1. Introduction. Continued fractions [8,14] are a useful tool in many number theoretical problems and in numerical computing. It is well known that the simple continued fraction expansion of a single real number gives the best solution to its rational approximation problem. Many people have contrived to construct multidimensional continued fractions in dealing with the rational approximation problem for multi-reals. One construction is the Jacobi-Perron algorithm (JPA) (see [1]). This algorithm and its modifications have been extensively studied [6,7,10,13]. These algorithms are adapted to study the same problem for multi-formal Laurent series [2,4,11,12]. But none of them guarantees the best rational approximation in general. In this paper, we deal with the multi-rational approximation problem over the formal Laurent series field F ((z −1 )): given an element r ∈ F ((z −1 )) m , find p ∈ F [z] m and q ∈ F [z] such that p/q approximates r as close as possible while deg(q) is bounded.We propose a new continued fraction algorithm for multi-formal Laurent series. It is proved that this algorithm guarantees best rational approximations for multi-formal Laurent series.The paper is organized as follows: Section 2 deals with the indexed valuation of F ((z −1 )) m . Section 3 contains the detailed definition of the problem of optimal rational approximation of multi-formal Laurent series. Section 4 proposes an algorithm called multidimensional continued fraction algorithm (m-CFA, for short), which produces a multi-continued fraction expansion C(r) for any given multi-formal Laurent series r. Section 5 shows that C(r) satisfies three basic conditions. Section 6 states the main results of this pa-
Predicate encryption provides fine-grained control over encryption and decryption. In a predicate encryption scheme, message is encrypted with a set of attributes, and decryption key is incorporated with a predicate. Only when the predicate evaluates to true over these attributes, the message can be correctly decrypted. We propose an extension to predicate encryption named delegate predicate encryption. In our extension, a user generates a encryption capability of a set of attributes, then sends the capability to a encryption proxy. Using this capability, the proxy can encrypt arbitrary message with these attributes without knowing anything about these attributes. We give a concrete construction from a predicate encryption scheme which supports the widest range of predicate known to date. Then, using our extension, we are able to construct an anonymous authentication scheme. In our anonymous authentication scheme, authentication rules can be described in an very expressive way, while the anonymity of user is protected and the authentication rules are also hidden.
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