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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919 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 ...
10
votes
3answers
2k views

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 ...
10
votes
2answers
422 views

How to determine the order of an elliptic curve group from its parameters?

Let $\quad E:\; y^2 = x^3 + ax + b \quad$ be an elliptic curve defined over a finite field $\mathbb F_q$ where $q = p^n$, $a,b \in \mathbb F_q$ and $p \neq 2, 3$. By Hasse's theorem we know that the ...
8
votes
2answers
2k 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 ...
7
votes
1answer
321 views

Logjam: “composite order subgroups” explained for TLS developers and system admins?

I have read the recent logjam paper Imperfect Forward Secrecy: How Diffie-Hellman Fails in Practice. On page 11 in the Recommendations section, they state: ...
6
votes
1answer
100 views

Is every point on an elliptic curve of a prime order group a generator?

If the order of elliptic group is prime then every point is a generator of that group. I tested the above statement on some elliptic curves and found it true. Does that really work on all curves? Is ...
6
votes
1answer
164 views

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 ...
5
votes
2answers
1k views

What are Cryptographic Multi-linear Maps?

I've encountered this term many times in the fields of Fully-Homomorphic Encryption and Obfuscation. I want to learn those subject and Cryptographic Linear Maps seems to be an obstacle in the way. ...
5
votes
1answer
384 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 ...
5
votes
2answers
253 views

lcm versus phi in RSA

In textbook RSA, the Euler $\varphi$ function $$\varphi(pq) := (p-1)(q-1)$$ is used to define the private exponent $d$. On the other hand, real-world cryptographic specifications require the ...
5
votes
2answers
395 views

in Bilinear pairings, what is the difference between Type 2 and Type 3?

in Bilinear pairings, what is the difference between Type 2 and Type 3? I understand in Type 2, there exists an efficiently computable homomorphic function $\phi : G_2 \rightarrow G_1$ , which is not ...
5
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1answer
222 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? ...
5
votes
1answer
160 views

How can I find the order of the group that an elliptic curve is defined over?

I have a Weierstrass elliptic curve ($y^2=x^3+a \times x+b \mod p $) How can I find the order of the group itself? I have seen Mathematica has a GroupOrder[] ...
5
votes
1answer
188 views

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 ...
4
votes
3answers
583 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 ...
4
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2answers
267 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_(...
4
votes
2answers
193 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 $a$?
4
votes
3answers
781 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 ...
4
votes
2answers
778 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 ...
4
votes
1answer
205 views

Discrete logarithm problem is easy in a cyclic group of order a power of two

Let $G=\langle g\rangle$ be a cyclic group of order $2^{k}$ and let $h\in G$. I have read that it is easy to find $\log _{g} h$, but I haven't been able to figure out how. Do you know why this can be ...
4
votes
1answer
123 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 $\mathbb{G}...
4
votes
1answer
231 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 ...
4
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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 chosen-...
3
votes
2answers
346 views

Why is “multiplying” $g^x$ and $g^y$ not possible?

The computational Diffie-Hellman problem states that for a cyclic group $G$ of order $p$ and a generator $g$, it is hard to find the value $g^{xy}$ given only $g^x$ and $g^y$ (but easy if either $x$ ...
3
votes
2answers
225 views

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?
3
votes
1answer
56 views

Do $v_1=\alpha\cdot r_1$ and $v_2=\alpha\cdot r_2$ leak information about $\alpha$

Please consider we have finite field $\mathbb{F}_p$ for large prime number $p$. We have a fixed field element $\alpha$. By $r_i\leftarrow \mathbb{F}_p$ we mean we pick $r_i$ uniformly random from the ...
3
votes
1answer
141 views

Can we reduce Diffie-Hellman problem to “Discrete-log inversion” problem?

Let $G$ be a cyclic multiplicative group of order $n$. Let $g$ be a (public) generator of $G$. The Diffie-Hellman (DH) problem asks: Given $g^x, g^y\in G$ for $x, y\in \mathbb{Z}^*_n$, to compute $g^{...
3
votes
3answers
37 views

What are possible caveats when generating a group for use as parameters for Diffie-Hellman key exchange?

As reusing a widely used group for Diffie-Hellman key exchanges might lead to far easier third-party key discovery through precomputation for that specific group, I would like to know what can ...
3
votes
2answers
171 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 '$a$' ...
3
votes
1answer
322 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 ...
3
votes
1answer
234 views

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 $...
3
votes
1answer
293 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 ...
3
votes
1answer
128 views

Given $g,g^t$ in a cyclic group of order $pq$, is it hard to compute $g^{t^{-1}}$?

Suppose we have a group $G$ cyclic of order $pq$ , where $p,q$ are primes. Let $g$ be a generator of $G$ and $t\in \mathbb{Z}_{pq}$. Having $g$ and $g^t$, it seems to be very hard to find $g^{t^{-1}}$,...
3
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1answer
65 views

How is a group element converted into a key?

I've only just started research on cryptography so I apologize if this is a basic question or I'm getting terms confused. I'm researching braid group cryptography and currently looking at the Anshel-...
3
votes
0answers
90 views

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 ...
2
votes
2answers
283 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 ...
2
votes
2answers
217 views

Given $g$, $b$, $g^{ab}$, is finding $g^a$ a hard problem?

As in the title, given $g$, $g^{ab}$ are big elements in a prime group $Z_p$ and $b$ in prime group $Z_r$ ($p > r$, $g$ is one generator of $Z_p$). $a$ is unknown and also in $Z_r$, is finding $g^a$...
2
votes
2answers
448 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 ...
2
votes
1answer
105 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 ...
2
votes
3answers
788 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, $\phi(N)...
2
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1answer
48 views

Number generation for Fujisaki-Okamoto commitment scheme parameters

I need to implement the Fujisaki-Okamoto commitment scheme for a project such that I can demonstrate performance of various zero-knowledge proofs in relation to one another, for example Boudot's "...
2
votes
1answer
298 views

Diffie-Hellman on additive group

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 $a$ ...
2
votes
1answer
68 views

Understanding the Hidden Subgroup Problem specific to Integer Factorization

I've been reading about the Hidden Subgroup Problem (HSP), specifically trying to understand how it is related to the integer factorization problem. I've read What exactly is the impact of the hidden ...
2
votes
1answer
163 views

Modular Arithmetic in RSA

Consider the following the following RSA public key $pk = (N, e) = (1457, 1307)$. (a) Knowing that $187^2 \equiv 1 \pmod {1457}$ find the factorization of $N$. (b) Given the factorization of $N$ ...
2
votes
1answer
61 views

In a additive group is it hard to calculate $bg$ given $ag, g, abg$

The ECDH problem defined that given $g,ag,bg$ it is difficult to calculate $abg$. But it is also difficult to calculate $bg$ given $ag,g,abg$. where $g$ is generator and a,b are elements of group.
2
votes
1answer
35 views

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: ...
2
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1answer
143 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?
2
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1answer
72 views

Elliptic Curve ElGamal and DSA - smooth group order and element of large prime order

In regular ElGamal and DSA, we choose large primes $p$ and $q$ such that $p\equiv 1\pmod{q}$, and a group element $g$ of order $q$ by computing $a^{(p-1)/q}$ for some random $a$. This is to prevent ...
2
votes
0answers
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 ...
1
vote
2answers
152 views

Factoring large $N$ given oracle to find square roots modulo $N$

When $p$ and $q$ are distinct odd primes and $N = pq$, the points in $\mathbb Z_N^\ast$ have either zero or four square roots. A quarter of the points have four square roots; the rest have no square ...