The cryptography tag has no wiki summary.

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### Canonical lifts from $\mathbb F_q$ and CM-theory

One knows that (ordinary) Jacobians of hyperelliptic curves over a finite field $\mathbb F_q$ (mostly of genus 1 (elliptic curves) and 2) are extensively studied by cryptographers, as a platform for ...

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58 views

### Genus 2 hyperelliptic cryptography : typical discriminant and class number

As far as I know, there is no standard yet for cryptography based on the DLP over Jacobians of genus 2 curves. Yet, what can we say about the class number, and the discriminant of the complex ...

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51 views

### Encrypting the same message using different schemes

$E_1$ and $E_2$ are IND-CPA secure encryption schemes.
$E$ is defined as:
$k_1,k_2 \leftarrow K_1 \times K_2$ .
$E_{k_1,k_2}(m) \leftarrow E_{1,k_1}(m)||E_{2,k_2}(m)$.
Hope the notations are in an ...

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65 views

### Connection between inf-entropy rate and min-entropy

I am reading the paper "Generating random bits from an arbitrary source: fundamental limits" by Vembu and Verdu. This paper is written in the language of information theory, however, I need to ...

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138 views

### Anomalous elliptic curves over finite rings

I was wondering if it is possible to solve the discrete logarithm on an Elliptic Curve E(Z/nZ) (defined over the ring of integers modulo a composite n) with #E(Z/nZ)=n by applying a method analogous ...

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26 views

### largest size for a randomness extractor

I am not so expert in theoretical computer science, so sorry if the question is trivial, i just could not find it in literature.
Suppose we have a source $X$ with min-entropy $\ell$, the randomness ...

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47 views

### Private Randomness extractor

Suppose we are given two random variables $X$ and $Y$ with fixed marginal and joint distribution. What is the maximum randomness that we can extract from $Y$ that is independent from $X$, that is, if ...

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268 views

### 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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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 ...

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433 views

### 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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119 views

### Lattice basis reductions and finding minimal values

While reading several articles about lattice basis reduction I am left with a few questions.
For one, I came across this piece of text
Let $\alpha$ and $\beta \in \mathbb{R}$. Also let $X>0$ and ...

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**1**answer

379 views

### 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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63 views

### Collision resistance of hash functions after permuting one hash digest

Given a hash function H and a fixed permutation pi of the digest set. Consider "collisions" of the form H(x) = pi(H(x')). How is resistance against this kind of ...

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1k views

### 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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**2**answers

14k views

### 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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**1**answer

679 views

### Pairing on elliptic curve

Let $E(\mathbb{F_q})$ - elliptic curve.
$G_1 = E(\mathbb{F_q})[r]$. $|G_1| = r$.
$k$ is minimal such $r | q^k - 1$.
$\pi_q$ - $q$-power Frobenius endomorphism.
$G_2 = E(\mathbb{F_{q^k}})[r] \cap ...

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900 views

### 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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217 views

### Is Guillou-Quisquater existentially unforgeable against adaptive message attack under a random oracle model?

First of all, the Guillou-Quisquater digital signature scheme is:
Note everything is $\bmod n$. Message is denoted by $m$.
Private key: $s$
Public key: Hash function $H$, $e$, ...

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**1**answer

329 views

### 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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**2**answers

280 views

### cryptographic primitive process

Is there a cryptographic primitive process/method for creating cryptographic tools like symmetric encryption/decryption, Hash code generator, MAC generator and Random number generator?
...

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113 views

### Knowing md5(c+x), is it possible to find md5(x)?

Suppose:
md5(c1 + x) = c2
md5(x) = y
Is it possible to find y, if c1 and c2 are known and x is uknown? Basically, I know md5(salt + key) and I want to find md5(key).

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190 views

### Cryptography and Availability

Hi,
Here is a question in cryptography which is probably naive, and a reference request.
Suppose I have 3 matrices(I1, I2, and I3 -same size) that I want to combine them some how(? do not know yet) ...

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1k views

### 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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**0**answers

95 views

### Is there a security analysis of the GQ digital signature scheme?

I'm doing summer cryptography research and I am have been looking for a security analysis of the Guillou-Quisquater (GQ) digital signature scheme, but I have been unable to find one.
Since this is ...

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**1**answer

319 views

### 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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274 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 ...

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**3**answers

274 views

### Cryptography and iterations

Hi,
Here is a question in cryptography which is probably naive, and a reference request.
I was wondering about the following key-exchange scheme, which is a variant on Diffie-Hellman. Consider a ...

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**1**answer

309 views

### 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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177 views

### Asymtotic Complexity Analysis using logarithms and binomial coefficients

On page 11 of "Smaller decoding exponents: ball-collision decoding" by Berstein et.al. they have the formula \begin{equation}\lim_{n \rightarrow \infty} ...

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616 views

### Weil pairing, Kummer theory, help to decrypt what Wikipedia says

I do not quite understand the sentence in the Wikipedia article:
http://en.wikipedia.org/wiki/Weil_pairing
Section "Formulation" line 3:
"... for given points $P,Q \in E(K)[n]$, where $E(K)[n]=\{T ...

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190 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 ...

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217 views

### Is it (believed to be) possible to algorithmically generate Diffie-Hellman tuples without “being able to know” one of the discrete logs involved (formal definition given in question)?

Is it (believed to be) possible, in the various standard examples of groups in which discrete log/Diffie Hellman are hard (including multiplicative groups in modular arithmetic and elliptic curves, ...

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528 views

### 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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427 views

### Factoring and Index Calculus and duality between DL and factoring via compuational problems made easy through them

If factoring is in $P$ (with a blazing fast polynomial time in $P$), would it affect the index calculus algorithm used for Discrete Log calculation in any serious way?
Other connections
$1.)$ ...

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541 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 ...

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281 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 $|a{b^{c}} - n|$ ...

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1k 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
...

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194 views

### Oracle separating FIP for bounded-depth Frege from FIP for Frege (and hardness conditions on DDH)

Is there an oracle such that in the relativized world, bd-Frege (bounded depth Frege propositional proof system) has FIP (feasible interpolation property) but Frege does not have FIP?
Such an ...

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395 views

### Does this algorithm exist - a secret secret?

I'm not quite sure how to phrase this question mathematically, so I am going to express it in words first:
Let us suppose I have a secret $m_1$ and a plausible innocent secret $m_2$. Is there an ...

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**2**answers

832 views

### Weil pairing and Miller's algorithm

I'm studying Weil pairing and its applications in cryptography. I already know that it can be defined like this:
$$w(P, Q) = (-1)^n\frac{f_P(Q)}{f_Q(P)}\frac{f_Q}{f_P}(\mathcal{O})$$
where
...

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1k views

### 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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2k views

### 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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**1**answer

584 views

### Elliptic curve over finite field: scalar multiplication

I'm implementing arithmetics for elliptic curves over secp256r1 as a homework assignment.
For scalar multiplication, the assignment specifically specifies that $k$ is "any hexidecimal encoded ...

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608 views

### 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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171 views

### AES key schedule: round 1 and round 0 equal?

is there a key that gives round 1 and round 0 that are equal?
(a0,b0,c0,d0) = (a1,b1,c1,d1)
how many keys like that exist?
can it continue to round 2? round3...?

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452 views

### 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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599 views

### 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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2k 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 ...

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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 ...

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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 ...