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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Generator for Group $QR_{N}$

Let $N=PQ$, where $P=2p+1$ and $Q=2q+1$. $P,Q,p,q$ are prime numbers. $QR_{N}$ is the set of quadratic residues modulo $N$. Please help me to prove $QR_{N}$ is a cyclic group. Note: $QR_{P}$ and ...
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Why does the new encryption scheme proposed by authors stop an adversary from guessing the subspace of the secret key?

In this paper, the authors construct an encryption scheme that is supposed to be resilient to tampering and leaking (as opposed to just leaking). Specifically this scheme: If you look at the ...
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435 views

What does the linear assumption over bilinear groups mean?

In the abstract of "Cryptography with Tamperable and Leaky Memory", at the end of the 3rd paragraph, the authors say: In both schemes we rely on the linear assumption over bilinear groups. What ...
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456 views

How is multiplication inverted in IDEA's decryption round?

As we can see in the picture we have a multiplication in this algorithm, we know that two 16 bit inputs should have a 32 bit output, but here we just use 16 bits of the 32 bit output. For ...
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170 views

Create a field in PBC

Edited (I removed the emphasize on Integers): My question is partly cryptography and partly programming, I would appreciate any help on any aspect of it :) I want to use PBC library to do the ...
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170 views

Finite fields in elliptic curve

I have an elliptic curve defined over finite field where $S_1=aP$ . Is it valid to say that $S_1P$ can also be computed. $P$ is the generator of the group. What my real question is that. Should '$a$' ...
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121 views

Hardness of CDH in different groups

What is the difference of the CDH problem in different groups? In particular, given a group $\mathbb{G}_1$ of order $q$ that is a subgroup of $\mathbb{Z}_q^*$, $q$ prime, and another group ...
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227 views

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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97 views

BLS signatures in the G-valued Random Oracle Model

This paper on semi-generic algorithms considers "non-standard properties of the employed hash function". For BLS signatures whose main group is $G$, I'm curious what can be shown when the hash ...
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120 views

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 ...
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140 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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572 views

Block cipher fixed points (plaintext equal to ciphertext)

A block cipher is a bijective map from the set of possible plaintexts to the set of ciphertexts, which are the same size and might as well be considered the same thing: $\theta: S\to S$. In this there ...
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365 views

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 ...
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1answer
150 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 ...
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202 views

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 ...
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311 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 ...
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221 views

Why work in a subgroup QR(n) of an RSA group $Z^*_n$?

I sometimes read in papers that a (sub-)group generator $g$ is taken from $\mathrm{QR}(n)$ instead of $\mathbb{Z}^*_n$, where $n = p \cdot q$ and $p$ and $q$ are prime. Is there a reason for this? ...
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279 views

Subgroups generators with respect to group generators of composite order

If I have a group $\mathcal{G}$ of order $N=npq$ and subgroups $\mathcal{G_n,G_p,G_q}$ of order $n$, $p$, $g$ respectively and if $g$ is a generator of $\mathcal{G}$ why then $g^{nq}$ is a generator ...
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756 views

When do we need composite order groups for bilinear maps and when prime order?

Why we need bilinear groups of composite order? What's the special security property of the composite order group in comparison with one of prime order? To put it in another way when do we need ...
3
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290 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 ...
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747 views

RSA finding the inverse of the public exponent

I have a very basic doubt in RSA key generation and its usage. In RSA key generation you choose two large prime numbers of a very large order. Then you multiply them.(eq $p \cdot q = N$) Now, ...
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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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754 views

How does the wider cryptographic community view non-abelian group based cryptography?

Is there perhaps some neural expository article on crypto systems based on non-abelian groups? I've gleaned that Anshel–Anshel–Goldfeld key exchange is the most well-known cryptographic algorithm ...
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372 views

Must the order of the groups in a bilinear map be the same?

I've been reading up on bilinear maps and their application to cryptography and one thing I keep seeing hasn't yet clicked. If $e:G_1\times G_2\to G_n$ is a bilinear map, $G_1,G_2,G_n$ are always ...
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Is there a group of prime order which could fit the CT-Computational Diffie-Hellman assumption?

I'm trying to choose a group that is hard under the Chosen-Target Computational Diffie-Hellman assumption, according to the definition in this paper, in order to implement the oblivious transfer ...
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Would the ability to efficiently find Discrete Logs have any impact on the security of RSA?

This answer makes the claim that the Discrete Log problem and RSA are independent from a security perspective. RSA labs makes a similar statement: The discrete logarithm problem bears the same ...
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890 views

What exactly is the impact of the hidden subgroup problem on cryptography?

I understand my group theory (allegedly), so I can make partial sense of The Hidden Subgroup problem: Given a group $G$, a subgroup $H \leq G$, and a set $X$, we say a function $f : G \Rightarrow ...