Questions tagged [bilinear-pairing]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
3 votes
0 answers
111 views

Will a semi-hyperelliptic pairing be used in real-world cryptography if it is faster than state-of-the-art elliptic pairings?

Let $\mathbb{G}_1$, $\mathbb{G}_2$, $\mathbb{G}_T$ stand for three groups of the same large prime order $r$. I invented a pairing $e\!: \mathbb{G}_1 \times \mathbb{G}_2 \to \mathbb{G}_T$ (with ...
user avatar
1 vote
1 answer
334 views

Why the definition of bilinearity property is different in cryptography compared to mathematics?

Background: In Wikipedia (bilinear map definition), a condition listed as the following: For any $\lambda \in F, {\displaystyle B(\lambda v,w)=B(v,\lambda w)=\lambda B(v,w)}$ In a ...
user avatar
0 votes
0 answers
26 views

Multiplicities of poles of a divisor of a rational function w.r.t. an elliptic curve

I am reading Sec 5.8.2 in the textbook Introduction to Mathematical Cryptology (Hoffstein, Pipher and Silverman), a precursor to introducing the structure of Weil pairing. It first defines a rational ...
user avatar
  • 249
0 votes
0 answers
51 views

A doubt in pairing based cryptography

I have seen authors taking $G_1=G_2=G_T=G$ to be the same group of prime order $q$. What I know is that for pairing of type $$e:G_1\times G_2\rightarrow G_T,$$ size of the element in the target group ...
user avatar
1 vote
1 answer
55 views

Properties of the bilinear pairing groups?

I stumbled across this correctness of a scheme: $e(g^r, H(id)^x) = e(g^x, H(id))^r = e(g^x, H(id))^r$ and have a hard time following the properties of the bilinear pairing. Does anyone know the "...
user avatar
  • 35
0 votes
0 answers
28 views

Two different bilinear mappings in PBC

In pairings-based cryptography, are there any examples of systems where they use two different bilinear mappings. That is, they make use of both $e_1$ and $e_2$ where $e_1$ is a symmetric bilinear ...
user avatar
  • 249
1 vote
1 answer
58 views

Proof of knowledge of constant discrete log in the bilinear setting

Consider a pairing $\mathbb{e}: \mathbb{G}_1\times \mathbb{G}_2\longrightarrow \mathbb{G}_T$ with generators $g_1$, $g_2$ for $\mathbb{G}_1$, $\mathbb{G}_2$ respectively. The groups $\mathbb{G}_1$, $\...
user avatar
3 votes
0 answers
55 views

pairings and clifford algebra connection

Pairing notation seems to suggest that bilinear pairings could be related to Clifford Algebra (ie: Geometric Algebra); and we only have an odd choice of notation that hides this fact. For example, if ...
user avatar
  • 319
2 votes
1 answer
253 views

What is a function on a Line or a Curve?

I am reading up on Pairings using Elliptic curves & all the texts talk about functions on a Curve. I am finding it difficult to even figure out what they mean by "function on a curve" or ...
user avatar
  • 1,647
0 votes
0 answers
24 views

Can I map a message M to a target group of bilinear pairs

Consider a bilinear pairing $e(G_1,G_2)=G_T$, let $G_1,G_2,G_T$ be multiplicative cyclic groups of order p. If I have a message $m$, can I map $m$ to $G_T$, and encrypt $m$ as $m \times G_T$?
user avatar
  • 1
1 vote
0 answers
56 views

Multiplication of pairings vs. exponentiation of the group elements

Assume that we have a pairing as $e:G_1\times G_2\rightarrow G_T$. such that $g_1$ and $g_2$ are the generator of $G_1$ and $G_2$ respectively. In a protocol I have $A=\prod_{i=1}^n e(H(i),pk_i)$ ...
user avatar
  • 325
3 votes
1 answer
92 views

Strong Diffie Hellman in bilinear groups

The $n$-strong Diffie Hellman assumption state that given the subset $\{g, g^s,\cdots,g^{s^n}\} \subseteq \mathbb{G}$ in a cyclic group $\mathbb{G}$ of prime order $p$, a PPT algorithm cannot output $...
user avatar
1 vote
0 answers
74 views

What's the difference between Optimal ate pairing and R-ate pairing?

I compare the algorithm description of Optimal ate pairing and R-ate pairing, it turns out to me that the formulas are the same. So I'm a little confused, what's the difference between them? or is it ...
user avatar
4 votes
1 answer
235 views

Elliptic curve bilinear pairing parameters for 80-bit security level

I am reading a paper based on elliptic curve bilinear pairing groups. The author has defined the size of private key, public key etc in terms of $|\mathbb{G}_1|, |\mathbb{G}_2|$ and $|\mathbb{G}_T|$. ...
user avatar
3 votes
0 answers
56 views

Decisional Diffie-Hellman Assumption on Pairing Friendly Curves

It is known that the Decision Diffie-Hellman (DDH) problem can be easily solved over groups on pairing friendly curves (that is: one can use pairing to tell if $g^x$ and $g^y$ and $g^z$ forms a DH ...
user avatar
  • 75
2 votes
0 answers
81 views

Cryptographic invariant maps

In [BGK+18] in section 4, Boneh et al. write that: For any choice of ideal classes $\mathfrak{a}_1,\dots,\mathfrak{a}_n,\mathfrak{a}_1',\dots,\mathfrak{a}_n'$ in ${Cl}(\mathcal{O})$, the abelian ...
user avatar
  • 173
3 votes
1 answer
273 views

Is this pairing-based signature scheme secure?

There are a number of signature schemes on small domains based on bilinear pairings which do not use random oracles. Examples are the Boneh-Boyen schemes and an interesting one from Okamoto which ...
user avatar
  • 346