# Questions tagged [group-theory]

Groups are an abstract algebraic concept based on a set and a group law (a binary function which closes the set).

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### Do Gap-CDH groups exist?

A Gap-CDH group is such that, given group elements $g, a = g^x, b = g^y$, it is hard to compute $g^{xy}$, but, given a group element $c$, easy to verify if $c = g^{xy}$. While such groups have been ...
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### Show How to Efficiently Solve the Computational Diffie-Hellman Assumption given an Algorithm that Solves the Square-DH Problem

Let $q$ prime number, $G$ a cyclic group with order $q$ and let $g \in G$ be a generator of $G$. Suppose that you have an algorithm $A$ who takes input the element $g^a$ of $G$ and gives as output the ...
401 views

### “Order” in cryptographic terms for generators

Frequently I have seen people use the term order in cryptography (the group theoretic one). I have a mathematical background and order (say for prime modulus $p$) is defined as the smallest integer ...
607 views

### How to compute two EC point multiplication?

I would like to know how to compute multiplication of two valid EC points over a curve E with generator G. i.e. Given only P and Q points then how to compute R = P * Q where $P = p G$, $Q = q G$ and ...
157 views

### Working on subgroup of $\mathbb{Z}^*_p$ in practice

It is said that, given a group $\mathbb{Z}^*_p$, we can always have a subgroup whose order is prime. To this end, for a safe prime $p=2q+1$, compute $x_i^2 \bmod p$ for all $x_i \in \mathbb{Z}^*_p$. ...
442 views

### In a group, is it hard to calculate the base $g$ given $g^a$ and $a$?

Discrete logarithm, that is: calculate $a$ given $g$ and $g^a$, is assumed to be a hard problem in some groups. Is it also hard to calculate $g$ given $g^a$ and $a$?
179 views

### Hash “Preimage by product” resistance

Let H() be a hash function that achieves collision resistance as well as first and second preimage resistance. Let's equip the output set of H of a multiplicative group structure, more precisely a ...
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### Is the CONF key sharing Problem equivalent to discrete log problem?

If there a proof in the literature which says the CONF Problem is equivalent to solving the discrete log ? Let $g$ be a generator of a cyclic group $\mathbb{G}$ of prime order $q$ CONF problem: ...
344 views

### Can anyone give an example where (asymmetric) crypto can go wrong due to selection of wrong groups?

Basically the title says it all. It would be great if someone could tell give an example using provable security. More information about groups can be found at: https://en.wikipedia.org/wiki/Group_(...
4k views

### Cycle attack on RSA

I originally posted this question in the mathematics section, you can see it here. Let $p$ and $q$ be large primes, $n=pq$ and $e : 0<e<\phi(n), \space gcd(e, \phi(n))=1$ the public encyption ...
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### $f : \mathbb{Z}_n \rightarrow \mathbb{Z}^\times_n$?

Is there any function $f : \mathbb{Z}_n \rightarrow \mathbb{Z}^\times_n$ that is invertible? By invertible, I mean it given $y \in \mathbb{Z}^\times_n$, it should be easy to find $x \in \mathbb{Z}_n$ ...
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### Are there groups where the computational Diffie Hellman problem is easy but the discrete log problem is hard?

I know that there are elliptic curve groups, used in pairing-based cryptography, where the decisional Diffie Hellman problem (ie. given $g$, $g^a$, $g^b$ and $c$, determine if $c = g^{ab}$ is easy but ...
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### How to understand the Bilinear mapping with an example [duplicate]

An efficient bilinear map is given by $e$: $G_{1}$ × $G_{1}$ → $G_{T}$. How can i prove this equation with the help of an example.
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### Diffie-Hellman insecure on addition modulo $n$

Assume that the group $G$ is the set $\mathbb{Z}_{n} = \{0,\ldots, n-1\}$ for a 1024 bit integer and $+$ is addition modulo $n$. Then why would Diffie-Hellman key exchange in this group be insecure?
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Given the finite cyclic, additive group (G, +), with |G| = n and generator = g, what are the computations and exchanged messages for Diffie-Hellman? What I tried myself: Alice chooses a private $a$ ...
365 views

### Best group if one wants the discrete log problem to be hard?

Suppose one is implementing a cryptographic scheme over a group where one needs the discrete logarithm to be hard - what is the recommended group to use? I'm looking for a group where calculations are ...
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### What type of groups does Microsoft's U-Prove use (Schnorr… etc?)

I'm trying to learn more about the Subgroups implementation of Microsoft UProve. I'm unsure if they are Schnorr Groups or use a different foundation? Can anyone point me to the technical reading ...
375 views

### Is Triple-DES a group?

I know for a fact that DES is not a group, but are any of the Triple-Des versions a group? Why, or why not?
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### ECC Point Multiplication of Product

I can calculate $Q = a\,b\,G$ in several ways: $Q = a \, (b \, G)$ or $Q = b \, (a \, G)$. These give the same result, as expected. But if I do $c = (a \, b) \bmod n$ where $a \, b$ is much greater ...
164 views

### Does Linear Cramer-Shoup have pseudo-random ciphertexts?

"Linear Cramer-Shoup" is defined on pages 4 and 5 of $\:$ eprint.iacr.org/2007/074.pdf . Are the ciphertexts in Linear Cramer-Shoup computationally indistinguishable from uniform under a chosen-...
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### Is solving a modular linear equation a hard problem when the coefficient is not an invertible element?

Assume that we have a linear equation like this: $$ax=b \pmod n$$ when $x$ is the unknown, and $a$ is not an invertible element in $n$. is finding $x$ a hard problem? (by solving I mean finding an ...
758 views

### How to find an element of high-order in an RSA group?

Is this even possible? The RSA group is not cyclic, so usually you wouldn't find a generator for accessing all group elements. What happens if you use the RSA group in a scenario where you want that ...
498 views

### Finding a solution to a (sort of) discrete logarithm by asking questions

Let $p = kq + 1$ and $q$ be primes such that $log$ $q = n$, $log$ $k = n$ and such that the bit size of every prime factor of $k$ is bounded by $log$ $n$. Let $g$ be a generator of the unique ...
2k views