Non-commutative cryptography is the area of cryptology where the cryptographic primitives, methods and systems are based on algebraic structures like semigroups, groups and rings which are non-commutative. One of the earliest applications of a non-commutative algebraic structure for cryptographic purposes was the use of braid groups to develop cryptographic protocols. Later several other non-commutative structures like Thompson groups, polycyclic groups, Grigorchuk groups, and matrix groups have been identified as potential candidates for cryptographic applications. In contrast to non-commutative cryptography, the currently widely used public-key cryptosystems like RSA cryptosystem, Diffie–Hellman key exchange and elliptic curve cryptography are based on number theory and hence depend on commutative algebraic structures.
Non-commutative cryptographic protocols have been developed for solving various cryptographic problems like key exchange, encryption-decryption, and authentication. These protocols are very similar to the corresponding protocols in the commutative case.
Some non-commutative cryptographic protocols
[edit]In these protocols it would be assumed that G is a non-abelian group. If w and a are elements of G the notation wa would indicate the element a−1wa.
Protocols for key exchange
[edit]Protocol due to Ko, Lee, et al.
[edit]The following protocol due to Ko, Lee, et al., establishes a common secret key K for Alice and Bob.
- An element w of G is published.
- Two subgroups A and B of G such that ab = ba for all a in A and b in B are published.
- Alice chooses an element a from A and sends wa to Bob. Alice keeps a private.
- Bob chooses an element b from B and sends wb to Alice. Bob keeps b private.
- Alice computes K = (wb)a = wba.
- Bob computes K' = (wa)b=wab.
- Since ab = ba, K = K'. Alice and Bob share the common secret key K.
Anshel-Anshel-Goldfeld protocol
[edit]This a key exchange protocol using a non-abelian group G. It is significant because it does not require two commuting subgroups A and B of G as in the case of the protocol due to Ko, Lee, et al.
- Elements a1, a2, . . . , ak, b1, b2, . . . , bm from G are selected and published.
- Alice picks a private x in G as a word in a1, a2, . . . , ak; that is, x = x( a1, a2, . . . , ak ).
- Alice sends b1x, b2x, . . . , bmx to Bob.
- Bob picks a private y in G as a word in b1, b2, . . . , bm; that is y = y ( b1, b2, . . . , bm ).
- Bob sends a1y, a2y, . . . , aky to Alice.
- Alice and Bob share the common secret key K = x−1y−1xy.
- Alice computes x ( a1y, a2y, . . . , aky ) = y−1 xy. Pre-multiplying it with x−1, Alice gets K.
- Bob computes y ( b1x, b2x, . . . , bmx) = x−1yx. Pre-multiplying it with y−1 and then taking the inverse, Bob gets K.
Stickel's key exchange protocol
[edit]In the original formulation of this protocol the group used was the group of invertible matrices over a finite field.
- Let G be a public non-abelian finite group.
- Let a, b be public elements of G such that ab ≠ ba. Let the orders of a and b be N and M respectively.
- Alice chooses two random numbers n < N and m < M and sends u = ambn to Bob.
- Bob picks two random numbers r < N and s < M and sends v = arbs to Alice.
- The common key shared by Alice and Bob is K = am + rbn + s.
- Alice computes the key by K = amvbn.
- Bob computes the key by K = arubs.
Protocols for encryption and decryption
[edit]This protocol describes how to encrypt a secret message and then decrypt using a non-commutative group. Let Alice want to send a secret message m to Bob.
- Let G be a non-commutative group. Let A and B be public subgroups of G such that ab = ba for all a in A and b in B.
- An element x from G is chosen and published.
- Bob chooses a secret key b from A and publishes z = xb as his public key.
- Alice chooses a random r from B and computes t = zr.
- The encrypted message is C = (xr, H(t) m), where H is some hash function and denotes the XOR operation. Alice sends C to Bob.
- To decrypt C, Bob recovers t as follows: (xr)b = xrb = xbr = (xb)r = zr = t. The plain text message send by Alice is P = ( H(t) m ) H(t) = m.
