The Sakai–Kasahara scheme, also known as the Sakai–Kasahara key encryption algorithm (SAKKE), is an identity-based encryption (IBE) system proposed by Ryuichi Sakai and Masao Kasahara in 2003. Alongside the Boneh–Franklin scheme, this is one of a small number of commercially implemented identity-based encryption schemes. It is an application of pairings over elliptic curves and finite fields. A security proof for the algorithm was produced in 2005 by Chen and Cheng. SAKKE is described in Internet Engineering Task Force (IETF) RFC 6508. As a specific method for identity-based encryption, the primary use case is to allow anyone to encrypt a message to a user when the sender only knows the public identity (e.g. email address) of the user. In this way, this scheme removes the requirement for users to share public certificates for the purpose of encryption.
Description of scheme The Sakai–Kasahara scheme allows the encryption of a message M {\displaystyle \mathbb {M} } to an receiver with a specific identity, I U {\displaystyle \textstyle I_{U}} . Only the entity with the private key, K U {\displaystyle \textstyle K_{U}} , associated to the identity, I U {\displaystyle \textstyle I_{U}} , will be capable of decrypting the message. As part of the scheme, both the sender and receiver must trust a Private Key Generator (PKG), also known as a Key Management Server (KMS). The purpose of the PKG is to create the receiver's private key, K U {\displaystyle \textstyle K_{U}} , associated to the receiver's identity, I U {\displaystyle \textstyle I_{U}} . The PKG must securely deliver the identity-specific private key to the receiver, and PKG-specific public parameter, Z {\displaystyle \textstyle Z} , to all parties. These distribution processes are not considered as part of the definition of this cryptographic scheme.
Preliminaries The scheme uses two multiplicative groups E {\displaystyle \textstyle E} and G {\displaystyle \textstyle G} . It is assumed:
The Diffie-Hellman problem is hard in E {\displaystyle \textstyle E} . Meaning that given two members of the group P {\displaystyle \textstyle P} and Q {\displaystyle \textstyle Q} , it is hard to find x {\displaystyle \textstyle x} such that [ x ] . P = Q {\displaystyle \textstyle [x].P=Q} . The Diffie-Hellman problem is hard in G {\displaystyle \textstyle G} . Meaning that given two members of the group g {\displaystyle g} and t {\displaystyle t} , it is hard to find x {\displaystyle \textstyle x} such that g x = t {\displaystyle \textstyle g^{x}=t} . There is a bilinear map, a Tate-Lichtenbaum pairing, e ( , ) {\displaystyle \textstyle e(,)} from E to G. This means that for P {\displaystyle \textstyle P} a member of E {\displaystyle \textstyle E} :
e ( P , [ x ] . P ) = e ( [ x ] . P , P ) = e ( P , P ) x {\displaystyle \textstyle e(P,[x].P)=e([x].P,P)=e(P,P)^{x}}
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