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

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Type 1 pairings and encoding function from $\mathbb G_T$ to $\mathbb G$

Let $e : \mathbb G \times \mathbb G \to \mathbb G_T$ be a Type 1 bilinear pairing. Is it possible to define an encoding function $E : \mathbb G_T \to \mathbb G$? If so, how could this encoding ...
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30 views

Bilinear pairing arithmetic - cryptographic accumulators

For calculating accumulated values for set of elements chosen randomly from say $ { e_1 ,e_2,...e_n}\varepsilon X $ we use the formula $acc= g^{f(e,s)}$ where $ f(e,s)= (e_1+s)(e_2+s).....(e_n+s)$ ...
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60 views

Bilinear pairing arithmetic

Is this $e(g^x,g^yH^z) = e(g^x,g^y)e(g^x,H^z)$ expression is true? where $ g$ is the generator and $ H \in G $
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109 views

security proof in pairing based cryptography

Let : $G_{0}$ and $G_{1}$ be two multiplicative cyclic groups of prime order $p$. $g$ is a generator of $G_{0}$ and $e$ is a bilinear map, $e : G0 × G0 → G1.$ $𝐶_{1} = 𝑔^{𝛽𝑠_{1}} $, $ 𝐶_{2} = ...
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53 views

Bilinear map + commitment

Let $\mathbb{G}_1,\mathbb{G}_2,\mathbb{G}_T$ be yclic group of the same order and $ e: \mathbb{G}_1 \times \mathbb{G}_2\rightarrow \mathbb{G}_T$, such that $u\in \mathbb{G}_1, g \in \mathbb{G}_2, ...
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42 views

security of pairing based cryptography

while going through some papers on cryptographic accumulators i find the following statement "several pairing-based accumulators have been proposed in the past. However, due to continuous and recent ...
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76 views

How to compute accumulated values in bilinear map accumulators

How to compute $ g^{1/(e_1+s)}$, where $g$ is the generator of group $\mathbb G$, and $e_1$ and $s$ are keys? I know only $s$ and $g^{e_1}$, not $e_1$. $\mathbb G$ has prime order for some prime $p$ ...
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Are there any asymmetric composite order group bilinear pairings?

Are there any asymmetric composite order group bilinear pairings? Is there a drawback of asymmetric over symmetric bilinear pairings of composite order either in efficiency or in security ?
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72 views

Decryption of message in IBE without random oracle using bilinear pairig reg

I find the following IBE scheme from the videos posted and i don't understand the decryption algorithm, will any one please elaborate the 6th step scheme Setup($\lambda$) : $(\mathbb{G}, ...
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44 views

Are Barreto-Naehrig Curves suitable for pairing-based cryptography?

If Barreto-Naehrig Curves are suitable for pairing-based cryptography, can I use the library available at Optimal ATE Pairing?
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Given $g^a, g^b, g^c, g^{1/b}$, is it hard to distinguish $e(g, g)^{abc}$ from a random value?

where $g$ is a group element in bilinear group $\mathbb{G}$. I understand it is very similar to the conventional DBDH problem, but $g^{1/b}$ is also known, possibly making it easier? Does anyone know ...
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62 views

Generating cyclic group for Ciphertext-Policy Attribute-Based Encryption [closed]

I am doing Project under the topic CP-ABE.I need to generate a symmetric bilinear group Go of prime order p and with generator g...Then how to choose random elements from Zp....kindly anyone help ...
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1answer
60 views

Hardness assumptions on composite order bilinear groups [closed]

I am not at all knowledgeable in elliptic curve cryptography. So, here lies a couple of questions that I failed to find answers for to my satisfaction. Is there any known Type-III bilinear pairing ...
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2answers
208 views

Is it an example of bilinear pairing?

