**33**

votes

**7**answers

17k views

### 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 ...

**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**

votes

**4**answers

2k 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 ...

**18**

votes

**1**answer

545 views

### 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 ...

**16**

votes

**5**answers

640 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 ...

**14**

votes

**4**answers

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 ...

**13**

votes

**2**answers

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 ...

**12**

votes

**3**answers

2k 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 ...

**12**

votes

**2**answers

4k 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 ...

**12**

votes

**2**answers

640 views

### 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 ...

**12**

votes

**1**answer

439 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 ...

**11**

votes

**5**answers

2k views

### 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 ...

**11**

votes

**2**answers

496 views

### “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.
...

**10**

votes

**3**answers

646 views

### 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 ...

**9**

votes

**1**answer

631 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 ...

**9**

votes

**1**answer

17k 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 ...

**9**

votes

**1**answer

419 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
...

**9**

votes

**3**answers

375 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 ...

**8**

votes

**4**answers

1k 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
...

**8**

votes

**4**answers

1k views

### 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 ...

**8**

votes

**0**answers

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 ...

**7**

votes

**5**answers

1k views

### 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 ...

**7**

votes

**1**answer

557 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 ...

**7**

votes

**1**answer

313 views

### 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" ...

**6**

votes

**3**answers

415 views

### 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 ...

**6**

votes

**3**answers

2k 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
...

**6**

votes

**1**answer

172 views

### 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.
Is there a polynomial analog of this problem ...

**5**

votes

**3**answers

678 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 ...

**5**

votes

**1**answer

378 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 ...

**5**

votes

**1**answer

358 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 $u$ ...

**5**

votes

**0**answers

338 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 ...

**4**

votes

**5**answers

3k 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 ...

**4**

votes

**1**answer

513 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 ...

**4**

votes

**2**answers

2k views

### 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 ...

**4**

votes

**2**answers

298 views

### Good quality data/packages for statistical/structure analysis of words in the English language

From time to time I find myself wishing to calculate basic statistics on words in the English language. For example, today I found myself wanting a graph of the number of English words vs. their ...

**4**

votes

**0**answers

238 views

### Polynomial dynamical systems

The question is somewhat related to the theory of permutation polynomials. Let $\mathbb{F}_p$ be a finite field of $p$ elements ($p$ is prime) and $\mathcal{V} = \mathbb{F}_p^2 = \{ (t_1,t_2)\::\: ...

**4**

votes

**0**answers

203 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 ...

**3**

votes

**3**answers

281 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 ...

**3**

votes

**1**answer

134 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 ...

**3**

votes

**0**answers

171 views

### PRNG and coding theory

Let $k, n \in \mathbb{N}$, $k = (1 - \epsilon)n$ where $1 >\epsilon > 0$.
I want to find $f: \{0,1\}^k \to \{0, 1\}^n$
such that:
1) $f(a) \not= f(b)$ if $a \not=b $
2) for any $x \in ...

**3**

votes

**0**answers

415 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 ...

**2**

votes

**4**answers

1k 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 ...

**2**

votes

**1**answer

95 views

### Future-Proof Encrypt for Multiple Independent Parties

I have a secret message which I want to encrypt such that any of several different keys can open it independently. The keys can't know about each other and it has to be able to work completely ...

**2**

votes

**2**answers

1k 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
...

**2**

votes

**1**answer

482 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
$1.)$ ...

**2**

votes

**1**answer

143 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**

votes

**1**answer

129 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 ...

**2**

votes

**3**answers

284 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 ...

**2**

votes

**1**answer

241 views

### 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 ...

**2**

votes

**1**answer

109 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 ...