+7$
We got strong numerical evidence that primes of the form $p=27a^2+27a+7$
are unsafe for cryptographic purposes since they can be found in the factorization.
Consider the following generic factoring ...
- 25.4k
26
votes
4
answers
6k
views
Discrete logs vs. factoring
One thing that I've never quite understood is why computing discrete logarithms (in the multiplicative group mod p) and factoring seem to be so closely related. I don't think that there's a reduction ...
- 12.6k
23
votes
5
answers
1k
views
Securing privacy of "who communicates with whom" under Orwell-like conditions
Assume that there is a big and powerful country with an
information-greedy secret service which has backdoors to all internet nodes
throughout the world which permit him to observe all exchanged data ...
- 19.6k
20
votes
1
answer
1k
views
Cryptography and elliptic curves
Cryptography sometimes uses elliptic curves over finite fields. Does cryptography also use elliptic curves over $\mathbb{Q}$ or rational points on them?
- 281
16
votes
4
answers
3k
views
Zero-knowledge proof of positivity
If I have committed to a number x by revealing g^x mod p, can I prove that 0 < x mod (p-1) < (p-1)/2, i.e. that x is positive, without leaking any more information about x?
My bounty is ending ...
10
votes
4
answers
1k
views
Number theory in symmetric cryptography
One of the most famous application of number theory is the RSA cryptosystem, which essentially initiated asymmetric cryptography.
I wonder if there are applications of number theory also in symmetric ...
- 111
10
votes
3
answers
3k
views
Reduction from factoring to solving Pell equation
The paper Polynomial-Time Quantum Algorithms for Pell's Equation and the Principal Ideal Problem claims
There are reductions from factoring to solving Pell’s equation, and from solving Pell’s
...
- 25.4k
9
votes
3
answers
576
views
"Most Similar Vector Problem" on an Integer Lattice?
I am currently working on problem that I think could be expressed as an integer lattice problem.
Given $u \in \mathbb{R}^n$ and a bounded integer lattice $L = \mathbb{Z}^n \cap [-M,M]^n$ I would like ...
- 379
6
votes
5
answers
6k
views
Analog to the Chinese Remainder Theorem in groups other than Z_n.
The idea hit me when I was in my Elliptic Curve Cryptography class. $Z_n \leftrightarrow Z_{f_1} \times Z_{f_2} \times ...$ where $f_1 \times f_2 \times ... = n$ and $\{f_1, f_2, ...\}$ are pairwise ...
- 359
6
votes
1
answer
566
views
Public key cryptography based on non-invertible matrices?
Added Wed 13 Apr 2022
I have written a short note with experimental data,
which shows not all pseudo keys are good keys.
Public key cryptography based on non-invertible matrices
We got public key ...
- 25.4k
6
votes
1
answer
417
views
Groups in which Computational Diffie Hellman is in $P$ but Discrete Logarithm is not known to be in $P$
The Computational Diffie Hellman (CDH) problem is to compute $g^{XY}$ given $g^X$ and $g^Y$ where $g$ generates the group. The Discrete Logarithm (DLOG) problem is to compute $X$ given $g^X$. The ...
- 13.9k
5
votes
1
answer
359
views
Discrete logarithm and the sequence $a(n)=(g^n \bmod p)^{p-1} \bmod p^2$
Let $p$ be prime and $g,n$ integers.
Define $a(n)=(g^n \bmod p)^{p-1} \bmod p^2$
By mod p we don't mean congruence, but the reduction modulo $p$ operator. $A \bmod ...
- 25.4k
5
votes
0
answers
317
views
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
3
answers
5k
views
modular exponentation for RSA, why is 2^16 + 1 commonly chosen?
I know that the number 216 + 1 is commonly used for RSA, since 0b 1 0000 0000 0000 0001 only contains two 1 bits. Many sites explain that this makes modular exponentiation faster, but I haven't come ...
- 59
4
votes
2
answers
501
views
Breaking the RSA encryption based on a $(e,N)$ given an integer $w \neq 0$ such that $e^w = 1 \mod(N)$?
In his book 'Forcing with Random Variables and Proof Complexity' Jan Krajíček claims (p.154) that it is possible to break the RSA encryption with public key $(e,N)$ if one has has an integer $w \neq ...
- 43
4
votes
1
answer
362
views
What is meant by a meet-in-the-middle approach?
I'm studing C. Petit's work "Faster algorithms for isogeny problems using torsion point images" (link) and he talks about meet-in-the-middle approach/strategy for solve some isogenies ...
- 73
4
votes
1
answer
103
views
On the average density of non-zero digits of NAFs of fixed length
An NAF is a non-adjacent form of a positive integer $k$.
One of the five properties of NAFs is "The average density of non-zero digits among all NAFs of length $l$ is approximately $1/3$."
...
- 41
4
votes
1
answer
288
views
Is total degree version and $x,y$ degree version of Coppersmith's theorem correct?
The notes here https://web.eecs.umich.edu/~cpeikert/lic13/lec04.pdf have the note 'Small decryption exponent $d$: so far the best known attack recovers $d$ if it is less than $N^{.292}$. This uses a ...
- 13.9k
4
votes
1
answer
448
views
Density of integers with a large rough divisor
Let $N < 2^a$ be a positive integer chosen uniformly at random. Let $\tilde{N}$ be the result of removing from $N$ all its prime factors less than $2^b$. What is the probability that $\tilde{N}$ is ...
- 2,967
4
votes
1
answer
425
views
Is it hard to decide whether a matrix is a square of another matrix?