Consider a bilinear pairing $e: G_1 \times G_2 \rightarrow G_T$. Let's assume, $G_1 = G_2 = G_T = (\mathbb{Z}_n,+)$, i.e. additive group of integer modulo $n$ and $e(x,y) = xy$ mod $n$. Isn't it an ...
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123 views

Variant of the Decisional Bilinear Diffie Hellman problem

I am working on a cryptographic scheme and I need to rely on the following problem, which I have nicknamed the "Hybrid Decisional Bilinear Diffie Hellman (hDBDH)" problem: Let $e: \mathbb G_1 ...
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1answer
95 views

Hardness of problem related to bilinear pairings

Let $e: \mathbb G_1 \times \mathbb G_1 \rightarrow \mathbb G_T$ be an efficient bilinear pairing. Note that the pairing is symmetric (i.e., Type 1). The problem is, given $g \in \mathbb G_1$ and ...
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99 views

Group description in pairing based cryptography

Suppose we have a public key encryption scheme, in which public parameter contains $(p, G, G_t, e, g)$, where $p$ is prime number, $G$ is a (cyclic) group of prime order, $e:G \times G \mapsto G_t$. ...
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143 views

As a cryptographer, what are the things I should care about in my implementation of pairing functions?

As a beginner in cryptography, I do not know anything about different pairing types more than their names. So far, I know these names: Ate pairing, tate pairing, eta pairing, and r-ate pairing. I am ...
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1answer
71 views

Symmetric vs non-symmetric pairing based crypto

I am trying to find a comparison of how different pairing-friendly elliptic curves perform (in terms of security) in real world applications; I am using PBC library, and it has type A curves and type ...
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25 views

Help in uderstanding NIZKPoK notation (and how to code it) needed

Hello fellow cryptographers I have a rather silly question for you; I am aware of how Schnorrs NIZKPoK / SoK works when someone must prove knowledge about DL: y = g^x; (and how to code it) RFC ...
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DLOG in $\mathbb{F}_{p^n}^*$?

Assume that we are given an element $g\in \mathbb{F}_{p^n}^*$ and $g$ does not belong to any of the smaller subfields contained in $\mathbb{F}_{p^n}$. If the degree of $g$ is some number $q$, how much ...
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53 views

Does the following linear equation hold in bilinear pairings:

Does the following hold in bilinear pairings: $$e(g^{a_1x_1}g^{a_2x_2},g^{c_1}g^{c_2})=e(g^{x_1+x_2},g^{a_1a_2(c_1+c_2)})$$
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104 views

Generic group model: use of polynomials in the proof of the master theorem

I've been looking at the paper of Boneh, Boyen, Goh Hierarchical Identity Based Encryption with Constant Size Ciphertext which contains a general theorem (Theorem A.2) about the advantage of an ...
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45 views

Need: Fast bulk signature verification followed by fast non-interactive multisignature aggregation

Q: Is there an efficient way to batch-verify signatures (e.g. some may be incorrect) and then non-interactively aggregate the correct ones into a multisignature (they are of the same message) such ...
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99 views

Type G Bilinear Pairings

I was reading PBC and its implementations for finding pairing parameters. I am particularly interested in implementing a BLS signature scheme with 20-byte (160-bit) signatures ("short signatures"). ...
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2answers
75 views

linear computations over bilinear pairings

Does this hold in asymetric bilinear pairings? $e(x_1,x_2)e(x_3,x_4) = e(x_1x_3,x_2x_4)$, where $x_1,x_3 \in \mathbb{G}_1$ and $x_2,x_4 \in \mathbb{G}_2$ for a bilinear pairing $e$
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108 views

Hardware Implementation of Pairing over BN curves

I am in the middle of FPGA based Hardware architecture design for the computation of Pairing (particularly R-ate Pairing) over BN curves. Where, the point addition, and point doubling should be ...
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103 views

A question about elliptic curves and finite fields in bilinear pairings

Based on what mentioned in the paper "Pairings For Cryptographers" http://www.sciencedirect.com/science/article/pii/S0166218X08000449 the two inputs of a pairing map are two members of two additive ...
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pairing-based schemes

some authors claimed that computational performance of a pairing-fee scheme (based on scalar multiplication over an elliptic curve group) is about 1000% more efficient than a pairing based one I would ...
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102 views

Simplification of a pairing

The paper Expressive, Efficient, and Revocable Data Access Control for Multi-Authority Cloud Storage contains the following simplification: $$e(C_i,\text{GPK}_{uid})\cdot e(D_i, ...
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Type 2 to Type 1 pairing transformation - why not considered?

