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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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In consequence, when a linear next state function is used (linear feedback), due to the predictability of t he states evolution, there is a commitment in the output sequence. If a high order of correlation immunity is desired, the output sequence can be very well approximated by a sequence linearly generated. \par \tab \tab Nevertheless, the situation differs when a non-line ar feedback function is used. In this case, the state evolution of the generator cannot be predicted. If the non-linear feedback function is approximated by its BAA, every time that a difference is done, a propagation error process is done because the act ual state of the generator is different to the foreseen. \par \par }\pard \ql \li0\ri0\widctlpar\faauto\adjustright\rin0\lin0\itap0 {\fs20\lang2057\langfe1033\langnp2057 \par }\pard \qj \li0\ri0\widctlpar\faauto\adjustright\rin0\lin0\itap0 {\b\fs36\lang2057\langfe1033\langnp2057 3. Non\_linear feedback function analysis \par }{\fs20\lang2057\langfe1033\langnp2057 \par \tab \tab In this section, we will analyse the features of the feedback functions associated to the FSR (Feedback Shift Registers) that originate maximum period seq uences, called DeBruijn sequences. They have excellent statistical properties, which permits their use in a large variety of applications, from random bit generation to criptography. \par \tab \tab The DeBruijn sequences originally arise from the teleprinter\rquote s problem. \par \par }\pard \qj \fi-1134\li1134\ri0\widctlpar\faauto\adjustright\rin0\lin1134\itap0 {\i\fs20\lang2057\langfe1033\langnp2057 Definition}{\fs20\lang2057\langfe1033\langnp2057 : A DeBruijn sequence of order L is a cyclic sequence such that each subsequence of L symbols appears exactly once. \par }\pard \qj \li0\ri0\widctlpar\faauto\adjustright\rin0\lin0\itap0 {\fs20\lang2057\langfe1033\langnp2057 \par \tab \tab Since there are 2}{\fs20\lang2057\langfe1033\super\langnp2057 L}{\fs20\lang2057\langfe1033\langnp2057 distinct L-tuples formed from 0 and 1, the sequence is n=2}{\fs20\lang2057\langfe1033\super\langnp2057 L}{ \fs20\lang2057\langfe1033\langnp2057 bits long. It is not difficult to see that the alphabet size needs not be 2. It could be any number m. In that case, the maximum-length sequence is m}{\fs20\lang2057\langfe1033\super\langnp2057 L}{ \fs20\lang2057\langfe1033\langnp2057 symbols long. We will concentrate the study on the binary case for simplicity and practical reasons. \par \tab \tab It is well known that the number of all the different DeBruijn sequences of order L is }{\pard\plain \qj \li0\ri0\widctlpar\faauto\adjustright\rin0\lin0\itap0 \fs20\lang2057\langfe1033\cgrid\langnp2057\langfenp1033 {\object\objemb \objw639\objh340{\*\objclass Equation.3}{\*\objdata 01050000020000000b0000004571756174696f6e2e33000000000000000000000e0000 d0cf11e0a1b11ae1000000000000000000000000000000003e000300feff0900060000000000000000000000010000000100000000000000001000000200000001000000feffffff0000000000000000ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff 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These sequences are usually generated by FSR. Each DeBruijn sequence has associated a different feedback