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The most efficient identity-based encryption schemes are currently based on bilinear pairings on elliptic curves, such as the Weil or Tate pairings. The first of these schemes was developed by Dan Boneh and Matthew K. Franklin (2001), and performs probabilistic encryption of arbitrary ciphertexts using an Elgamal-like approach.
The Boneh–Franklin scheme is an identity-based encryption system proposed by Dan Boneh and Matthew K. Franklin in 2001. [1] This article refers to the protocol version called BasicIdent. It is an application of pairings (Weil pairing) over elliptic curves and finite fields.
The Weil pairing is an important concept in elliptic curve cryptography; e.g., it may be used to attack certain elliptic curves (see MOV attack). It and other pairings have been used to develop identity-based encryption schemes.
In mathematics, the Weil pairing is a pairing (bilinear form, though with multiplicative notation) on the points of order dividing n of an elliptic curve E, taking values in nth roots of unity. More generally there is a similar Weil pairing between points of order n of an abelian variety and its dual.
Recently, [when?] a large number of cryptographic primitives based on bilinear mappings on various elliptic curve groups, such as the Weil and Tate pairings, have been introduced. Schemes based on these primitives provide efficient identity-based encryption as well as pairing-based signatures, signcryption, key agreement, and proxy re-encryption.
Pairing-based cryptography is used in the KZG cryptographic commitment scheme. A contemporary example of using bilinear pairings is exemplified in the BLS digital signature scheme. [3] Pairing-based cryptography relies on hardness assumptions separate from e.g. the elliptic-curve cryptography, which is older and has been studied for a longer time.
This is because the Weil pairing or Tate pairing can be used to solve the problem directly as follows: given ,,, on such a curve, one can compute (,) and (,). By the bilinearity of the pairings, the two expressions are equal if and only if a b = c {\displaystyle ab=c} modulo the order of P {\displaystyle P} .
It has several applications on cryptography, as key exchange protocols, identity-based encryption, and broadcast encryption. There exist constructions of cryptographic 2-multilinear maps, known as bilinear maps, [ 1 ] however, the problem of constructing such multilinear [ 1 ] maps for n > 2 {\displaystyle n>2} seems much more difficult [ 2 ...