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.}}{\*\pnseclvl4\pnlcltr\pnstart1\pnindent720\pnhang{\pntxta )}} {\*\pnseclvl5\pndec\pnstart1\pnindent720\pnhang{\pntxtb (}{\pntxta )}}{\*\pnseclvl6\pnlcltr\pnstart1\pnindent720\pnhang{\pntxtb (}{\pntxta )}}{\*\pnseclvl7\pnlcrm\pnstart1\pnindent720\pnhang{\pntxtb (}{\pntxta )}}{\*\pnseclvl8 \pnlcltr\pnstart1\pnindent720\pnhang{\pntxtb (}{\pntxta )}}{\*\pnseclvl9\pnlcrm\pnstart1\pnindent720\pnhang{\pntxtb (}{\pntxta )}}\pard\plain \s16\qc \li0\ri0\widctlpar\faauto\adjustright\rin0\lin0\itap0 \fs48\lang2057\langfe1033\cgrid\langnp2057\langfenp1033 {Non\_Linear Feedback Shift Registers with Maximum Period \par }\pard\plain \ql \li0\ri0\widctlpar\faauto\adjustright\rin0\lin0\itap0 \fs24\lang3082\langfe1033\cgrid\langnp3082\langfenp1033 {\fs20\lang2057\langfe1033\langnp2057 \par }\pard \qc \li0\ri0\widctlpar\faauto\adjustright\rin0\lin0\itap0 {\fs28\lang2057\langfe1033\langnp2057 Daniela Ferrero, Ra\'fal Gonzalo, Miguel Soriano}{\fs20\lang2057\langfe1033\langnp2057 \par \par \par }{\b\fs28\lang2057\langfe1033\langnp2057 Abstract}{\fs20\lang2057\langfe1033\langnp2057 \par }\pard \qj \li1134\ri1134\widctlpar\faauto\adjustright\rin1134\lin1134\itap0 {\fs20\lang2057\langfe1033\langnp2057 \par \tab \tab Stream ciphering devices seem to be one of the best alternatives in order to provide confidentiality to high-speed transmissions. \par \tab \tab Several indexes on the security of stream ciphers have been proposed for guaranteeing their strength. According to the lite rature, linear complexity of the key stream, randomness and correlation immune attacks are of great importance. But the proposed indexes are not sufficient to guarantee the security of stream ciphers. It is possible that sequences with a high linear com plexity have a very bad \'93linear complexity stability\'94 , i.e., after changing a few bits of the original sequence, its linear complexity decreases or increases fast. In this case, a BAA attack (Best Affine Approximation) could be very successful; the sequenc es may be very well approximated by another one with very lower linear unpredictability. This problem is especially important when linear feedback shift registers are used. To solve this problem, non\_ linear next state functions are the possible solution. This paper shows some techniques in order to analyse non\_linear functions with maximum period and to set necessary conditions. \par \par }\pard \qc \li1134\ri1134\widctlpar\faauto\adjustright\rin1134\lin1134\itap0 {\fs20\lang2057\langfe1033\langnp2057 e-mail: telrgd@mat.upc.es, matdfc@mat.upc.es, soriano@mat.upc.es \par }\pard \qj \li0\ri0\widctlpar\faauto\adjustright\rin0\lin0\itap0 {\fs20\lang2057\langfe1033\langnp2057 \par \par }{\b\fs36\lang2057\langfe1033\langnp2057 1. Introduction}{\fs20\lang2057\langfe1033\langnp2057 \par \tab \par \tab \tab The recent growth of multimedia service s and supercomputing applications has led to a need for high speed networks which can carry all fundamental media streams: data, voice and video. The Asynchronous Transfer Mode (ATM) is the most promising technique for high speed networks in the near futu re, for the public WANs, the Broadband Integrated Services Digital Network (B-ISDN) and also private local networks like ATM LANs. \par \tab \tab Some applications and services for both scenarios require security services. First, it is necessary to control the access of users to the resources by means of smart cards, passwords, fingerprints, etc.. Mutual authentication is also required for communications over the network, between clients and servers. We have to follow an authentication protocol that uses some encryption mechanisms in order to achieve this authentication. Some applications can also negotiate session keys during or after the authentication that can later be used to cipher the communication, providing integrity and/or confidentiality. \par \tab \tab The services of integ rity and confidentiality must be applied for the bulk of the information. So, we need fast ciphering mechanisms for high-speed networks. With today's technology, hardware implementations of stream ciphers seem to be the best choice to encrypt at a rate of hundreds of Mbits/sec, and thus be compatible with high-speed networks. \par \tab \tab In the next section, the use of structures with a non\_linear feedback is justified. In the third section, we propose an algorithm to characterise the non\_linear functions by means o f which the shift registers present the maximum period. And finally, we present other results obtained when the problem is seen from the graph theoretical point of view. \par \par }\pard \ql \li0\ri0\widctlpar\faauto\adjustright\rin0\lin0\itap0 {\b\fs36\lang2057\langfe1033\langnp2057 2. 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