Scenario optimization: Difference between revisions

Revision as of 23:58, 14 January 2014

Anshel–Anshel–Goldfeld protocol, also known as a commutator key exchange, is a key-exchange protocol using nonabelian groups. It was invented by Drs. Michael Anshel, Iris Anshel, and Dorian Goldfeld. Unlike other group-based protocols it does not employ any commuting or commutative subgroups of a given platform group and can, in fact, use any nonabelian group with efficiently computable normal forms.

Description

Let G be a fixed nonabelian group called a platform group.

Alice's public/private information:

Alice's public key is a tuple of elements $\overline{a} = (a_{1}, \dots, a_{n})$ in G.
Alice's private key is a sequence of elements from $\overline{a}$ and their inverses: $a_{i_{1}}^{ε_{1}}, \dots, a_{i_{L}}^{ε_{L}}$ , where $a_{i_{k}} \in \overline{a}$ and $ε_{k} = \pm 1$ . Based on that sequence she computes the product $A = a_{i_{1}}^{ε_{1}} \dots a_{i_{L}}^{ε_{L}}$ .

Bob's public/private information:

Bob's public key is a tuple of elements $\overline{b} = (b_{1}, \dots, b_{n})$ in $G$ .
Bob's private key is a sequence of elements from $\overline{b}$ and their inverses: $b_{j_{1}}^{δ_{1}}, \dots, b_{j_{L}}^{δ_{L}}$ , where $b_{j_{k}} \in \overline{b}$ and $δ_{k} = \pm 1$ . Based on that sequence she computes the product $B = b_{j_{1}}^{δ_{1}} \dots b_{j_{L}}^{δ_{L}}$ .

Transitions:

Alice sends a tuple $\overline{b} = (A^{- 1} b_{1} A, \dots, A^{- 1} b_{n} A)$ to Bob.
Bob sends a tuple $\overline{b} = (B^{- 1} a_{1} B, \dots, B^{- 1} a_{n} B)$ to Alice.

Shared key:

The key shared by Alice and Bob is the group element $K = A^{- 1} B^{- 1} A B \in G$ called the commutator of $A$ and $B$ .

Alice computes $K$ as a product $A^{- 1} \cdot B^{- 1} a_{i_{1}}^{ε_{1}} B \dots B^{- 1} a_{i_{L}}^{ε_{L}} B$ .
Bob computes $K$ as a product $(A^{- 1} b_{i_{1}}^{ε_{1}} A \dots A^{- 1} b_{i_{L}}^{ε_{L}} A)^{- 1} \cdot B = A^{- 1} B^{- 1} A B$ .

References

I. Anshel, M. Anshel, and D. Goldfeld, An algebraic method for public-key cryptography, Math. Res. Lett. 6 (1999), pp. 287–291.

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+'''Anshel–Anshel–Goldfeld protocol''', also known as a '''commutator key exchange''', is a key-exchange protocol using [[nonabelian group]]s. It was invented by Drs. Michael Anshel, Iris Anshel, and [[Dorian M. Goldfeld|Dorian Goldfeld]].
+Unlike other group-based protocols it does not employ any commuting or commutative
+subgroups of a given platform group and can, in fact, use any nonabelian group with
+efficiently computable normal forms.
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+== Description ==
+Let ''G'' be a fixed nonabelian [[Group (mathematics)|group]] called a ''platform group''.
+'''Alice's public/private information:'''
+* ''Alice's public key'' is a tuple of elements <math>{\overline{a}}=(a_1,\ldots,a_n)</math> in ''G''.
+* ''Alice's private key'' is a sequence of elements from <math>{\overline{a}}</math> and their inverses: <math>a_{i_1}^{\varepsilon_1}, \ldots, a_{i_L}^{\varepsilon_L}</math>, where <math>a_{i_k}\in{\overline{a}}</math> and <math>\varepsilon_k=\pm 1</math>. Based on that sequence she computes the product <math>A = a_{i_1}^{\varepsilon_1} \ldots a_{i_L}^{\varepsilon_L}</math>.
+'''Bob's public/private information:'''
+* ''Bob's public key'' is a tuple of elements <math>{\overline{b}}=(b_1,\ldots,b_n)</math> in <math>G</math>.
+* ''Bob's private key'' is a sequence of elements from <math>{\overline{b}}</math> and their inverses: <math>b_{j_1}^{\delta_1}, \ldots, b_{j_L}^{\delta_L}</math>, where <math>b_{j_k}\in{\overline{b}}</math> and <math>\delta_k=\pm 1</math>. Based on that sequence she computes the product <math>B = b_{j_1}^{\delta_1} \ldots b_{j_L}^{\delta_L}</math>.
+'''Transitions:'''
+* Alice sends a tuple <math>{\overline{b}}=(A^{-1}b_1A,\ldots,A^{-1}b_nA)</math> to Bob.
+* Bob sends a tuple <math>{\overline{b}}=(B^{-1}a_1B,\ldots,B^{-1}a_nB)</math> to Alice.
+'''Shared key:'''
+The key shared by Alice and Bob is the group element <math>K = A^{-1} B^{-1} A B \in G</math> called the [[commutator]] of <math>A</math> and <math>B</math>.
+* Alice computes <math>K</math> as a product <math>A^{-1} \cdot B^{-1}a_{i_1}^{\varepsilon_1}B\cdots B^{-1}a_{i_L}^{\varepsilon_L}B</math>.
+* Bob computes <math>K</math> as a product <math>(A^{-1}b_{i_1}^{\varepsilon_1}A \cdots A^{-1} b_{i_L}^{\varepsilon_L}A)^{-1} \cdot B = A^{-1} B^{-1} A B</math>.
+==See also==
+* [[Group-based cryptography]]
+== References ==
+* I. Anshel, M. Anshel, and D. Goldfeld, ''[http://www-cs.engr.ccny.cuny.edu/~csmma/MRLpap.pdf An algebraic method for public-key cryptography]'', Math. Res. Lett. 6 (1999), pp.&nbsp;287–291.
+{{DEFAULTSORT:Anshel-Anshel-Goldfeld key exchange}}
+[[Category:Cryptographic protocols]]

Scenario optimization: Difference between revisions

Revision as of 23:58, 14 January 2014

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