6 research outputs found
Syntax-Directed Translations and Quasi-alphabetic Tree Bimorphisms - Revisited
Quasi-alphabetic tree bimorphisms [Steinby, Tîrn uc :
Dening syntax-directed translations by tree bimorphisms. Theor. Comput.
Sci., to appear. http://dx.doi.org/10.1016/j.tcs.2009.03.009,
2009] are reconsidered. It is known that the class of (string) translations
dened by such bimorphisms coincides with the class of syntaxdirected
translations. This result is extended to a smaller class of tree
bimorphisms namely (linear and complete) symbol-to-symbol tree bimorphisms.
Moreover, it is shown that the class of simple syntax-directed
translations coincides with the class of translations dened by alphabetic
tree bimorphisms (also known as nite-state relabelings). This proves
that alphabetic tree bimorphisms are not suciently powerful to model
all syntax-directed translations. Finally, it is shown that the class of tree
transformations dened by quasi-alphabetic tree bimorphisms is closed
under composition. The corresponding result is known in the variable-free
case
Key agreement: security / division
Some key agreement schemes, such as Diffie--Hellman key agreement, reduce to Rabi--Sherman key agreement, in which Alice sends to Charlie, Charlie sends to Alice, they agree on key , where multiplicative notation here indicates some specialized associative binary operation.
All non-interactive key agreement schemes, where each peer independently determines a single delivery to the other, reduce to this case, because the ability to agree implies the existence of an associative operation. By extending the associative operation’s domain, the key agreement scheme can be enveloped into a mathematical ring, such that all cryptographic values are ring elements, and all key agreement computations are ring multiplications. (A smaller envelope, a semigroup instead of a ring, is also possible.)
Security relies on the difficulty of division: here, meaning an operator
such that . Security also relies on the difficulty of the less
familiar wedge operation .
When Rabi--Sherman key agreement is instantiated as Diffie--Hellman key agreement: its multiplication amounts to modular exponentiation; its division amounts to the discrete logarithm problem; the wedge operation amounts to the computational Diffie--Hellman problem.
Ring theory is well-developed and implies efficient division algorithms in some specific rings, such as matrix rings over fields. Semigroup theory, though less widely-known, also implies efficient division in specific semigroups, such as group-like semigroups.
The rarity of key agreement schemes with well-established security suggests that easy multiplication with difficult division (and wedges) is elusive.
Reduction of key agreement to ring or semigroup multiplication is not a panacea for cryptanalysis. Nonetheless, novel proposals for key agreement perhaps ought to run the gauntlet of a checklist for vulnerability to well-known division strategies that generalize across several forms of multiplication. Ambitiously applying this process of elimination to a plethora of diverse rings or semigroups might also, if only by a fluke, leave standing a few promising schemes, which might then deserve a more focused cryptanalysis