Questions tagged [pairings]

Pairing-based cryptography uses bilinear maps to create a gap group that allows efficient constructions of certain primitives.

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How to prove in zero-knowledge that the attributes of Pointcheval Sanders signature is the opening of a commitment?

In anonymous credentials schemes, it is possible to anonymously prove knowledge of a signature. Proposals for anonymous credentials with attributes also include a method for proving statements about ...
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52 views

Can you use ECDSA on pairing-friendly curves?

I'm learning about Elliptic curve cryptography. If I understand right, ECDSA and other algorithms used in ECC are dependent on the curve chosen. So, before you want to use ECDSA, you first have to ...
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Is BBS+ 1. a multi-messages signing protocol or 2. a group signature signed by a member of a group anonymously?

There's no doubt that the BBS Signature was born from this classic Short Group Signature paper in 2004. It's capable of Zero-knowledge. In the paper, section 5, it describes how a member of a group, ...
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Importance of supersingularity of elliptic curves

I'm struggling to understand the high-level idea of "Verifiable Delay Functions from Supersingular Isogenies and Pairings" (https://eprint.iacr.org/2019/166.pdf) by De Feo et al. I will ...
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In q-SDH problem, where are those points $\frac{1}{\beta+x}g$ or $g_1^\frac{1}{x+c}$ on elliptic curve?

For the q-SDH problem, given the generator $g_1$ as a point on the elliptic curve, I can picture the $\beta g_1, \beta^2g_1, ..., \beta^qg_1$ since we can simply do the point adding $g_1$ multiple of $...
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How to achieve identity authentication without revealing credentials

I am looking at a scenario where I would like to claim to an authority (call it A) that I am indeed me without revealing my identity documents. I am guessing some zero knowledge protocol has to be ...
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68 views

Exponentiation Problem of G2 in MNT curve

I made a simple python program in the Charm framework (https://github.com/JHUISI/charm): ...
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Where can I find references to the theory behind PBC library?

I don't know if this is the right place to ask this type of question. Anyway, I'm using PBC library for a project, but I'm a very newbie for what concerns pairing based cryptography. Then I ask some ...
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what is the non_residue value of BN462

I found an algorithm to find the frobenius coeffs here. Here, the non_residue of BLS12-381 is -1, And non_residue of BLS12-377 is written -5. How can I find this value? And what is the non_residue ...
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Can I convert $F_{q^{12}}$ to $F_q$?

I seeing a paper about Elliptic curve based proxy re-encryption. And I want to implement this through BLS12-381 Curve. However, When looking at the documentation for paring or the library, the value ...
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What is different between G1×G1→GT and G1×G2→GT in the bilinear pairing?

It is an implementation of the bls12-381 algorithm known as pairing-friendly, at GitHub. Looking at this, the pairing parameters are $G_1$ and $G_2$, $G_1$ is the point of $F_q$, $G_2$ is the point of ...
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How to estimate the computation overhead of ECDSA?

I am using ECDSA as a digital signature scheme. Using Charm, I got the timing for the multiplication, exponentiation, and pairing operations; they take 0.005, 9, and 4.4 ms respectively. I want to ...
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Fixed variable in Groth16

In the paper On the Size of Pairing-based Non-interactive Arguments by Jens Groth, it is always referred in the equations to satisfy that $a_0 = 1$ and the others $a_1, ..., a_m \in \mathbb{F}$. I am ...
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Can Pedersen commitment be used in pairing groups?

For bilinear groups: $(p,\mathbb{G}_1,\mathbb{G_2},\mathbb{G}_T,e,g_1,h_1,g_2,h_2)$, where $\mathbb{G}_1,\mathbb{G_2},\mathbb{G}_T$ are groups of prime oder $p$. $g_1,h_1$ are generators of $\mathbb{G}...
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Current situation of bilinear pairing protocols

The bilinear pairings are considered as the key enabler for many novel cryptographic protocols, such as three-party one round DH[1], shorter signatures and certificateless (ID-based) crypto[3] , which ...
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Understanding the groups used in bilinear Ate-pairing

