BLS (Cryptography)

In cryptography, the Boneh-Lynn-Shacham signature scheme allows a user to verify that a signer is "authentic". The scheme uses a pairing function for verification and signatures are group elements in some elliptic curve. Working in an elliptic curve provides defense against index calculus attacks against allowing shorter signatures than FDH signatures. Signatures are often referred to as "short signatures", "BLS short signatures", or simply "BLS signatures". The signature scheme is provably secure (that is, the scheme is existentially unforgeable under adaptive chosen-message attacks) assuming both the existence of random oracles and the intractability of the computational Diffie-Hellman problem.cite journal
author = Dan Boneh, Ben Lynn, and Hovav Shacham
title = Short Signatures from the Weil Pairing
journal = Journal of Cryptology
volume = 17
date = 2004
pages = 297-319]

Pairing functions

A gap group is a group in which the computational Diffie-Hellman problem is intractable but the decisional Diffie-Hellman problem can be efficiently solved. Non-degenerate, efficiently computable, bilinear pairing functions permit such groups.

Let $ecolon\; G\; imes\; G\; ightarrow\; G\_T$ be a non-degenerate, efficiently computable, bilinear pairing function where $G$, $G\_T$ are groups of prime order, $r$. Let $g$ be a generator of $G$. Consider an instance of the CDH problem, $g^x$, $g^x$. Intuitively, the pairing function $e$ does not help us compute $g^\{xy\}$ a solution to the CDH problem. It is conjectured that this instance of the CDH problem is intractable. Given $g^z$, we may check to see if $g^\{xy\}=g^z$ without knowledge of $x$, $y$, and $z$, by computing $e(g^x,g^y)=e(g,g^z)$.

By using the bilinear property $x+y+z$ times, we see that if $e(g^x,g^y)=e(g,g)^\{xy\}=e(g,g)^\{z\}=e(g,g^z)$, then since $G\_T$ is a prime order group, $xy=z$.

The scheme

A signature scheme consists of three functions, "generate", "sign", and "verify"

Key Generation

The key generation algorithm selects a random integer $x$ in the interval [0,r-1] . The private key is $x$. The holder of the private key publishes the public key, $g^x$.

Signing

Given the private key $x$, and some message $m$, we compute the signature by hashing the bitstring $m$, as $h=H(m)$. We output the signature $sigma=h^x$.

Verificaion

Given a signature $sigma$ and a public key $g^x$, we verify that $e(sigma,g)=e(H(m),g)$.

References