In cryptography, the Boneh–Lynn–Shacham signature scheme allows a user to verify that a signer is authentic. The scheme uses a bilinear pairing 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. ^[1]

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 pairings permit such groups.

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

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

The scheme

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

Key generation

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

Signing

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

Verification

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

References

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Summary

DescriptionTetrationConvergence2D.png	English: Graph of tetrations ${\displaystyle {}^{n}x}$, converging to the function ${\displaystyle {\frac {W(-\ln(z))}{-\ln(z)}}}$ in black between ${\displaystyle e^{-e}}$ and ${\displaystyle e^{1/e}}$. Notice the convergence in ${\displaystyle [0,e^{-e}\approx 0.066]}$, oscillating between the two branches of the bifurcation. Selfmade with MuPAD.
Date	19 July 2008 (original upload date)
Source	Transferred from en.wikipedia to Commons.
Author	Sam Derbyshire at English Wikipedia

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