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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100 views

Double discrete logarithm on elliptic curve

Background: I am attempting to implement the paper Publicly Verifiable Secret Sharing. I managed to get it working using modular groups, but when I want to make it more efficient by transferring to ...
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74 views

Why do we use prime numbers apart from hard factorization?

Why are prime numbers important in cryptographic constructs? I am not interested in RSA examples where factorization is the hard problem itself, that makes sense. However wherever I go I encounter ...
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115 views

Calculation of the order of the cosets used in defining the Tate Pairing

I'm working through Pairings for Beginners by Craig Costello, and am trying to understand the preamble to the Tate pairing. (See p. 70 ff., section 5.2 of of the PDF.). I'm having trouble following a ...
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61 views

How to prove element equality in G1 using Groth-Sahai proofs

In the BeleniosRF e-voting scheme, Groth-Sahai proofs with the "Instantiation based on the SXDH assumption" are used (see https://eprint.iacr.org/2007/155, version 20160411:065033, page 24). In the ...
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260 views

Proof one-time pad is perfectly secret with eavesdropping game definition

I have the following definition of perfect secrecy (please assume that the probabilistic version is not available): If we consider the eavesdropping game given by: $$\begin{array}{|r | r|} \hline ...
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Is this problem same as discrete logarithm?

Given $g,h\in\mathbb Z_p$ where $g$ generates $\mathbb Z_p^\star$ Discrete logarithm problem is to find $z$ such that $g^z\equiv h\bmod p$ holds. Take the problem given $g,g',h$ where $g^z\equiv h\...
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142 views

Sum probability

if $a,b,c$ are selected at random from a large cyclic group, then what could be the probability that $g^{a+b}$=$g^c$? it simply corresponds to the probability that multiplication of two random ...
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496 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 ...
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320 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 ...
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582 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 ...
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881 views

ElGamal and Schnorr groups

As I gather, a normal practice for choosing a cyclic group for ElGamal key generation is to find a safe prime $p$ and use a multiplicative cyclic group with modulus $p$ and order $q = (p-1)/2$. ...
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Equality with Bilinear Maps

Let e be a bilinear pairing function and $g_1$ and $g_2$ be the generators of $G_1$ and $G_2$ Given $e(a,g_2^x), e(b,g_2^y), g_2^x,g_2^y$ is there a way to find out if $a = b$?
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Schoofs Algorithm

I studied Schoofs Algorithm described by Washington. On page 125 he says that we could write $y'/y$ as a function of $x$, which makes sense since earlier on the page he denotes $y'= r_{2,j}(x)y$. But ...
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A related question to “relationship between generating elements given by cycles in Cayley graphs”

I am writing this question with reference to the post at Relationship between generating elements given by cycles in Cayley graph When considering the generating elements $g_qg_p$, does it have the ...
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184 views

Steps to determine the single element generators for a multiplicative group

As student I've been asked the following question: Consider the specific prime $p=17$. Determine the single element generators (by hand or by Java program) of $F^*_{17}$. Recall $F^*_{17}$ is the ...
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FLT is partly applying to RSA equation, and also relation between ED mod phi and Phi + 1 mod N

After numerous attempts from myself and all of you guys, I finally came to understand RSA. I can now prove it and understand how I got there. But I still have some very few polishing questions. 1) We ...
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Proof of two pairs with same exponent

Lets assume we have a group $G$ with unknown order. And we have a pair $(A_1,K_1), (A_2,K_2)$ in which all $A_1,K_1,A_2,K_2$ are group elements. The claim is $A_1= K_1 ^ x$ and $A_2 = K_2 ^ x$. or ...
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How do pairings behave on G2/twist points off the prime order subgroup?

$\newcommand{\F}{\mathbb{F}}$ Consider the ate pairing defined on a curve $G_1 = E(\F_q)$ and $G_2 = E'(\F_{q^r})$ where $E'$ is a twist of $E$ with the twisting isomorphism defined over $\F_{q^r}$. ...
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How is the generator used in Feldman's Verifiable Secret Sharing scheme determined? [duplicate]

According to the Wikipedia description of Feldman's VSS scheme First, a cyclic group G of prime order p, along with a generator g of G, is chosen publicly as a system parameter. (Typically, one ...
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112 views

Drawbacks of Schnorr Authentication that require Fiat-Shamir and Random Oracles?

I've been going through G. Maxwell's paper on the Borromean Ring Signature, and I don't fully understand this part on Schnorr Signature. If some could explain it more intuitively thank you. "...
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discrete logarithm vs normal logarithm [duplicate]

Crypto schemes normally use discrete logarithm instead of normal logarithm. I think this has to do with the fact that discrete logarithm is hard to solve while normal logarithm isn't. Can someone ...
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Can Bitcoin mining solve Graph Isomorphism-related problems?

