Continued fractions of Laurent series with partial quotients from a given set
1. Introduction. Van der Poorten and Shallit's paper [10] begins: " It is notorious that it is damnably difficult to explicitly compute the continued fraction of a quantity presented in some other form ". The quantity is usually presented either as a power series or as the root of a specific equation. There has been some success in the former case for continued fractions of real numbers, such as Euler's famous continued fraction for e [11] and more recent work [10] on " folded " continued fractions; however, other than the well-known results for quadratic real numbers, the only success with the latter has been for continued fractions of Laurent series rather than real numbers. In this paper we continue this line of investigation. We consider families of continued fractions of Laurent series whose partial quotients all lie in a given set. Following ideas of Baum and Sweet [2], we show that one may describe the zeros of certain collections of equations in terms of such families. The paragraphs that follow introduce the notation and definitions necessary to give a fuller description of our results. Let F q be the finite field with q elements and L q denote the field of formal Laurent series in x
Towards factoring in SL ( 2 , F 2 n )
The security of many cryptographic protocols relies on the hardness of some computational problems. Besides discrete logarithm or integer factorization, other problems are regularly proposed as potential hard problems. The factorization problem in ﬁnite groups is one of them. Given a ﬁnite group G , a set of generators S for this group and an element g ∈ G , the factorization problem asks for a “short” representation of g as a product of the generators. The problem is related to a famous conjecture of Babai on the diameter of Cayley graphs. It is also motivated by the preimage security of Cayley hash functions , a particular kind of cryptographic hash functions. The problem has been solved for a few particular generator sets, but essentially nothing is known for generic generator sets. In this paper, we make significant steps towards a solution of the factorization problem in the group G := SL ( 2 , F 2 n ) , a particularly interesting group for cryptographic applications. To avoid considering all generator sets separately, we ﬁrst give a new reduction tool that allows focusing on some generator sets with a “nice” special structure. We then identify classes of trapdoor matrices for these special generator sets, such that the factorization of a single one of these matrices would allow efﬁciently factoring any element in the group. Finally, we provide a heuristic subexponential time algorithm that can compute subexponential length factorizations of any element for any pair of generators of SL ( 2 , F 2 n ) . Our results do not yet completely remove the factorization problem in SL ( 2 , F 2 n ) from the list of potential hard problems useful for cryptography. However, we believe that each one of our individual results is a significant step towards a polynomial time algorithm for factoring in SL ( 2 , F 2 n ) . any matrix in the group. We also provide sufﬁcient conditions for a reduction to Tillich–Zémor generators.
Urban Identities and Catastrophe: Izmir and Salonica at the End of the Ottoman Empire*
Two huge fires dramatically influenced the urban development of Salonica and Izmir, in 1917 and in 1922, respectively. These catastrophes occurred after the Ottoman Empire fell, and Salonica and Izmir had shifted into new national contexts. The fires mainly destroyed the districts that were transformed during the late‐Ottoman period. These districts became the cosmopolitan façades of modern Izmir and Salonica. The post‐fires' tabula rasa provided an opportunity for Greek and Turkish politicians and European planners to change the urban identity of both cities. Moreover, the Lausanne Treaty of 1923 transformed their multicultural societies also. Reconstruction plans had been thought to de‐Ottomanize and remove the previous idea of the towns and their multiplicity, interpreting new cultural and nationalist feeling. Here, I emphasize how modernity was interpreted before and after the fires, and point out contradictions between ideological aspects of planning and how the first urban districts were built during the 1920s.