How come that in various articles about pairings I never saw anybody mention that there is a possibility to turn Type 2 pairing into Type 1 by setting $e' : G_2 \times G_2 \rightarrow \mu, (P,Q) ...
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116 views

Bilinear pairing

I am working on Efficient Construction of Pairings which are being realized by Miller's algorithm. In this algorithm the basic steps are point doubling and line function computation point addition ...
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1answer
74 views

Pairing Field size as security parameter

I have read Pairings for cryptographers: It states that the groups $G_1$ and $G_2$ are groups of points on the curve and the group GT is a subgroup of the multiplicative group of a related finite ...
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in Bilinear pairings, what is the difference between Type 2 and Type 3?

in Bilinear pairings, what is the difference between Type 2 and Type 3? I understand in Type 2, there exists an efficiently computable homomorphic function $\phi : G_2 \rightarrow G_1$ , which is not ...
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307 views

Is pairing based cryptography ready for productive use?

I'm currently testing one among those many interesting cryptographic protocols based on bilinear maps. It's quite hard to understand the underlying fundamentals, especially since there are several ...
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134 views

Pairings in Identity-based encryption vs. Attribute-based encryption

The bilinear map in Identity-based encryption should satisfy $e(aP,bQ)=e(P,Q)^{a\cdot b}$ whereas Attribute-based encryption schemes use $e(P^a,Q^a)=e(P,Q)^{a\cdot b}$ with $a,b\in\mathbb{Z}_p$ and ...
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Why does computing g^a * g^{-a} with the PBC library result in zero?

My example code is as follows: /* * Example 1 * 1) Calculate g^a * 2) calculate g^{-a} * 3) multiply g^a * g^{-a} * */ Note: here ...
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Weil pairing implementation - low level programming language

I'm just started to studying elliptic curve cryptosystems. My one month-goal is to write a simple signature system based on the Weil-pairing. Some parts of it can be written without deeper ...
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Sextic twist of BN pairing parameters vs security

I've previously asked questions on BN pairing parameters. Here's one more. In the BN construction, one is working in a subgroup of a curve over an extension field $\mathbf{F}_{p^{12}}$ for some ...
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Sextic twist optimization of BN pairing - cubic root extraction required?

I found the following paper really interesting: http://www.researchgate.net/publication/220378229_A_family_of_implementation-friendly_BN_elliptic_curves/file/79e4150b3a773beecd.pdf It allows ...
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For $e(g, d) = c $, can we compute $d$, given others

Given $$e(g, d) = c $$ where, $e$ is bilinear pairing function chosen by the user/attacker, the values of $g$ and $c$ are known $g, d ∈ \mathbb{G}_1$ , $c$ depends upon the $e$ can we somehow ...
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Generating bilinear pairing parameters - running time of finding member of p-torsion group

Update: Question completely rephrased. I want to create the parameters for a bilinear pairing (the Tate pairing in this case). In case you're interested I'm following this thesis, specifically the ...
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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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939 views

How to Mathematically Prove the Bilinear Pairing Properties [closed]

I am currently working on Bilinear Pairing.To start my work i need to find the mathematically prove of three properties of bilinear pairing. Let $ G_{1} $ and $ G_{T} $be a cyclic multiplicative ...
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56 views

Type A Curves of Supersinglular

In the Type A curve used in Pairing based Cryptography, what is meant by sign0 & sign1 in the expression below? ...
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410 views

Advantages of bilinear map

Consider the pairing $e: G_1*G_2 \to G_t$. Why we are mapping element from group $G_1$ and group $G_2$ to an element in $G_t$. How are they used in cryptography? What advantages do they provide?
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Why does the new encryption scheme proposed by authors stop an adversary from guessing the subspace of the secret key?

In this paper, the authors construct an encryption scheme that is supposed to be resilient to tampering and leaking (as opposed to just leaking). Specifically this scheme: If you look at the ...
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1answer
195 views

Why is a simple hash into $G_2$ for (certain) pairing based crypto not possible?

In the paper Pairings for Cryptographers we read about what the authors call a type 2 pairing in which we have a "pairing friendly curve $E$ over $\mathbb{F}_q$ with embedding degree $k>1$ and ...
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2answers
322 views

What does the linear assumption over bilinear groups mean?

In the abstract of "Cryptography with Tamperable and Leaky Memory", at the end of the 3rd paragraph, the authors say: In both schemes we rely on the linear assumption over bilinear groups. What ...
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387 views

Blockwise Montgomery multiplication

I have to implement a 256*256 bit Montgomery multiplier for pairing computations. The straightforward approach is to use a bit-serial version, but I would like to utilize the built-in 64*64 bits ...