Questions tagged [bilinear-pairing]

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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 ...
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 ...
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 ...
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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 ...
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 "...
• 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 ...
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1 vote
1 answer
58 views

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1 vote
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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 ...
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 $|\mathbb{G}_T|$. ...
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 ...
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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 ...
• 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 ...
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