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

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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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52 views

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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84 views

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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147 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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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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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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47 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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127 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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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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understanding pairing $e:G \times G \to G_T$ and ( Decision)BDH assumption

From DrLecter's comment, I know that DDH problem can be efficiently solved with this $$e(g^a,g^b)\stackrel{?}{=} e(g,g^z).$$ I have some trouble to understand this map $e:G \times G \to G_T$. Am I ...
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196 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 ...
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254 views

Smart Card Basics

I want to implement some of the basic encryption algorithms on smart card, could any body guide me, how to program a smart card, which tools (hardware and software) i should have ,and are these tools ...
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331 views

Security of pairing-based cryptography over binary fields regarding new attacks

In the last week, the discrete logarithm problem was broken for the binary fields $\mathbb{F}_{2^{(14 \times 127)}}$ and $\mathbb{F}_{2^{(27 \times 73)}}$. Pairing-based cryptography using binary ...
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85 views

Create a field in PBC

Edited (I removed the emphasize on Integers): My question is partly cryptography and partly programming, I would appreciate any help on any aspect of it :) I want to use PBC library to do the ...
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what is pairing in cryptography? [closed]

Let $G_1, G_2$ be additive groups and $G_T$ a multiplicative group, all of prime order $p$. Let $P $ in $G_1, Q $ in $G_2$ be generators of $G_1$ and $G_2$ respectively. A pairing is a map: $e:( ...
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207 views

Must the order of the groups in a bilinear map be the same?

I've been reading up on bilinear maps and their application to cryptography and one thing I keep seeing hasn't yet clicked. If $e:G_1\times G_2\to G_n$ is a bilinear map, $G_1,G_2,G_n$ are always ...
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1answer
137 views

Generating non-supersingular elliptic curves for symmetric pairings

I am looking into the application of pairings in CPABE in particular. I've notice that the scheme uses a supersingular curve as the basis of the pairing. Looking through Ben Lynn's thesis for the ...
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81 views

BLS signatures in the G-valued Random Oracle Model

This paper on semi-generic algorithms considers "non-standard properties of the employed hash function". For BLS signatures whose main group is $G$, I'm curious what can be shown when the hash ...
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184 views

Can somebody explain the major contributions of the tenants of the Gödel Prize 2013?

As you may know, the Gödel Prize 2013 will be awarded this year to cryptographers (see this ACM press release). The people awarded are Antoine Joux, the team of Dan Boneh and Matthew K. Franklin. Can ...
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106 views

Complex Numbers on Elliptic Curves & Usage in Tate Pairing

I'm working with understanding the internals of the Tate Pairing. I was going through an example of the curve $E: y^2 = x^3 + 3x$ over $\mathbb{F_{11}}$. The author is showing the computation of ...
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Hardware implementation of pairing

I would like to design a hardware-based accelerator for a pairing algorithm to make it faster. I know that I need to do arithmetic over a base and an extension field. Could anyone would suggest which ...
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1k views

Simple example for CP-ABE (Ciphertext policy attribute-based encryption)

I'm currently working on Ciphertext Policy Attribute-Based Encryption (CP-ABE). So far I'm only using it with a basic understanding how it actually works. Now I want to understand it a bit better, but ...
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Modulus for elliptic curve point multiplication

I want to implement a point multiplication ($k \cdot P$) operation on FPGA. I have a BN curve $y^2=x^3+2$, and a scalar value $k$. The $x$ and $y$ coordinates of point $P$ are of 256 bits. In the ...
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107 views

Exponentiation In PBC library

I need to compute a function $h^l$, where h is an element of G2 and l is a rational number. How can this be done using the PBC library? I have converted the h to ...
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1answer
164 views

How are Elliptic Curve Cryptography and Pairing Based Cryptography related?

I have been doing a project that uses the PBC library developed by Ben Lynn. But I am still not clear on how PBC is related to ECC. I know that this is a site for complex crypto QA, but I did not know ...
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59 views

Is it possible to create a Bilinear Function with Already Assigned “Multiplicative” Input Groups?

Assume that we have an already assigned Multiplicative Cyclic Group $\mathbb Z_p^*$ with order $q=p-1$, and $p$ is a prime number, is it possible to create a bilinear function $\hat{e}: \mathbb Z_p^* ...
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89 views

Discrete logs on elliptic curve with embedding degree 3 with the 'MOV' attack

The curve $E(\mathbb{F}_{47}):y^2=x^3+x+38$ has order $61$ and $61|47^3-1$ so the embedding degree of $E$ is $3$ and therefore the MOV attack, presumably using some sort of distortion map and a ...
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104 views

Using pairings to verify an extended euclidean relation without leaking the values?

Let $P_i(x)$ be polynomials $i=1,...,n$, $s$ some value, and $g$ a generator of a group $G$ where the discrete logarithm is hard. Assume a prover wants to convince a verifier having access to the ...
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275 views

when do we need composite order groups for bilinear maps and when prime order?

Why we need bilinear groups of composite order? What's the special security property of the composite order group in comparison with one of prime order?To put it in another way when do we need ...
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why $e(g,g)^N=1$ in bilinear pairings holds?

I can't get the point of prime order bilinear pairings:$\mathbb{G}\times\mathbb{G}\rightarrow\mathbb{G}_T$,$g=$ generator of $\mathbb{G}$ , $N=p*q$, $p$ and $q$ primes and $e(g,g)^N=1$. why ...
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210 views

Why pairing based crypto is suitable for some particular cryptographic primitives?

Why pairing based crypto is being widely used in some special crypto primitives as ID based crypto and variations of standard signatures? I mean taking as deep as possible what makes it suitable for ...
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213 views

Does Identity-Based Encryption actually solve any problem?

Identity based encryption schemes [*] seem to have great potential in high-latency Delay-Tolerant and mobile, ad-hoc networks since they apparently seem to avoid the need for key negotiation and ...
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246 views

Useful pairings for cryptography

I've recently looked a bit at pairing based cryptography and I was wondering what properties the groups involved should have in order to be useful for cryptographic purposes? Has anything more exact ...
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364 views

Alternatives to FHE for secure function evaluation

As a followup to a previous question I asked which was more related to Fully Homomorphic Encryption (FHE), what other cryptographic methods are available for computing a private function on public ...
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363 views

Pairing-friendly curves in small characteristic fields

There are several well-known techniques to generate pairing-friendly curves of degrees 1 to 36 on prime fields GF(p): Cocks-Pinch, MNT, Brezing-Weng, and several others. In extension fields GF(p^n), ...
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874 views

What is Identity-Based Encryption (IBE) and why is it “better”?

Most CS/Math undergrads run into the well-known RSA cryptosystem at some point. But about 10 years ago Boneh and Franklin introduced a practical Identity-Based Encryption system (IBE) that has ...
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Mapping points between elliptic curves and the integers

My primary question is: Is there an easy way to create a bijective mapping from points on an elliptic curve E (over a finite field) to the integers (desirably to $\mathbb{Z}^*_q$ where $q$ is the ...