The bilinear ate pairing $e:G_1\times G_2 \rightarrow G_T$ is defined over the following groups: \begin{equation} \begin{aligned} & G_1 = E(\mathbb{F}_p)[r] \cap Ker(\pi_p-[1]), \\ & G_2 = E(...
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Groth-Sahai Proof of multi-exponentiation

I want to create a NIZK proof of a multi-exponentiation via Groth-Sahai proofs. In the lectures of Jens Groth in a winter school of Bar-Ilan University, he shows a way to transform multi-...
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pairing teta function calculation

I am struggling with some pairing calculation techniques and i do not know how to approach them I would be glad if anyone can explain them in an easy way. For example i have these two formulas 1) ...
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Is the k-sum attack efficient in bilinear pairing variants?

From the k-sum definition, the objective is to solve something like: $\sum_{i=1}^n c_i = v_1$ where $v_1$ is known. If we use Elliptic Curve Points ($C_i$ and $G$) we can define a bilinear pairing ...
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106 views

Bilinear Map over group of unknown order

Is it possible to build a bilinear map where the underlying group is of unknown order? To maintain context, the original question appears below. As per poncho's excellent answer, my original idea is ...
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What are the BLS12-381 settings?

I can't find the exact type and settings for the BLS12-381 curve. Is this type-3 in Symmetric XDH settings ?
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Can BLS aggregate signatures be merged?

In some non-interactive, pairing-friendly signature scheme, such as BLS12-381, is it possible to merge partially-overlapping aggregate signatures? For example, say you have two aggregate signatures, $...
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Bilinear pairings with groups of safe prime order

I am looking for a curve with a bilinear pairing of safe prime order. I.e. a bilinear map $G \times G \rightarrow G_T$ where $|G| = p$, $p = 2q +1$, and $p,q$ are both prime. Is it feasible to find ...
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Pair-friendly elliptic curves vs non friendly

Group law operations on pair-friendly elliptic curves are slower than in non friendly elliptic curves, but how much slower? Can't seem to find a performance comparison between the two for a given ...
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Schnorr signature verification via Pairing-Based Cryptography?

Let $e: G^{\dagger}_{1} \times G_{2} \mapsto G_{T}$ define the pairing. Assuming an output $\sigma = (c, p)$ similar to a Schnorr signature: $s \times Y \mapsto Y_{s}$ and $m \times G \mapsto M$ $c = ...
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Are all MNT curves assumed to hold XDH?

For one of my projects, I need a pairing group which holds the External co-Diffie-Hellman assumption. I am trying to implement it using Charm crypto python modules which provides support for MNT ...
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Variants of Bilinear Diffie-Hellman Assumption

Could someone point me to the paper/reference where the following variant of q-strong Bilinear Diffie-Hellman assumption was used? Given $s \in \mathbb{Z}_p^*$ and $g, g^{\frac{1}{s}}, g^{s}, g^{s^2},...
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What to learn in order to understand Weil Divisors

My current role as a software developer is leading into areas of research in elliptic-curve-cryptography that keeps bringing me back to bilinear pairings. This includes BLS Signatures, CL Signatures ...
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Group Signatures with VLR Revocation Check

I was reading the paper by Boneh et al. Link. The scheme describes what is called a Verifier Local Revocation Technique. The Scheme assumes that a Revocation List [RL] allows each verifier to check ...
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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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Pairing-free group based encryption scheme

Can someone explain what is a pairing-free group ? There is encryption schemes, for example 1, that qualify there proposed construction as pairing-free group. However, they use bilinear group in the ...
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What's the difference between symmetric and asymmetic multilinear map?

In pairing-based cryptography, we have 3 types, namely Type-I, where $\mathbb{G}_1 = \mathbb{G}_2$, and in Type-II and Type-III we have that $\mathbb{G}_1 \neq \mathbb{G}_2$, however in Type-II we ...
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New Pairing Friendly Curves for SM9

A paper was presented at the INSCRYPT 2018 called: "Searching BN Curves for SM9 (Pay-Walled)" SM9 is a Chinese National Identity Based Cryptography Standard and was originally published using a 256-...
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187 views

Do Weil, Tate, and Ate pairings exist on all elliptic curves?