function. \par \tab \tab Let L denote the length of the FSR (it will also be the order of the DeBruijn sequence generated). The sequence bits will be calculated from lower to higher index, i.e. s}{\fs20\lang2057\langfe1033\sub\langnp2057 k}{ \fs20\lang2057\langfe1033\langnp2057 = f (s}{\fs20\lang2057\langfe1033\sub\langnp2057 k-1}{\fs20\lang2057\langfe1033\langnp2057 ,...,s}{\fs20\lang2057\langfe1033\sub\langnp2057 k-L}{\fs20\lang2057\langfe1033\langnp2057 ), where f(\'b7 ) is the feedback function. The vector of the register will be denoted as this: x}{\fs20\lang2057\langfe1033\sub\langnp2057 L-1}{\fs20\lang2057\langfe1033\langnp2057 ...x}{\fs20\lang2057\langfe1033\sub\langnp2057 1}{\fs20\lang2057\langfe1033\langnp2057 x}{\fs20\lang2057\langfe1033\sub\langnp2057 0}{\fs20\lang2057\langfe1033\langnp2057 . The most significant bit (MSB) will be the (L-1)-th bit and the LSB will correspond to the 0 tap of the FSR. \par }\pard\plain \s15\qj \li0\ri0\widctlpar\faauto\adjustright\rin0\lin0\itap0 \fs24\lang3082\langfe1033\cgrid\langnp3082\langfenp1033 {\fs20\lang2057\langfe1033\langnp2057 \tab \tab When the register is in the x}{\fs20\lang2057\langfe1033\sub\langnp2057 L-1} {\fs20\lang2057\langfe1033\langnp2057 ...x}{\fs20\lang2057\langfe1033\sub\langnp2057 1}{\fs20\lang2057\langfe1033\langnp2057 x}{\fs20\lang2057\langfe1033\sub\langnp2057 0}{\fs20\lang2057\langfe1033\langnp2057 state, the next state will be the vector x}{ \fs20\lang2057\langfe1033\sub\langnp2057 L-1}{\fs20\lang2057\langfe1033\langnp2057 ...x}{\fs20\lang2057\langfe1033\sub\langnp2057 1}{\fs20\lang2057\langfe1033\langnp2057 x}{\fs20\lang2057\langfe1033\sub\langnp2057 0}{\fs20\lang2057\langfe1033\langnp2057 , where x}{\fs20\lang2057\langfe1033\sub\langnp2057 i}{\fs20\lang2057\langfe1033\langnp2057 = x}{\fs20\lang2057\langfe1033\sub\langnp2057 i+1}{\fs20\lang2057\langfe1033\langnp2057 for every i from 0 to L-2, and x}{ \fs20\lang2057\langfe1033\sub\langnp2057 L-1}{\fs20\lang2057\langfe1033\langnp2057 is calculated with the expression given by f(\'b7). Graphically, \par }\pard\plain \qj \li0\ri0\widctlpar\faauto\adjustright\rin0\lin0\itap0 \fs24\lang3082\langfe1033\cgrid\langnp3082\langfenp1033 {\fs20\lang1024\langfe1024\noproof\langnp1033 {\shp{\*\shpinst\shpleft5820\shptop219\shpright6504\shpbottom675\shpfhdr0\shpbxcolumn\shpbxignore\shpbypara\shpbyignore\shpwr3\shpwrk0\shpfblwtxt0\shpz3\shplid1026{\sp{\sn shapeType}{\sv 202}}{\sp{\sn fFlipH}{\sv 0}}{\sp{\sn fFlipV}{\sv 0}} {\sp{\sn lTxid}{\sv 65536}}{\sp{\sn fShadow}{\sv 1}}{\sp{\sn f3D}{\sv 0}}{\sp{\sn fLayoutInCell}{\sv 0}}{\sp{\sn fLayoutInCell}{\sv 0}}{\shptxt \pard\plain \qc \li0\ri0\widctlpar\faauto\adjustright\rin0\lin0\itap0 \fs24\lang3082\langfe1033\cgrid\langnp3082\langfenp1033 {\fs20 x}{\fs20\sub 0}{\fs20 \par }}}{\shprslt{\*\do\dobxcolumn\dobypara\dodhgt8195\dptxbx\dptxlrtb{\dptxbxtext\pard\plain \qc \li0\ri0\widctlpar\faauto\adjustright\rin0\lin0\itap0 \fs24\lang3082\langfe1033\cgrid\langnp3082\langfenp1033 {\fs20 x}{\fs20\sub 0}{\fs20 \par }}\dpx5820\dpy219\dpxsize684\dpysize456\dpfillfgcr255\dpfillfgcg255\dpfillfgcb255\dpfillbgcr255\dpfillbgcg255\dpfillbgcb255\dpfillpat1\dplinew15\dplinecor0\dplinecog0\dplinecob0}}} {\shp{\*\shpinst\shpleft4281\shptop219\shpright5820\shpbottom675\shpfhdr0\shpbxcolumn\shpbxignore\shpbypara\shpbyignore\shpwr3\shpwrk0\shpfblwtxt0\shpz2\shplid1027{\sp{\sn