Bilinear groups are a set of three abstract algebraic groups, G1, G2 and GT , together with a deterministic function e, called a bilinear map, that takes as input one element from G1 and one element from G2 and outputs an element in GT . Suppose all three groups have order Q, element g1 generates group G1, and element g2 generates group G2. Then, one special property called bilinearity of the map e is that for all a, b < Q, we have that e(g1^a , g2^b) = e(g1, g2)^ab. This new element, e(g1, g2)^ab, is in group GT . The key observation is what happens to the exponents, a and b, during the mapping: they are multiplied. The group GT is distinct from both G1 or G2; thus the output of the mapping cannot be fed back into the map e as input.Elliptic curves are generally used for the groups, although bilinear schemes in at least some other algebras are also possible.
Two of the main applications of bilinear groups are proxy re-signatures and proxy re-encryption. In proxy re-signatures, a semi-trusted party transforms Alice's public key signature into Bob's. The proxy does not have, cannot derive, and thus cannot sign with either Bob's secret key or with Alice's, but can only transform Alice's signature into Bob's. The proxy re-signer is thus "semi-trusted" -- it is trusted with some things we normally would trust a proxy signer with, but not with others. For example it is not trusted with either Alice's or Bob's private key, only with a special key that allows the signature transformation.
The target signature could also be a group signature. Thus, for example, Alice could sign her e-mail with her own digital signature, and a proxy re-signer sitting on the corporate e-mail firewall could re-sign the e-mail with the corporate group signature.
Proxy re-signers can be chained in a series, so that signature A is transformed by proxy AB into signature B, which is transformed by proxy BC into signature C, and so on. The last signature Z proves that the message was signed by each proxy in the chain in order. Proxy re-signers can also be chained together in a tree or directed acyclic graph. (Note that threshold signatures by contrast do not require or prove that the signatures took place in a particular order).
Proxy re-encryption is the same idea for public key encryption, with the added bonus that the re-encryptor can't read the message. So, for example, we could have the following scheme to restrict the distribution of content:
(1) Content owner Alice encrypts her content with her public key and publishes it to proxies P1, P2, etc., along with re-encryption keys AC1, AC2, etc. for each customer.
(2) Proxy allows customer to access the content only if paid. When paid, the proxy re-encrypts to the customer using the re-encryption key for that customer.
The proxies themselves are trusted neither with an ability to view the content nor with the discretion to distribute to additional customers not desired by content owner Alice. The proxy is trusted only to restrict access to a customer. (I present this scheme mainly just to illustrate what proxy re-encryption does. As an application, this particular content distribution scheme seems to me to only be useful if it somehow lowers transaction costs to route all payments through proxies rather than paying Alice directly, the latter which could be done by normal public-key cryptography, and of course it doesn't protect against a cheating customer re-publishing the content to the world).
I suspect proxy re-encryption could simplify the design of digital mix schemes like onion routing -- this is left as an exercise for the cryptographically inclinded reader.
This thesis is my source for most of this blog post; it discusses bilinear group cryptography for proxy re-encryption, proxy re-signing, and for reducing the trust needed for blinded offline digital cash.
Legal caveat: many, if not most protocols based on bilinear groups seem to have been recently patented.
How can one patent math?
Step one: Apply for a patent on math.
Step two: Accept patent for math from the Patent Office.
It's really pretty simple, Anonymous.
Well, perhaps that Anonymous is in a country that forbid this kink of patent. For example, in theory, European countries should not allow them.
How is this functionally different from simply appending signatures, i.e., Alice signs her document, then Alice's Corp signs her signed document, etc.?
I understand that the transform is a more efficient way to do the above, but functionally, is there any difference? Anything one scheme can accomplish that the other cannot?
See my paper, Elemental Subject Matter for how software (including math) run on computers or stored in memory came to be patentable, even though algorithms and other math are not supposed to be patentable. The basic problem is one of counting gratuitous elements -- elements that do not add anything to the novelty or nonobviousness of a patent, such as a general-purpose memory or CPU, for the purposes of determining whether the subject matter is patentable.
jojo: One basic difference is that when the proxy transforms Alice's signature into Bob's, the proxy does not have to be Bob. Another is that the setup requires cooperation between Alice, Bob, and the proxy during key generation, so that (depending on what out-of-band semantics is imputed to the signature) one can impute some kind of authorization or consent by the parties to the re-signing.
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