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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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 ...
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29 views

Which is the fastest way to find a member of a subgroup with known size modulo prime $P$, with know factorization of $P-1$?->$x$ with $x^s \mod P = 1$

Assuming you know the factorization of used prime $P-1$ $P-1 = s \cdot f_2\cdot f_3...f_i$ Now you want to find a member of a subgroup $\mathbb{Z}_s$. This means any $x$ with $x^s \equiv 1 \mod P $ ...
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119 views

Can we prove $(a,b,c,d)$ is of form $(g,g^x,g^y,g^{xy})$ without knowing $\phi (N)$?

assume $N=pq$ where $p$ and $q$ are two large prime numbers and all computations are modulo $N$. given $\phi(N)$, one can prove that the tuple $(a,b,c,d)$ is of the form $(g, g^x,g^y, g^{xy})$, i.e....
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57 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 ...
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129 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 $x_i \in \mathbb{Z}^*_p$. ...
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900 views

How to perform homomorphic multiplication in ElGamal?

How can I compute homomorphic multiplication in ElGamal? That is: Given two ciphertexts $(R_1,c_1)$ and $(R_2,c_2)$ corresponding to plaintexts $m_1$ and $m_2$ under some public key; how can I compute ...
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242 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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26 views

Is there any property of the product you can predict before using $n$-times generator $g$ $\mod P$? Can any $n$'th element have a certain property?

Given a value $v$ which is in same group as the generator $g$ modulo prime $P$. The group size is a prime $s$. $v = g^a \mod P$ Only known values are $v,g,P,s$. Some (possible) computation of other ...
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How does the order of a group, it's torsion subgroup and the co-factor link?

Given an elliptic curve that defines some group of non-prime order, with co-factor h. Would it then have a h-torsion subgroup? What are the implications for ECC ...
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42 views

What are some potential applications of trilinear mappings?

In "Applications of Multilinear Forms to Cryptography" Boneh and Silverberg give "one-round n-way Diffie-Hellman key exchange protocol" as a potential application of n-multilinear mappings. If we ...
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127 views

Proof in RSA encryption over multiplicative group

I everyone, I am considering an RSA encryption over the multiplicative group $G = (Z/nZ)$ of the ring $Z/nZ$, where $n = pq$, and $p$ and $q$ are distinct odd primes. First, I want to prove that $H=...
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35 views

Attack against factorization of $p-1$ of $\mathbb{Z}_p^*$ group

It is said that for the group $\mathbb{Z}_p^*$, the factorization of $p-1$, is critical. If $p-1$ has some small factors $q_1, q_2, q_3, q_4$, then when we transmit $g^x \bmod p$ where $g$ is a ...
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38 views

Algorithm to compute DLOG for elliptic curve $E(F_p)$ with order p

I was reading about elliptic curves in this pdf. Page 55 of the pdf states that if number of points on elliptic curve #$E(F_p) = p$, then there exists a p-adic logarithmic map that homomorphically ...
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174 views

Size of group for Elliptic curves vs RSA for equal security

For my research, I would like to compare the efficiency of a scheme when instantiated with Elliptic curves and RSA. So, I would like to know a "latest" comparison (as of 2018) on what group sizes of ...
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277 views

Number of generators of an elliptic curve

Consider the elliptic curve E:$y^2 = x^3 + 3x + 11\,\, mod\,\, 19$. Two questions: Let the cardinality of the set of points on the elliptic curve( including $O$ ) be $|E| = 25$. How many points are ...
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280 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......
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62 views

What is the difference between these cyclic groups

$\mathbb{Z}^*_p$ vs $\mathbb{Z}^*_{p-1}$ vs $\mathbb{Z}^*_{p^2}$ vs $\mathbb{Z}^+_{p^2}$ I know $p$ is the value. The value create must be coprime to $p$. Does that mean that the value create must be ...
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82 views

$f : \mathbb{Z}_n \rightarrow \mathbb{Z}^\times_n$?

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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97 views

find additive inverse of modular arithmetic [closed]

For a set to be called as a ring, it should have the following properties closed commutative associative Identity existence Inverse existence but how is Z7 a ring, as there aren't any inverse ...
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212 views

Abelian groups in Elliptic curves [closed]

Do every elliptic curve defined over a prime field forms an abelian group?
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255 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: ...
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113 views

Meaning of Multiplicative Group to the power n i.e. Z^n [closed]

What does $Z_q^n$ mean in this notation?      References: on the 2nd paragraph of page: http://en.wikipedia.org/wiki/Learning_with_errors
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99 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?
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150 views

Symmetry for finite cyclic groups (Z/pZ)∗

How well is it known that for $i$ such that $1 \leq i \leq \frac{p − 1}2$: $$ g^{i+(p−1)/2} = g^{i−1+(p−1)/2} − g^i + g^{i−1} \pmod p $$ Whilst working in the finite cyclic group of prime moduli $(Z/...
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2answers
40 views

How many different subgroups exist in composite order groups? [closed]

For a composite order group $G$ of order $N$, s.t. $N = pq$ where $p$ and $q$ are prime numbers. By the Lagrange's theorem, the order of every element should either be $p$ or $q$ or $N$. Every such ...