I don't know much about the math behind elliptic curves. Do Weil, Tate and Ate pairings exist on all elliptic curves? If the answer is negative, then what pairings do MNT, BN and SS curves have? ...
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405 views

What does the number 256 in pairing curve BN256 indicate?

There are many pairing based elliptic curves like MNT curves, BN curves, SS curves etc., When we say BN256 curve, what does the number 256 indicate? Is it some group order or number of bits required ...
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102 views

Is there a concept of embedding degree for non-pairing based elliptic curves?

From this post, I learned the concept of embedding degree. Intuitively, if embedding degree of an elliptic curve $E(F_p)$ is $k$, it means there is a way to transform points in $E(F_p)$ to $F_{p^k}$. ...
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Does this equation hold in a bilinear map?

I would like to verify whether or not the following equation holds: $e(a,c)^{c1\cdot c2\cdot c3}e(b,c)^{c1\cdot c2\cdot c4}==e(a,c)^{c2\cdot c3}e(b,c)^{c1^2\cdot c2\cdot c4}$ for appropriately ...
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If curve bn256/bls12 support the isomorphism from $G_2$ to $G_1$?

Is bn256 or bls12 a type-2 pairing-friendly curve? As Dan Boneh said here While in many pairing instantiations this ψ exists naturally, in some instantiations it does not. However I can not find ...
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141 views

Are there any NIST curves with pairings?

NIST FIPS.186-4 has standardized 5 ECC curves on field 𝔽𝑝 (P-192, P-224, P-256, P-384, P-521) and 10 elliptic curves on binary fields. None of them seem to have pairings. Are there any standard ...
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Computing a sextic twist

Let $(x,y) \in E'_{\mathbb{F}_{p^2}}$ be a point of the sextic twist. I am currently trying to compute: $\psi : (x, y) \leftarrow (\mu^2x,\mu^3y)$ with $\mu \in \mathbb{F}_{p^{12}}$ the root of $(Y^...
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Security strength of JPBC Type A curve compared to SecP curve

I recently encountered some problems when learning about the JPBC library. Does the curve generated by (J)PBC using the method typeAcurvegenerator(160,512) and the ...
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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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Elliptic Curves for Pairings

how can is possible to know if an elliptic curve is suitable for protocols that adopt pairing? For example, in Certificateless Cryptography with pairing, is it possible to know if all elliptic curves ...
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Prove that a given signature scheme is secure under random message attacks

This is a follow up to my previous question. Consider the following signature scheme: $\operatorname{KeyGen} (1^k$) : On input of a security parameter $k$, choose a symmetric bilinear group with $e : ...
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Prove that a signature scheme is RUF-NMA and not EUF-CMA

I am working on the following exercise: Now, assume the following signature scheme: $\operatorname{KeyGen} (1^k$) : On input of a security parameter $k$, choose a symmetric bilinear group with $e : G ...
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Is this an error in the Pinocchio Protocol paper

I am going through the Pinocchio protocol paper and I need 2 clarifications in the section Protocol 1 (Verifiable Computation from strong QAP). The part that explains the Verify process, which ...
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138 views

How to apply lagrange interpolation on bilinear pairings?

I have seen in some places applications of Shamir secret sharing and lagrange interpolation mixed with bilinear pairings, however I fail to understand how this works. For instance, here I find the ...
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Is there a bilinear map on a non abelian group or non cyclic group?

I've recently been studying a pairing map on cryptography. In usual definition, a pairing map is always defined on the cyclic group G. Is it possible to construct a bilinear map on a non-abelian group ...
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How to change the following commitment scheme to Pedersen commitment?

In my question here Zero knowledge set membership protocol The suggested solution allows a prover to choose a commitment $C$. Then, A trusted third party ($T$) can validate if $C$ is valid or not. ...
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Does this pairing-based signature scheme work?

Suppose $g$ is a pairing-friendly elliptic curve with subgroup generators $G_1$ and $G_2$. Suppose also that $M$ is the message I want to sign. Setup Compute $A = a \cdot G_1$ and $P = p \cdot G_2$, ...

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