Clear DefinitionsPairing-based Cryptography
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Pronunciation
[pair-ing beyst krip-tog-ruh-fee]
Analogy
Imagine you have two separate secret gardens (elliptic curve groups). Pairing-based cryptography gives you a magical bridge (the bilinear map) that lets you combine an element from the first garden with an element from the second garden to produce an element in a *third*, different kind of magical space. This 'bridging' operation has unique properties that enable new types of cryptographic tricks not possible with standard ECC alone, like letting Alice and Bob combine their secrets with Charlie's public info to all arrive at a shared secret without Charlie needing to do much.
Definition
A branch of cryptography that uses a bilinear map (a pairing) on elliptic curve groups (or other suitable groups) to enable advanced cryptographic schemes. Pairings allow for three-party Diffie-Hellman key exchanges, identity-based encryption, and efficient zero-knowledge proofs, among other applications.
Key Points Intro
Pairings extend the capabilities of elliptic curve cryptography to enable novel cryptographic protocols.
Key Points
Utilizes a bilinear map e: G1 x G2 -> GT, where G1, G2 are (often elliptic curve) groups and GT is a target multiplicative group.
The bilinear property means e(aP, bQ) = e(P, Q)^(ab) for scalars a, b and points P, Q.
Enables protocols like Identity-Based Encryption (IBE), Attribute-Based Encryption (ABE), short digital signatures (BLS signatures), and efficient SNARKs (zero-knowledge proofs).
Often involves more complex mathematics and potentially larger computational overhead than standard ECC.
Example
Identity-Based Encryption (IBE) schemes, made practical by pairings, allow someone to encrypt a message for a recipient using their publicly known identity (like an email address) as the public key, without needing to first look up a digital certificate for that recipient. The recipient uses a master secret key (held by a trusted authority) corresponding to their identity to decrypt.
Technical Deep Dive
Common pairings used in cryptography include the Weil pairing, Tate pairing, and Ate pairing, defined on specific types of elliptic curves (e.g., supersingular curves or pairing-friendly ordinary curves). The choice of curve and pairing type affects security and efficiency. The security of pairing-based schemes often relies on the hardness of problems like the Bilinear Diffie-Hellman (BDH) problem or variants. While powerful, pairings can be computationally more expensive than standard ECC operations and the underlying curves might be more complex to select and implement securely.
Security Warning
Implementing pairing-based cryptography is complex and requires careful selection of secure elliptic curves and pairing parameters. Vulnerabilities can arise from improper curve choices or implementation errors. The security assumptions are also different and sometimes stronger than those for standard ECC.
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