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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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23 views

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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81 views

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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1answer
33 views

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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37 views

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 $...
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30 views

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 ...
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53 views

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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1answer
56 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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53 views

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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1answer
50 views

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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13 views

Are compressed pairings used for Barreto-Naehrig curves in practice?

In 2009 Galbraith and Lin wrote the article "Computing Pairings Using x-Coordinates Only" https://link.springer.com/article/10.1007/s10623-008-9233-3, where they proposed to compute pairings on ...
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2answers
72 views

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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1answer
77 views

Can we use PHE or SWHE instead of bilinear pairings in ZK-SNARKS?

In ZK Snarks bilinear pairings are used to do "encrypted computation". I was wondering if we can use Partial Homomorphic Encryption or Somewhat Homomorphic Encryption instead of bilinear pairings. Can ...
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65 views

NIST ECC Curves without pairings

NIST FIPS.186-4 has standardized 5 ECC curves on field $\mathbb{F}_p$ (P-192, P-224, P-256, P-384, P-521) and 10 elliptic curves on binary fields. I need to use ECC curves without pairings for my ...
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1answer
97 views

Order of twisted curve in pairings

We are doing optimal Ate pairings using a Barreto-Naehrig curve, and I am trying to make sure that an observation I made generalizes. We define $E$ as $y^2 = x^3 + 3$ and use the tower of extensions \...
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1answer
83 views

Pairing-based cryptography VS ZK-proofs: what's more efficient for threshold systems?

In some papers (e.g. 1, 2) the authors approve that pairings are more efficient than classic zk-proofs (e.g. proof of discrete logarithm knowledge) for the described applications (threshold encryption,...
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71 views

Security of BLS under additional information on the secret key

Question A Is the BLS signature scheme still secure if an adversary in addition to the public key $ pk = g_2 \, sk \in \mathbb{G}_2 $ also obtains additional information on the private key $ sk $, ...
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The multi-linear maps are prone to zeroizing attack. Can anyone explain how the zeroizing attack breaks the computational assumptions?

I read that many cryptographic schemes based on multi-linear mapping are not secure because the zero testing procedure and zero encoding help an adversary to reveal the secrets and because of it, many ...
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1answer
260 views

Simple explanation of Miller's algorithm

Could someone explain to me in few lines (even one sentence) what Miller's algorithm computes? Without talking about divisors and all the other concepts, I would like to be able to explain it to ...
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54 views

What should I change in my implementation to pass from a curve $y^2=x^3+1$ to $y^2=x^3+x$

I tried to change my implementation of pairing in curves $y^2 = x^3 + 1$ to use curves of the type $y^2 = x^3 + x$ but it didn't work. I thought the only thing I had to change in my code was the ...
3
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1answer
173 views

How to find a primitive cube root of unity

I would like some help to understand how to compute a distortion map and the result of a pairing. I know that with this equation : $E : y^2 = x^3 + 1$ over some $\mathbb{F}_p$ we have a distortion ...
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149 views

Proving multiple products “in the exponent”

I'm trying to come up with a small-sized (non-interactive) proof for a Diffie-Hellman-like statement. I'll start by giving an example. The prover has $g^a, g^b, g^c, g^{ac}, g^{ab}, g^{bc}, g^{abc}$. ...
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110 views

Pairing in implementation of a simple digital signature

I'm trying to implement in python the 1st scheme from the paper "Efficient Identity Based Signature Schemes Based on Pairings" by Hess to learn elliptic curve cryptography. It's a simple digital ...
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1answer
224 views

Why “pairings on elliptic curve” are used?

I'm just curious why do we use pairings on elliptic curve cryptography. There is a lot of information about how to use it, but I cannot find information about why to use it. Can somebody help?
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1answer
131 views

Understanding the big picture with regard to how anonymous credentials work

I'm trying to learn about anonymous credential systems. I started with Lysyanskaya's thesis, which gives an outline of how to build an AC system given some particular primitives (signature scheme, ...
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1answer
215 views

Weil pairing of P and Q

it the following example for page 56 of the book http://www.craigcostello.com.au/pairings/PairingsForBeginners.pdf I could not understand how the author found the Weil pairing of two points p and Q. I ...
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2answers
183 views

DDH and pairings are not contradictory in RingCT 2.0?

As I know DDH assumption and bilinear pairings are contradictory, but I see this in a paper, RingCT 2.0. How could this be ok? Linkable ring signature will be attacked by bilinear pairings.
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Complexity lower bound in Uber-Assumption family

I try to wrap my head around calculating of complexity lower bound using Corollary 1 from The Uber-Assumption Family by X.Boyen (http://www.academia.edu/download/30698012/...
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1answer
114 views

co-DHP* in G2 for type 3 pairing

In type 3 pairing ($e:G_1 \times G_2\rightarrow G_\tau$ where $G_1\ne G_2$ and no isomorphism from $G_2$ to $G_1$ is known) we have co-DHP* problem: Given $H, aP \in G_1$ and $aQ\in G_2$ calculate $...
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1answer
328 views

Pairing in Cryptography [closed]

I am having problems in doing pairing in cryptography. I have tried the PBC library but seems like it doesn't work and I do not understand the documentation because it uses C++ language. Is there a ...
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0answers
306 views

Functional signatures with ECDSA Verifier

Functional signatures use some master key to create a functional signature key. The functional key can be used to sign messages, but only if the message is accepted by some predicate function. For ...
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79 views

Pairings over elliptic curves on rings

Looking at this presentation, Boneh says that elliptic curves could be defined over $\mathbb{Z}/n\mathbb{Z}$ and not necessarily over a prime field $\mathbb{F}_p$ and hence we could define pairings ...
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1answer
377 views

Can the precompiles in Ethereum Byzantium for pairings be used for implementation of BLS verification?

