A trusted party gives each participant a secret key and a public identifier, which enables any two participants to independently create a shared key for communicating. However, if an attacker can compromise the keys of at least k users, he can break the scheme and reconstruct every shared key. Blom's scheme is a form of threshold secret sharing.
The key exchange protocol involves a trusted party (Trent) and a group of users. Let Alice and Bob be two users of the group.
Inserting a new participant
New users Alice and Bob want to join the key exchanging group. Trent chooses public identifiers for each of them; i.e., k-element vectors:
Trent then computes their private keys:
Using as described above:
Each will use their private key to compute shared keys with other participants of the group.
Now Alice and Bob wish to communicate with one another. Alice has Bob's identifier and her private key .
She computes the shared key , where denotes matrix transpose. Bob does the same, using his private key and her identifier, giving the same result:
They will each generate their shared key as follows:
In order to ensure at least k keys must be compromised before every shared key can be computed by an attacker, identifiers must be k-linearly independent: all sets of k randomly selected user identifiers must be linearly independent. Otherwise, a group of malicious users can compute the key of any other member whose identifier is linearly dependent to theirs. To ensure this property, the identifiers shall be preferably chosen from a MDS-Code matrix (maximum distance separable error correction code matrix). The rows of the MDS-Matrix would be the identifiers of the users. A MDS-Code matrix can be chosen in practice using the code-matrix of the Reed–Solomon error correction code (this error correction code requires only easily understandable mathematics and can be computed extremely quickly).
- Rolf Blom. Non-public key distribution. In Proc. CRYPTO 82, pages 231–236, New York, 1983. Plenum Press
- R. Blom, "An optimal class of symmetric key generation systems", Report LiTH-ISY-I-0641, Linköping University, 1984