Given a cryptographic hash $H:\{0,1\}^*\mapsto\{0,1\}^N$ and data $D\in\{0,1\}^*$, the Hashcash/Bitcoin Proof-of-Work entails finding a nonce $x$ such that $H(x\Vert D)$ begins with $d$ leading zeros, ...
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Is there a flaw in this ring signature scheme? [closed]

Having read some papers about RSA accumulators applied to ring signatures schemes, I ended up thinking why would we need to accumulate all the members public keys for our specific use case. So I came ...
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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 $...
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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 ...
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Converting a number to a member of a multiplicative cyclic group

I am currently trying to make an implementation of the ElGamal encryption for educational purposes. As I understand it, when using the encryption with multiplicative cyclic groups, one generates a ...
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66 views

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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378 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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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 ...
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Why is ElGamalEngine in bouncy castle restricting input data to be less than the length of the group parameter p?

I am currently using the ElGamalEngine of bouncy castle to implement exponential ElGamal for the purpose of making ElGamal additively homomorphic. To do this i raise the message to be encrypted to the ...
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If I know an element and it's inverse, can I learn the modulus?

If I know an element $x$ in a group, and it's inverse $x^{-1}$, can I guess the modulo, or with a probability?
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Why are some group representations much easier to compute discrete logarithm for? [duplicate]

The multiplicative group mod $p$ is isometric to the additive group mod $p-1$, yet computing discrete logarithms in the additive group is easy and completing discrete logarithms in the multiplicative ...
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A Question about Notations and Groups

Please consider the following question: Determine the order of all the elements of the following multiplicative groups. You can write a C or Java program to do this. a. $Z_{21}^*$ b. $Z_{23}^*$ Now ...
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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 ...
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What is Group in Diffie-Hellman?

I understand how Diffie-Hellman key-exchange works. Mainly, two parties agrees in a prime $p$ and a generator $g$. Then one party selects its private exponenet $x$, computes its public value $g^x \...
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Are there any zero knowledge protocols which do not rely on a Group?

To me (new), it seems that a lot of cryptography relies on group theory. Are there any zero knowledge protocols which do not rely on a group?
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Key exchange protocols non-reducible to groups

Some questions I have ended up wondering about while reading through many of key-exchange protocols are: Is there an intrinsic reason, why most key exchange protocols use group-based approaches apart ...
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What does the “description of group $G$” includes?

I was reading here:second discrete log meaning in the solution and also here:key generation, first point where the say given $G$ (or its description). My question is what does this description ...
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275 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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Why is Multiplicative group used in RSA or Euler's theorem but not additive?

I am on the verge to understanding RSA, but suddenly a question popped into mind. When we are calculating $U(N)$ i.e $U(PQ)$, we are taking invertible elements that are co-prime to $N$. For example, $...
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What's the number of unique possible Cayley tables in a 16*16 grid for XOR'ing single hex characters?

A few days ago, I designed and s-box then derived the following Cayley table of all possible XOR outputs of hex digits in the range of ${2^4}$ and was curious how many such "valid" possible ...
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Elliptic curves - operations in larger groups - performance

According to my measurements and to this work, it seems that operations, for example scalar multiplication, are more expensive in larger groups. If I have, for example, an 80-bit elliptic curve and an ...
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282 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 ...
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Pollard's Rho - Constructing the random function

Suppose we are aiming to solve the discrete logarithm problem $\alpha^x=\beta$ in some cyclic group $G=<\alpha>$. Then we are looking for a (uniformly) random sequence of elements of the form $\...
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Elliptic curve group over a prime finite field $F_p$

If $p$ is a big prime, and the elliptic curve $E$ is defined over $F_p$ by the equation $y^2=x^3+ax+b$ where $a,b\in F_p$. The point on $E/F_p$ together with the infinite point $\mathcal{O}$ form a ...
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Collusion resistance in proxy re-encryption scheme depending on bilinear map rules

A proxy re-encryption scheme is collusion resistant, if the proxy and a delegatee are not able to recover the secret key of the delegator. For example, when we have a message that was originally ...
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Rounding down in the exponent of group element

I have been struggling to find the algorithm $\mathcal{A}$ in the following. Let $(G,g,q)$ be the group parameter, $p << q$, $x\in \mathbb{Z}_q$, can we build the following algorithm: $$\...
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Is modulus a prime number important for non-symmetric cryptology?

From this link Generation of a cyclic group of prime order we know how to generate a prime order group. This illustrates why a prime order group is important. But why is modulus a prime number ...
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370 views

Is that possible to calculate modular inverse of a point on elliptic curves?

Imagine that you are given a point $P$ so that $P=a\times G$. If you have no knowledge of $a$ is that possible to calculate point $I$ so that $I$ is the modular inverse of $P$? We know that over ...
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Operation Table for Permutation Group

I cannot seem to figure out how the operation table for this permutation is formed. Is it multiplying each index and doing modulus? I can't seem to figure out. This is a Table 4.2 found in "...