Questions tagged [cryptography]
Questions concerning the mathematics of secure communication. Relevant topics include elliptic curve cryptography, secure key exchanges, and public-key cryptography (eg. the RSA cryptosystem).
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What computational problems would be good proof-of-work problems for cryptocurrency mining?
What computational mathematics problems that could be used as proof-of-work problems for cryptocurrencies? To make this question easier to answer, I want proof-of-work systems that work in ...
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Example of a good Zero Knowledge Proof
I am working on my zero knowledge proofs and I am looking for a good example of a real world proof of this type. An even better answer would be a Zero Knowledge Proof that shows the statement isn't ...
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1
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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 ...
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4
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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 ...
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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 ...
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answer
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Is hyperelliptic cryptography "practical"?
Previosly my impression on this subject was that hyperelliptic cryptography systems (as well as other possible cryptosystems based on abelian varieties of dimension $>1$) have no advantages over ...
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answer
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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?
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Bitcoin Research
I have recently been assigned to advise a student on a senior thesis. She has taken linear algebra, introductory real analysis, and abstract algebra. Her interest is in cryptography. And she has a ...
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Mathematics of privacy?
I wonder to which extent the current public debate on privacy issues (not only by state sniffing, but e.g. by microtargetting ads too an issue) offers interesting questions in mathematics?
Can we ...
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Which hard mathematical problems do you have to solve to earn bitcoins ?
A virtual currency called bitcoins has been in the news recently. It is said that in order to "mine" bitcoins, you have to solve hard mathematical problems.
Now, there are two kinds of mathematical ...
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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 ...
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Factorization when a factor is partially known
Let's say that I have a very large number of the order ($10^{250+}$) which is composite. I have been given one of its factor partially to a significant amount of digits (say 75+). Then, how can I ...
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Will quantum computing kill cryptography ? [closed]
I apologize as this question is not really mathematical, and therefore perhaps not
well-suited for this site. Please feel free to close it if you think it is not. My reason
for asking it here is that ...
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votes
2
answers
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Encrypting a message for multiple recipients
Let $m$ be a secret message that needs to be sent to $n >1$ recipients. Let each recipient $r_i$ have a public key $p_i$ and private key $s_i$. Is there a scheme such that we can encrypt the ...
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Introducing Cryptology to Undergraduates
This summer I am going to give some lectures to some REU students. I am still tossing around ideas for what I am going to talk about, but one thing I would at least like to give one or two lectures on,...
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2
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“The Two Sheriffs” puzzle -2: threshold for security
I've already asked a question “The Two Sheriffs” puzzle with wrong assumption. Yoav Kallus in his amazing answer using Fano plane showed that the problem has a solution in the case of seven suspects.
...
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12
votes
1
answer
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Are there very strongly pseudorandom permutations?
A pseudorandom permutation can be defined formally as a function $\phi$ from $\{0,1\}^k\times\{0,1\}^n$ to $\{0,1\}^n$ such that for every $x\in\{0,1\}^k$ the function $\phi_x:y\mapsto\phi(x,y)$ is a ...
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Zero knowledge proof of equality
Alice and Bob each secretly chooses an integer between 1 and 10, a and b. They want to know (with high probability) whether or ...
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answers
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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 ...
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1
answer
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What can I say about the permutation $\alpha\beta$ if I know the permutation $\beta\alpha$?
I'm looking into a secret sharing scheme that has a secret permutation $\theta$ which has the cycle structure (n/2)+(n/2) (i.e. two (n/2)-cycles).
The permutation $\theta$ is decomposed into two ...
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votes
2
answers
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Knot Diffie–Hellman
Here's an idea for a knot-based Diffie–Hellman exchange:
Public: random (oriented) knot $P$.
Private: random (oriented) knots $A$ and $B$.
Exchange: Alice sends (randomized or canonical ...
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votes
1
answer
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Cryptographic Secret Santa
Is there a protocol for conducting a Secret Santa without a central authority? Precisely, we want to sample uniformly a permutation that has no one-cycles and reveal to each member his or her ...
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Attack on CRT-RSA
The survey paper of Prof. Dan Boneh entitled
"Twenty years of attacks on the RSA cryptosystem" mentioned that (Page 5)
one can attack CRT-RSA in square root of decryption exponent. However no
...
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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
...
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votes
1
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Discrete logarithm for polynomials
Let $p$ be a fixed small prime (I'm particularly interested in $p = 2$), and let $Q, R \in \mathbb{F}_p[X]$ be polynomials.
Consider the problem of determining the set of $n \in \mathbb{N}$ such that $...
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votes
4
answers
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The "interplay" between additive and multiplicative structure in a field
A field is an ordered triple $(F, +,\cdot)$ of a set $F$ and binary operations $+,\times$ on $F$ such that $(F,+)$ and $(F\backslash 0,\times)$ are abelian groups satisfying the distributive laws
$\...
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3
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"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 ...
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votes
1
answer
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Any nice examples of small cancellation theory appearing in applied mathematics?
Are there any nice discussions of applications of small cancellation theory, or other cases of the word problem, in applied mathematics or algorithms for seemingly non-group theoretic problems?
I ...
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votes
3
answers
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Predicting if something is a code
I'm trying to help a non-mathematical friend by posting a question of his here. He studies literature and has come across a book which is written in a made-up language. The book is hundreds of pages,...
