We’re rewarding the question askers & reputations are being recalculated! Read more.

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).

Filter by
Sorted by
Tagged with
-1
votes
1answer
231 views

Abelian groups in Elliptic curves [closed]

Do every elliptic curve defined over a prime field forms an abelian group?
9
votes
1answer
344 views

How hard is the DLP in that simple group?

Let $p=4q-1$ be a prime with $q$ an odd prime. Let $G=\{0,1,\dots,p-1,\infty\}$. The following law $*$ makes $(G,*)$ a commutative group of order $4q$ with neutral element $\infty$: $$a*b=\begin{cases}...
0
votes
2answers
130 views

Can someone explain the definition of four square roots as it pertains to groups in Z*p?

So I'm given the following as a problem: When $p$ and $q$ are distinct odd primes and $N = pq$, the points in $Z^∗_N$ have either zero or four square roots. A quarter of the points have four square ...
-1
votes
1answer
114 views

Order of the curve and generator

Does the order of the curve and the order of generator should be coprime for an elliptic curve defined over a prime field?
7
votes
1answer
966 views

Groups for which DDH is easy but CDH is hard

For prime p, is $\mathbb{Z}^{*}_{p}$ a group for which the Decision Diffie-Hellman problem is easy (because one can compute the Legendre symbol of ($g^{ab}$) while CDH is thought to be hard? Of course,...
3
votes
1answer
461 views

What's dh-composite test on badssl.com?

The site badssl.com provides examples of bad (red icon) and good (green icon) uses of TLS for the purpose of testing TLS implementations. I'm a bit confused by the test called dh-composite. This ...
4
votes
1answer
162 views

$L^3$ Grover search of NTRU variants

I was reading a text on cryptology by Wayne Patterson and came across the $L^3$ algorithm which reduces integer lattices with respect to their base. I've also read on the NIST CFP A8 that attacks ...
2
votes
1answer
4k views

How to find the generator of an elliptic curve? [duplicate]

If the elliptic curve has prime order of points, then all of its points are generator. Is this true? If so, how can I find the optimized generator(which generates more number of points) among them?
9
votes
2answers
1k views

Why are elliptic curves constructed using prime fields and not composite fields?

I come across this: Numbers mod composite number does not form a field rather it forms a ring and every number has a multiplicative inverse under integer mod prime Maybe these are the reasons ...
2
votes
1answer
1k views

why are non singular curves used in elliptic curve cryptography?

It is not possible to draw tangent at all the points of a singular curve. What is the specialty of this and how it is related to cryptography and elliptic discriminant?
-1
votes
1answer
290 views

Elliptic curve points addition is not associative [closed]

I've found an article that says how to add points in projective coordinates.But in my implementation these points don't form a group. Fields: ...
0
votes
2answers
181 views

Pairing - Is it possible to map two $r$-torsion points to a $r^2$-torsion point?

Let $E(\mathbb F_{q^k})$ be an elliptic curve on finite field $\mathbb F_{q^k}$, where $\mathbb F_{q^k}$ is an extension of $\mathbb F_q$ with $k>1$. Let $e: G_1 \times G_2 \rightarrow G_t$ be a ...
1
vote
1answer
284 views

Generating Diffie-Hellman parameters

I'm trying to implement a diffie-hellman key exchange in c++, and I'm struggling with my missing understanding of math / group theory. Let's say I found a large prime number p - how can I find a ...
2
votes
2answers
319 views

How to determine proportion of quadratic residues in elliptic curve group?

I'm using a 'try and increment' method to hash to an Elliptic Curve point, explained below. With security parameter $k$, EC equation $y^2 = x^3 + ax + b \mbox{ mod } q$, we have: $ u = sha256(\mbox{...
2
votes
1answer
77 views

LSB of the Exponent in the DL Problem Can Be Efficiently Computed for Groups of Even Order

I am studying a script on the mathematical foundations of cryptography as part of which I am currently trying to wrap my head around some basic cryptographic reductions. I am stuck on one problem that ...
0
votes
1answer
294 views

Generating cyclic group for Ciphertext-Policy Attribute-Based Encryption [closed]

I am doing Project under the topic CP-ABE.I need to generate a symmetric bilinear group Go of prime order p and with generator g...Then how to choose random elements from Zp....kindly anyone help me......
6
votes
2answers
2k views

Why must an elliptic curve group for ECC have prime order?

What is the deeper reason, a group must have prime order for usage in cryptography?
2
votes
1answer
486 views

Why is only one generator stated in literature for elliptic curve group P-256?