Protocols for authentication
[edit]Let Bob want to check whether the sender of a message is really Alice.
- Let G be a non-commutative group and let A and B be subgroups of G such that ab = ba for all a in A and b in B.
- An element w from G is selected and published.
- Alice chooses a private s from A and publishes the pair ( w, t ) where t = w s.
- Bob chooses an r from B and sends a challenge w ' = wr to Alice.
- Alice sends the response w ' ' = (w ')s to Bob.
- Bob checks if w ' ' = tr. If this true, then the identity of Alice is established.
Security basis of the protocols
[edit]The basis for the security and strength of the various protocols presented above is the difficulty of the following two problems:
- The conjugacy decision problem (also called the conjugacy problem): Given two elements u and v in a group G determine whether there exists an element x in G such that v = ux, that is, such that v = x−1 ux.
- The conjugacy search problem: Given two elements u and v in a group G find an element x in G such that v = ux, that is, such that v = x−1 ux.
If no algorithm is known to solve the conjugacy search problem, then the function x → ux can be considered as a one-way function.
Platform groups
[edit]A non-commutative group that is used in a particular cryptographic protocol is called the platform group of that protocol. Only groups having certain properties can be used as the platform groups for the implementation of non-commutative cryptographic protocols. Let G be a group suggested as a platform group for a certain non-commutative cryptographic system. The following is a list of the properties expected of G.
- The group G must be well-known and well-studied.
- The word problem in G should have a fast solution by a deterministic algorithm. There should be an efficiently computable "normal form" for elements of G.
- It should be impossible to recover the factors x and y from the product xy in G.
- The number of elements of length n in G should grow faster than any polynomial in n. (Here "length n" is the length of a word representing a group element.)
Examples of platform groups
[edit]Braid groups
[edit]Let n be a positive integer. The braid group Bn is a group generated by x1, x2, . . . , xn-1 having the following presentation:
Thompson's group
[edit]Thompson's group is an infinite group F having the following infinite presentation:
Grigorchuk's group
[edit]Let T denote the infinite rooted binary tree. The set V of vertices is the set of all finite binary sequences. Let A(T) denote the set of all automorphisms of T. (An automorphism of T permutes vertices preserving connectedness.) The Grigorchuk's group Γ is the subgroup of A(T) generated by the automorphisms a, b, c, d defined as follows:
Artin group
[edit]An Artin group A(Γ) is a group with the following presentation:
where ( factors) and .
Matrix groups
[edit]Let F be a finite field. Groups of matrices over F have been used as the platform groups of certain non-commutative cryptographic protocols.
Semidirect products
[edit]See also
[edit]References
[edit]- ^ Habeeb, M.; Kahrobaei, D.; Koupparis, C.; Shpilrain, V. (2013). "Public Key Exchange Using Semidirect Product of (Semi)Groups". Applied Cryptography and Network Security. ACNS 2013. Lecture Notes in Computer Science. Vol. 7954. Springer. pp. 475–486. arXiv:1304.6572. CiteSeerX 10.1.1.769.1289. doi:10.1007/978-3-642-38980-1_30. ISBN 978-3-642-38980-1.
Further reading
[edit]- Myasnikov, Alexei; Shpilrain, Vladimir; Ushakov, Alexander (2008). Group-based Cryptography. Birkhäuser Verlag. ISBN 9783764388270.
- Cao, Zhenfu (2012). New Directions of Modern Cryptography. CRC Press. ISBN 978-1-4665-0140-9.
- Benjamin Fine; et al. (2011). "Aspects of Nonabelian Group Based Cryptography: A Survey and Open Problems". arXiv:1103.4093 [cs.CR].
- Myasnikov, Alexei G.; Shpilrain, Vladimir; Ushakov, Alexander (2011). Non-commutative Cryptography and Complexity of Group-theoretic Problems. American Mathematical Society. ISBN 9780821853603.