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1
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Is this obfuscation scheme unbreakable?
I've just come across this popular article about a breakthrough (which can be purchased here), published in Foundations of Computer Science (FOCS), 2013 IEEE 54th Annual Symposium by a team of ...
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votes
1
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Inverting a function
I posted this question on crypto.SE but got no answer:
Let $w = a_0 \cdot a_1 \cdots a_{n-1} $ be a word from $ \{0,1\}^n $, $|w| = n$
Let $m = \sum_{i=0}^{n-1}{ a_i \cdot 2 ^ {n-1-i} } $ be the ...
user6671
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answers
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Is there a two-party multiplicative and additive secret sharing scheme ?
A secret sharing scheme such as Shamir's secret sharing allow to perform addition and multiplication for secret values so far as there is at least 3 participants. Addition of two secret values is done ...
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Are there any unitary matrices which satisfy the Yang-Baxter equation which are universal for quantum computation?
Let $H$ be a finite dimensional hilbert space. Let $L:H\otimes H\rightarrow H\otimes H$ be a unitary transformation. Then the equation
$$(L\otimes I)(I\otimes L)(L\otimes I)=(I\otimes L)(L\otimes I)(I\...
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answers
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Question on randomness extractors
Person A has a source $W$ with min-entropy($W$) = $k$. He also has an extra piece of information about the random source, denoted with $y$, such that min-entropy($W|y$) = $k/3$.
The adversary doesn't ...
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A balls and urns model for a hashing problem
Fix $N \in \mathbb{N}$. Suppose we throw $N$ numbered balls into $N$ numbered urns, so that for each $b \in \{1,\ldots,N\}$, ball $b$ lands in urn $j$ with equal probability $1/N$. Choose a number $c \...
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1
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Modular polynomials for elliptic curves point counting
The Schoof-Elkies-Atkin (SEA) algorithm (for counting points on elliptic curves over a finite field) performs computations over polynomials modulo some modular polynomials. Originally the "classical" ...
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Zero-knowledge proofs for answers to the $P=NP$ question
Are there zero-knowledge proofs for every answer to the $P=NP$ question?
For instance, if you have a polynomial-time algorithm of moderate complexity for the graph-coloring problem, then it is easy to ...
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7
votes
0
answers
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Polynomial representation of modular arithmetic in finite fields
Let $n \in \mathbb{N}$ be a predefined integer. Consider the following bijection (between the ring of integers modulo $2^n$ and finite field with $2^n$ elements:
$$ \phi: \mathbb{Z}_{2^n} \to \mathbb{...
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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 ...
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Computing the correlation between two vectors without divulging them
Alice and Bob respectively know a vector of $N$ real numbers $u$ and $v$. They would both like to know $\rho = \langle u,v \rangle/N$ but Alice does not want Bob to gain anymore information about $u$ ...
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Minimum number of operations necessary to arrive at any configuration
Let $k \geq 2$ and $N_1, N_2, ..., N_k$ be positive integers.
Let $S=\{(a_1,a_2,...,a_k) \in \mathbb{Z}^k:1 \leq a_i \leq N_i\}$ and $A=\{1,2,...,\prod_{i=1}^{k} N_{i}\}$.
Given a bijective map $f:...
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Shortest vector problem over polynomials
In shortest vector problem, given a lattice in $\Bbb Z^n$, we seek the shortest non-zero vector in the lattice. This problem is computationally difficult.
Answer in Evidence for integer factorization ...
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1
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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 ...
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3
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Torus based cryptography
In cryptography one needs finite groups $G$ in which the discrete logarithm problem is infeasible. Often they use the multiplicative group $\mathbb{G}_m(\mathbb{F}_p)$ where $p$ is a prime number of ...
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A silly question: is the number of points on a Jacobian (of a curve, over a finite field) known?
In a cryptography book I read that people does not known how to compute the number of points on a Jacobian of a hyperelliptic curve $C$ over a finite field $F_q$? Is this true? It seems easy to ...
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Whitening a random bit sequence
Given an (infinite) stream of uncorrelated random bit with a known "reasonable" bias (say 15-85% 1's) I want to whiten it, e.i. produce a shorter stream of bits that has no bias. The restriction is ...
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Diffie Hellman cryptography based on graph isomorphism?
We got a cryptographic algorithm and computer implementation
based on graph isomorphism.
An isomorphism between two graphs is a bijection between their vertices that pre
serves the edges.
For a graph $...
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5
votes
2
answers
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Maximum number of vectors with upper bound on pairwise inner products
I have a collection $\{v_1,...,v_k\}$ of vectors in $\{\pm 1\}^n$ with the property that for all $i\neq j$ we have $\langle v_i, v_j \rangle \le c\log_2(n)$. I am looking for an upper bound on $k$ in ...
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Using High Level Probability Theory (eg Markov Chain Mixing) in Cryptography/Cryptanalysis
I'm currently doing a PhD in probability theory, specifically (discrete space) Markov chains and their mixing properties. As well as my current main project, I'm looking to have a side project, eg to ...
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Fastest algorithm to compute (a^(2^N))%m?
Hi.
There are well-known algorithms for cryptography to compute modular exponentiation $a^b\%c$ (like Right-to-left binary method here : http://en.wikipedia.org/wiki/Modular_exponentiation).
But do ...
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