I refer to elliptic curve groups over prime fields and their application in cryptography. If the order of a group is prime, it follows (am I wrong?) that: the group is cyclic and every element ...
1
vote
1answer
240 views

Discrete logarithm problem in subgroup of index 2. ElGamal

I need some insight for the following problem in ElGamal encryption procedure. It is stated that ElGamal problem in a group $\mathbb{Z}_p^*$ becomes easier in subgroups. Assume I have a subgroup of ...
3
votes
1answer
571 views

Diffie-Hellman key exchange using addition instead of multiplication?

I know that the additive finite group $(Zp,+)$ of prime order $p$ when I perform the Diffie-Hellman key exchange (DHKE) protocol is insecure. I didn't however find many sources online explain why it ...
5
votes
1answer
710 views

How is the order of a point calculated for elliptic curves over GF(p)

My question is about elliptic curves over $GF(p)$: How is the order of a generating element $G$ (which is to my knowledge also the order of the cyclic subgroup $G^n$) calculated? Taking P-256 as an ...
4
votes
2answers
3k views

Find the generators of multiplicative group of units efficiently?

Say you're give some prime numbers $p_{1},p_{2},p_{3}, p = 2p_{1}p_{2}p_{3} + 1$ (which is assumed to be also prime) and a list of numbers $L$ and you're asked to find the generators of the ...
2
votes
0answers
140 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 ...
2
votes
1answer
249 views

Additive proof of discrete log?

Let's say someone has $K$, and I want to prove to them that I know $k$ such $K = k*g$ (where $g$ is some sort of group element). I can use a Schnorr signature. Is there some protocol for mergable ...
1
vote
1answer
85 views

Permutations coming from RSA

The RSA algorithm can be used to generate a permutation. Given two prime numbers $p$ and $q$, the key length is $n=pq$. If $a$ is the private key, then $b$ is the public key, where $ab \equiv 1 \...
3
votes
1answer
483 views

How to construct a hash function into a cyclic group such that its discrete log is intractable?

From the Linkable Ring Signatures paper: Let $G = \langle g\rangle$ be a cyclic group of prime order $q$ such that the underlying discrete logarithm problem (DLP) is hard. Let $H_1 : {0, 1}^∗ \to \...
0
votes
1answer
58 views

How to find g in Factoring-Based Trapdoor Hash Function

Please explain how to find a value of $g$ if $p,q$ are safe primes having $p'=(p-1)/2$ and $q'=(q-1)/2$ are also primes $n=p*q$ $\lambda(𝑛) = \operatorname{lcm}(𝑝 − 1, 𝑞 − 1) = 2𝑝'𝑞'$. How to ...
1
vote
0answers
56 views

Hash function notations and their corresponding existing cryptographic hash algorithm

I am implementing Role-based access control by referencing the paper Enforcing Role-Based Access Control for Secure Data Storage in the Cloud. In page 6, they have mentioned to choose hash functions $...
0
votes
1answer
124 views

Derive $x$ when given $g,g^x$ and $g^{(1/x)}$?

If an adversary has access to the generator g of a group G and is given access to $g^{x}$ and $g^{(1/x)}$, will it make it any easier to derive the value of $x$ compared to when he had access to only $...
11
votes
1answer
1k 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 ...
4
votes
1answer
319 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 ...
5
votes
2answers
508 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 $x$ ...
4
votes
1answer
1k 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 ...
2
votes
2answers
2k 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 ...
2
votes
1answer
270 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 $g^{t^{-1}}$,...
4
votes
3answers
90 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 ...
2
votes
1answer
215 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 ...
3
votes
1answer
436 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 $N$ ...
3
votes
1answer
126 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-...
7
votes
3answers
2k 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
votes
1answer
298 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 "...
1
vote
0answers
160 views

What is the hardness in Decisional Linear Assumption (DLIN)?

I had understood what does the DLIN assumption means and here is a related question. But I fail to understand the 'real hardness' in this problem. I would be grateful if someone can help me to ...
4
votes
1answer
75 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 ...
0
votes
1answer
101 views

Confusion regarding computing Multiplicative Inverse Modulo P?

May be a silly doubt, please rectify my confusion regarding below problem: For concreteness assume $g=2, p=11, a=6$ and $x=9$ $$A = g^a \bmod p = 2^6 \bmod 11 = 9$$ $$X = g^x \bmod p = 2^9 \bmod 11 ...
6
votes
3answers
1k 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
votes
1answer
394 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 ...
0
votes
1answer
210 views

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 ...
1
vote
1answer
378 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 ...
2
votes
2answers
568 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 ...
0
votes
1answer
145 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 $x_i \in \mathbb{Z}^*_p$. ...