According to the well-know quadratic residue (QR) theory over integers, we know that it is hard to decide whether a given integer $m\in\mathbb Z_N$ is a quadratic residue (i.e., a square of another ...
- 475
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
views
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
3
answers
4k
views
Recovering $\Phi(n)$ from a multiple?
I've been attending a series of lectures on Cryptography from an engineering perspective, which means that most of the assertions made are supplied without proof... here's one that the lecturer couldn'...
- 873
3
votes
2
answers
360
views
Reference describing supersingular elliptic curves over algebraically closed field in characteristic 2
I'm looking for a reference for the fact that over an algebraically closed field of characteristic two, there is (essentially) only one supersingular elliptic curve.
This fact appears on Wikipedia, ...
- 31
3
votes
1
answer
138
views
Given $n, c$ find $a>1,b$ such that $b ^ a \equiv c \pmod n$
Given a natural number $n$ (of unknown factorization) and an arbitrary number $c \in \mathbb{Z}^*_n$ (the set of natural numbers smaller than $n$ and coprime to it), is there an efficient algorithm ...
- 271
3
votes
1
answer
382
views
Equivalence between Diffie Hellman and Discrete Log
For which non-trivial groups, do we know that the Diffie Hellman problem and the Discrete Log are equivalent?
Is there any group for which we suspect them to be different?
Could there be a finite ...
user76479
3
votes
1
answer
266
views
p-adic logarithms with fixed precision
Probably this is easy, but we would like to see it on paper.
Let $p$ be prime and $D,g,n$ positive integers.
Let $A=g^n \bmod p^D$.
Let $\log(p,a,D)$ be the p-adic logarithm with precision $D$.
In ...
- 25.4k
3
votes
1
answer
137
views
Subexponential algorithms that apply only one of factoring and discrete logarithm?
Shor (quantum polynomial), Number Field Sieve (subexponential), Pollard rho (square root) all have both factoring and discrete logarithm over $\mathbb F_p^*$ variants.
What are the subexponential ...
- 13.9k
3
votes
1
answer
164
views
Mestre-type algorithm for higher-genus curves?
Is there an analogous algorithm for genus $g>2$ curves that, given a complete set of invariants, outputs a curve with those invariants?
(I'm interested in particular in $g=3$.)
Any references ...
- 61
3
votes
0
answers
85
views
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
1
answer
172
views
On roots of irreducible quadratics modulo composites
Assume factorization of $N$ is unknown. What is the best complexity we know to find roots of the irreducible equation $$ax^2+bx+c\equiv0\bmod N?$$
Is this problem equivalent to any hardness results?
- 13.9k
2
votes
2
answers
235
views
On fixed point probability in discrete logarithm
Fix an integer $n>2$.
Question. What is the probability that, for a given $h\in\Bbb Z_n,$ there is no $$x\in[0,\varphi(n)-1]\cap\Bbb Z$$ such that
$h^{x\bmod\varphi(n)}\equiv x\bmod n$?
- 13.9k
2
votes
1
answer
162
views
DL-problem on abelian variety
Let $A$ be an abelian variety over $\mathbb{F_q}$ with dimension $n$. Let $q$ be a constant.
Is there polynomial algorithm of finding discrete logarithm in $A$?
UPD: really I don't undestend: can we ...
- 2,576
2
votes
3
answers
398
views
Generating a set of integer passwords that can be securely authenticated
First, apologies for the title. This is an odd question, and I couldn't come up with a simple title for it.
My question is as follows.
Given a positive integer $k$, determine a set of properties $S$ ...
- 135
2
votes
1
answer
262
views
Are there any homomorphic analog error correction code?
Are there any analog error correction codes that are additively and multiplicatively homomorphic?
- 41
2
votes
1
answer
258
views
Does this modification of the General Number Field Sieve factor integers?
The General Number Field Sieve
factors composite $n$ basically this way.
Select homogeneous polynomials with integer coefficients $f(x,y),g(x,y)$
s.t. $f(x,1),g(x,1)$ have common root modulo $n$ but ...
- 25.4k
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
views
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
views
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
views
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
votes
0
answers
145
views
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
2
votes
0
answers
688
views
Elliptic Curves and cryptography. Recommended Reading [closed]
I have been studying RSA cryptography and want to extend this to ECC. I am interested in any books on the topic, that start off with basic principles of elliptic curves as I have almost zero knowledge ...
- 133
1
vote
1
answer
138
views
How to compute the Müller modular polynomials?
According to R.A.Kazmi's dissertation "Isogenies and Cryptography" (Page 22), given an isogeny degree $l$, the Müller modular polynomials are defined as
$$G_l(x,y)=\sum_{r=0}^{l+1}\sum_{k=0}^{v}a_{...
- 475
1
vote
1
answer
142
views
Cryptography with general RSA type integers?
Denote $\mathcal N_r=\{n\in\mathbb Z:\exists\mbox{ distinct equal bit primes }p_1,\dots,p_r:n=p_1p_2\dots p_{r-1}p_r\}$.
$\mathcal N_1$ refers to primes and $\mathcal N_2$ referes to balanced ...
- 13.9k
1
vote
1
answer
184
views
Deduce kernel of isogeny from action on torsion points
I'm stuck with the following problem:
In Petit's work "Faster Algorithms for Isogeny Problems using Torsion Point Images", p. 8, he says that we can deduce $\ker \psi_{N_2}$ knowing the ...
- 73
1
vote
1
answer
799
views
Finding cyclic subgroups of points on elliptic curves for isogeny based cryptography
Isogeny based cryptography is one of the newest post-quantum cryptography. Hardness of this system is based on finding isogeny between two elliptic curves. Also this is a theorem:
Elliptic curves ...
