This volume contains the written versions of four courses plus the one seminar
talk given at the Lluis A. Santal´ o Summer School, entitled “Recent Trends in Cryp-
tography”, and organized by the Real Sociedad Matem´ atica Espa˜nola (RSME). One
of the objectives of the RSME is to present the challenges and opportunities that
the modern Information Society offer to the mathematical community. Following
this objective, the main goal of the Summer School was to give an overview of
some recent mathematical methods used in cryptography and cryptanalysis. There
were four courses given by well-known specialists plus several seminar talks and
two round table discussions with representatives of the industry, administration,
and research communities. The Summer School was held at the Universidad Inter-
nacional Men´ endez Pelayo (UIMP) at the Palacio de la Magdalena in Santander
(Spain) on July 11–15, 2005.
The program of the school consisted of four courses taught by Tor Helleseth,
Alfred Menezes, Phong Nguyen, and Igor Shparlinski. There were also talks by
Amparo uster, Jaime Guti´ errez, Ignacio Luengo, and two round table discussions.
The main topics were analysis of stream cyphers and public-key cryptography a
revolutionary idea introduced thirty years ago that has changed our lives in aspects
such as the Internet and electronic commerce. All topics were of great interest for
their applications to the different aspects of information security, such as authenti-
cation, digital signatures, smart cards, electronic voting, and the security of mobile
phones. The course of T. Helleseth gave a very nice introduction to stream cyphers
including the main mathematical aspect, such as boolean functions, and combina-
tion of stream cyphers, and algebraic and correlation attacks to such cyphers. The
course of A. Menezes gave an introduction to pairing-based cryptography with el-
liptic curves and his main applications, such as short signatures and identity based
encryption. This course presented some important developments in protocol design,
Tate pairing computations, and selection of elliptic curves. The course of Phong
Nguyen made a complete introduction to the mathematical techniques used in the
cryptanalysis of public-key cryptosystems. It included a description of the main
public-key cryptosystems, its security, and attacks. This course also presented an
introduction to lattices and more advanced attacks based on them. The course of
I. Shparlinski presented a survey of several recently suggested constructions of gen-
erating sequences of pseudorandom points on elliptic curves. Such constructions
are of interest for both classical and elliptic curve cryptography and are also of
intrinsic mathematical interest.
All this material is presented in this proceedings collection in an expository
manner with many examples and references and will be very useful to graduate
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