By Gian-Carol Rota

Combinatorics has lately woke up from a protracted shut eye, which started on the time of Euler. It has emerged as a brand new topic status on the crossroads among natural and utilized arithmetic, the guts of bustling task, a simmering pot of latest difficulties and intriguing speculations.The seven papers of this survey signify a large adequate sampling of present traits, from which the reader may well, a minimum of, extrapolate a number of the lacking fabric. They undergo in universal the only attribute of latest combinatorics: striving for normal new effects, whereas utilizing previous and new difficulties as a attempt of efficiency.The advent to matroid conception via Brylawski and Kelly describes a idea whose history — no less than from a distance — is the four-color conjecture, very similar to the history of algebraic quantity conception was once, a minimum of first and foremost, Fermat's conjecture. It issues little that neither idea has succeeded in fixing its motivating problem...

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**Example text**

Wk ) ∈ (X ∗ )k as a single word in the alphabet X = X ∪ {, }, which allows one to view the decision problems above as given, again, in the form (L, U ), with U ⊆ (X )∗ . A search computational problem can be described by a binary predicate R(x, y) ⊆ X ∗ × Y ∗ , where X and Y are ﬁnite alphabets. In this case, given an input x ∈ X ∗ , one has to ﬁnd a word y ∈ Y ∗ such that R(x, y) holds. For example, in the Diophantine Problem above x is a polynomial equation (or a “word” describing this equation) Ex (y) = 0 in a tuple of variables y, and the predicate R(x, y) holds for given values of x and y if and only if these values give a solution of Ex (y) = 0.

Since ai bi = bi ai in G, one has KA = KB = K (as an element of G), which is now Alice’s and Bob’s common secret key. We now discuss several modiﬁcations of the above protocol. , [65], [67]), according to computer experiments. Again, there is a public group G, and two public subgroups A, B ≤ G commuting elementwise. 1. Alice randomly selects private elements a1 ∈ A and b1 ∈ B. Then she sends the element a1 wb1 to Bob. 2. Bob randomly selects private elements b2 ∈ B and a2 ∈ A. Then he sends the element b2 wa2 to Alice.

These include braid groups, groups of matrices, small cancellation groups, and others. Chapter 4 Canonical Non-commutative Cryptography In this chapter, we discuss various cryptographic primitives that use noncommutative (semi)groups as platforms, but at the same time do not depart from the canonical paradigm of a public-key protocol based on a one-way function. We include here the ground-breaking Anshel-Anshel-Goldfeld protocol [1] as well as protocols that are closer in spirit to classical protocols based on commutative (semi)groups.