Antisymmetrizer: Difference between revisions
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'''Blom's scheme''' is a symmetric threshold [[key exchange]] protocol in [[cryptography]]. The scheme was proposed by the Swedish cryptographer Rolf Blom in a series of articles in the early 1980s.<ref>Rolf Blom. Non-public key distribution. In Proc. CRYPTO 82, pages 231–236, New York, 1983. Plenum Press</ref><ref>R. Blom, "An optimal class of symmetric key generation systems", Report LiTH-ISY-I-0641, Linköping University, 1984 [http://www.csl.mtu.edu/cs6461/www/Reading/blom-eurocrypt84.pdf]</ref> | |||
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 scheme|threshold secret sharing]]. | |||
Blom's scheme is currently used by the [[HDCP]] copy protection scheme to generate shared keys for high-definition content sources and receivers, such as [[HD DVD]] players and [[high-definition television]]s. | |||
==The protocol== | |||
The key exchange protocol involves a trusted party (Trent) and a group of <math>\scriptstyle n</math> users. Let [[Alice and Bob]] be two users of the group. | |||
===Protocol setup=== | |||
Trent chooses a random and secret [[symmetric matrix]] <math>\scriptstyle D_{k,k}</math> over the [[finite field]] <math>\scriptstyle GF(p)</math>, where p is a prime number. <math>\scriptstyle D</math> is required when a new user is to be added to the key sharing group. | |||
For example: | |||
<math>\begin{align} | |||
k &= 3\\ | |||
p &= 17\\ | |||
D &= \begin{pmatrix} 1&6&2\\6&3&8\\2&8&2\end{pmatrix}\ \mathrm{mod}\ 17 | |||
\end{align}</math> | |||
===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: | |||
<math>I_{\mathrm{Alice}}, I_{\mathrm{Bob}} \in GF(p)</math>. | |||
For example: | |||
<math>I_{\mathrm{Alice}} = \begin{pmatrix} 3 \\ 10 \\ 11 \end{pmatrix}, I_{\mathrm{Bob}} = \begin{pmatrix} 1 \\ 3 \\ 15 \end{pmatrix}</math> | |||
Trent then computes their private keys: | |||
<math>\begin{align} | |||
g_{\mathrm{Alice}} &= DI_{\mathrm{Alice}}\\ | |||
g_{\mathrm{Bob}} &= DI_{\mathrm{Bob}} | |||
\end{align}</math> | |||
Using <math>D</math> as described above: | |||
<math>\begin{align} | |||
g_{\mathrm{Alice}} &= \begin{pmatrix} 1&6&2\\6&3&8\\2&8&2\end{pmatrix}\begin{pmatrix} 3 \\ 10 \\ 11 \end{pmatrix} = \begin{pmatrix} 85\\136\\108\end{pmatrix}\ \mathrm{mod}\ 17 = \begin{pmatrix} 0\\0\\6\end{pmatrix}\ \\ | |||
g_{\mathrm{Bob}} &= \begin{pmatrix} 1&6&2\\6&3&8\\2&8&2\end{pmatrix}\begin{pmatrix} 1 \\ 3 \\ 15 \end{pmatrix} = \begin{pmatrix} 49\\135\\56\end{pmatrix}\ \mathrm{mod}\ 17 = \begin{pmatrix} 15\\16\\5\end{pmatrix}\ | |||
\end{align}</math> | |||
Each will use their private key to compute shared keys with other participants of the group. | |||
===Computing a shared key between Alice and Bob=== | |||
Now Alice and Bob wish to communicate with one another. Alice has Bob's identifier <math>\scriptstyle I_{\mathrm{Bob}}</math> and her private key <math>\scriptstyle g_{\mathrm{Alice}}</math>. | |||
She computes the shared key <math>\scriptstyle k_{\mathrm{Alice / Bob}} = g_{\mathrm{Alice}}^t I_{\mathrm{Bob}}</math>, where <math>\scriptstyle t</math> denotes [[matrix transpose]]. Bob does the same, using his private key and her identifier, giving the same result: | |||
<math>k_{\mathrm{Alice / Bob}} = k_{\mathrm{Alice / Bob}}^t = (g_{\mathrm{Alice}}^t I_{\mathrm{Bob}})^t = (I_{\mathrm{Alice}}^t D^t I_{\mathrm{Bob}})^t = I_{\mathrm{Bob}}^t D I_{\mathrm{Alice}} = k_{\mathrm{Bob / Alice}}</math> | |||
They will each generate their shared key as follows: | |||
<math>\begin{align} | |||
k_{\mathrm{Alice / Bob}} &= \begin{pmatrix} 0\\0\\6 \end{pmatrix}^t \begin{pmatrix} 1\\3\\15 \end{pmatrix} = 0 \times 1 + 0 \times 3 + 6 \times 15 = 90\ \mathrm{mod}\ 17 = 5\\ | |||
k_{\mathrm{Bob / Alice}} &= \begin{pmatrix} 15\\16\\5 \end{pmatrix}^t \begin{pmatrix} 3\\10\\11 \end{pmatrix} = 15 \times 3 + 16 \times 10 + 5 \times 11 = 260\ \mathrm{mod}\ 17 = 5 | |||
\end{align}</math> | |||
==Attack resistance== | |||
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). | |||
== References == | |||
{{refbegin}} | |||
* {{cite book | |||
| author = [[Alfred Menezes|Alfred J. Menezes]], [[Paul van Oorschot|Paul C. van Oorschot]] and [[Scott Vanstone|Scott A. Vanstone]] | |||
| year = 1996 | |||
| title = Handbook of Applied Cryptography | |||
| publisher = [[CRC Press]] | |||
| isbn = 0-8493-8523-7 | |||
| url = http://www.cacr.math.uwaterloo.ca/hac/ | |||
}} | |||
{{refend}} | |||
<references/> | |||
[[Category:Secret sharing]] | |||
Revision as of 12:50, 8 February 2013
Blom's scheme is a symmetric threshold key exchange protocol in cryptography. The scheme was proposed by the Swedish cryptographer Rolf Blom in a series of articles in the early 1980s.[1][2]
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.
Blom's scheme is currently used by the HDCP copy protection scheme to generate shared keys for high-definition content sources and receivers, such as HD DVD players and high-definition televisions.
The protocol
The key exchange protocol involves a trusted party (Trent) and a group of users. Let Alice and Bob be two users of the group.
Protocol setup
Trent chooses a random and secret symmetric matrix over the finite field , where p is a prime number. is required when a new user is to be added to the key sharing group.
For example:
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:
For example:
Trent then computes their private keys:
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:
Attack resistance
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).
References
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