I asked this in the Ethereum StackExchange yesterday but I figured it would be more appropriate to ask it here. I am looking into implementing some operations for the BLS signature scheme in ...
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42 views

Can someone give me an idea of active areas relating to pairing based cryptography, and what they involve?

Apologies if this isn't the place to ask this question. I'm an undergraduate math student and I've been reading about elliptic curves. I've learned about the Weil and Tate pairings as well as Miller's ...
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1answer
104 views

Linkable Ring Signatures on Pairing-Friendly curves

I'm working on a pairing-friendly (Barreto-Naehrig) elliptic curve. If I understand correctly, one consequence of this is that the Decisional Diffie-Hellman assumption no longer holds, but the ...
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198 views

What pairings are used in PBC and charm-crypto with the proposed elliptic curves?

I'm implementing a pairing-based signature scheme using Charm Crypto and PBC, and I'm struggling to understand the relationship between the curves proposed for use with pairings and the pairings ...
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1answer
73 views

Question about the location of the r-torsion in the quotient group used in the Tate pairing

I'm working through Pairings for Beginners by Craig Costello, and am trying to understand the Tate pairing. He defines $rE = \{r*P | P \in E(\mathbb{F}_{q^k})\}$ and then forms the quotient group $E(\...
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1answer
87 views

Usage of twists at pairing-based cryptography

First of all I would like to understand how twists are used in pairings. The 2nd step is, how to use them to improve the calculation speed? Say $E'(\mathbb F_{p^2})$ is a twist of $E(\mathbb F_p)$. ...
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1answer
373 views

Usage of pairings in proxy re-encryption algorithm

I am having trouble understanding one part of the AFGH algorithm for proxy re-encryption (my background in discrete mathematics is lacking a bit). The paper describes the algorithm setup the ...
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0answers
86 views

Certificate Master Public Key with Pairing

I am currently working on CipherText Policy Attribute Based Encryption (CP-ABE) scheme. If we see the scheme, we can notice that the Authority may generate a Master Public Key and a Master Private Key(...
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1answer
264 views

Can Curve25519 be used for pairing-based cryptography?

We usually need pairing-friendly curves in order to use them for bilinear pairing computation. Is Curve25519 a pairing-friendly curve? If not, can we still use it for pairings and how much more ...
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0answers
141 views

Frey-Rück Attack (FR-Reduction) - Tate Pairing

I am trying to understand the Frey-Rück attack and found different ways of a possible implementation. Since I am not yet very familiar with the Tate-Lichtenbaum pairing and the theory of divisors I ...
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1answer
184 views

Does the following Diffie-Hellman problem hold in bilinear groups $G\times G \rightarrow G_T$

For every PPT distinguisher A there exists a negligible function $neg(·)$ such that for all $\lambda$ $|\Pr[A(1^\lambda, g,g_1^a,g_1^b,g_1^{ab}) = 1] - \Pr[A(1^\lambda, g,g_1^a,g_1^b,g_1^z) = 1]| \...
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1answer
244 views

Bilinear Form: Weil pairing

I am doing an example on Weil pairings, and for that purpose I follow the thesis of Alex Edward Aftuck, The Weil Pairing on Elliptic Curves and Its Cryptographic Applications. By following his thesis,...
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0answers
177 views

Prime extension field encoding ASN.1

ASN.1 encoding for elliptic curve cryptography is recommended by Certicom, as explained at a related question, covering curves over prime fields and binary extension fields. I'm looking for known ...
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1answer
416 views

Elliptic curves with pairings at 128-bit security in libpbc?

I am using Ben Lynn's libpbc to implement a BLS threshold signature scheme and I am aiming for 128-bit security (i.e., a forgery attack should take around $2^{128}$ ...
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1answer
181 views

Are pairings still the most efficient implementation for identity and attribute-based encryption?

I read on Wikipedia: [...] pairings have also been used to construct many cryptographic systems for which no other efficient implementation is known, such as identity based encryption or attribute ...
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1answer
302 views

Pairing on FourQ

How would you define a pairing on the - so called - curve "Four$\mathbb Q$? Since FourQ is a twisted Edwards curve, given by $E/\mathbb F_{q}:\ -x^2+y^2 = 1+dx^2y^2$, where $d\in\mathbb F_p(i), q=p^2,...
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2answers
188 views

Special-purpose witness encryption without multilinear maps

In Witness Encryption and its Applications Garg et al describe "witness encryption" which allows one to encrypt some specified data to a NP problem, such that another party can decrypt iff they ...
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1answer
157 views

Is it safe to use pairing-based cryptography in a commercial product?

My findings don't show any standardization(NIST, FIPS?) of pairing-based cryptography. Would this mean that it is a bad idea to incorporate pairing-based cryptography(bilinear groups) within a ...
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1answer
78 views

Does pairings based cryptography inherently require a CRS/trusted setup?

In all algorithms I've seen that rely on pairings-based cryptography (some examples: snarks without PCPs, more snarks, sublinear ring signatures), a common reference string is required. Is this always ...