shapeType}{\sv 202}}{\sp{\sn fFlipH}{\sv 0}}{\sp{\sn fFlipV}{\sv 0}} {\sp{\sn lTxid}{\sv 131072}}{\sp{\sn fShadow}{\sv 1}}{\sp{\sn f3D}{\sv 0}}{\sp{\sn fLayoutInCell}{\sv 0}}{\sp{\sn fLayoutInCell}{\sv 0}}{\shptxt \pard\plain \qc \li0\ri0\widctlpar\faauto\adjustright\rin0\lin0\itap0 \fs24\lang3082\langfe1033\cgrid\langnp3082\langfenp1033 {\fs20 . . . . . \par }}}{\shprslt{\*\do\dobxcolumn\dobypara\dodhgt8194\dptxbx\dptxlrtb{\dptxbxtext\pard\plain \qc \li0\ri0\widctlpar\faauto\adjustright\rin0\lin0\itap0 \fs24\lang3082\langfe1033\cgrid\langnp3082\langfenp1033 {\fs20 . . . . . \par }}\dpx4281\dpy219\dpxsize1539\dpysize456\dpfillfgcr255\dpfillfgcg255\dpfillfgcb255\dpfillbgcr255\dpfillbgcg255\dpfillbgcb255\dpfillpat1\dplinew15\dplinecor0\dplinecog0\dplinecob0}}} {\shp{\*\shpinst\shpleft3597\shptop219\shpright4281\shpbottom675\shpfhdr0\shpbxcolumn\shpbxignore\shpbypara\shpbyignore\shpwr3\shpwrk0\shpfblwtxt0\shpz1\shplid1028{\sp{\sn shapeType}{\sv 202}}{\sp{\sn fFlipH}{\sv 0}}{\sp{\sn fFlipV}{\sv 0}} {\sp{\sn lTxid}{\sv 196608}}{\sp{\sn fShadow}{\sv 1}}{\sp{\sn f3D}{\sv 0}}{\sp{\sn fLayoutInCell}{\sv 0}}{\sp{\sn fLayoutInCell}{\sv 0}}{\shptxt \pard\plain \qc \li0\ri0\widctlpar\faauto\adjustright\rin0\lin0\itap0 \fs24\lang3082\langfe1033\cgrid\langnp3082\langfenp1033 {\fs20 x}{\fs20\sub L-2 \par }}}{\shprslt{\*\do\dobxcolumn\dobypara\dodhgt8193\dptxbx\dptxlrtb{\dptxbxtext\pard\plain \qc \li0\ri0\widctlpar\faauto\adjustright\rin0\lin0\itap0 \fs24\lang3082\langfe1033\cgrid\langnp3082\langfenp1033 {\fs20 x}{\fs20\sub L-2 \par }}\dpx3597\dpy219\dpxsize684\dpysize456\dpfillfgcr255\dpfillfgcg255\dpfillfgcb255\dpfillbgcr255\dpfillbgcg255\dpfillbgcb255\dpfillpat1\dplinew15\dplinecor0\dplinecog0\dplinecob0}}} {\shp{\*\shpinst\shpleft2913\shptop219\shpright3597\shpbottom675\shpfhdr0\shpbxcolumn\shpbxignore\shpbypara\shpbyignore\shpwr3\shpwrk0\shpfblwtxt0\shpz0\shplid1029{\sp{\sn shapeType}{\sv 202}}{\sp{\sn fFlipH}{\sv 0}}{\sp{\sn fFlipV}{\sv 0}} {\sp{\sn lTxid}{\sv 262144}}{\sp{\sn fShadow}{\sv 1}}{\sp{\sn f3D}{\sv 0}}{\sp{\sn fLayoutInCell}{\sv 0}}{\sp{\sn fLayoutInCell}{\sv 0}}{\shptxt \pard\plain \qc \li0\ri0\widctlpar\faauto\adjustright\rin0\lin0\itap0 \fs24\lang3082\langfe1033\cgrid\langnp3082\langfenp1033 {\fs20 x}{\fs20\sub L-1 \par }}}{\shprslt{\*\do\dobxcolumn\dobypara\dodhgt8192\dptxbx\dptxlrtb{\dptxbxtext\pard\plain \qc \li0\ri0\widctlpar\faauto\adjustright\rin0\lin0\itap0 \fs24\lang3082\langfe1033\cgrid\langnp3082\langfenp1033 {\fs20 x}{\fs20\sub L-1 \par }}\dpx2913\dpy219\dpxsize684\dpysize456\dpfillfgcr255\dpfillfgcg255\dpfillfgcb255\dpfillbgcr255\dpfillbgcg255\dpfillbgcb255\dpfillpat1\dplinew15\dplinecor0\dplinecog0\dplinecob0}}}}{\fs20\lang2057\langfe1033\langnp2057 \par }{\fs20\lang1024\langfe1024\noproof\langnp1033 {\shp{\*\shpinst\shpleft2400\shptop171\shpright2400\shpbottom1311\shpfhdr0\shpbxcolumn\shpbxignore\shpbypara\shpbyignore\shpwr3\shpwrk0\shpfblwtxt0\shpz10\shplid1030 {\sp{\sn shapeType}{\sv 20}}{\sp{\sn fFlipH}{\sv 0}}{\sp{\sn fFlipV}{\sv 1}}{\sp{\sn shapePath}{\sv 4}}{\sp{\sn fFillOK}{\sv 0}}{\sp{\sn fFilled}{\sv 0}} {\sp{\sn fArrowheadsOK}{\sv 1}}{\sp{\sn fLayoutInCell}{\sv 0}}{\sp{\sn fLayoutInCell}{\sv 0}}}{\shprslt{\*\do\dobxcolumn\dobypara\dodhgt8202\dpline\dpptx0\dppty0\dpptx0\dppty1140\dpx2400\dpy171\dpxsize0\dpysize1140 \dplinew15\dplinecor0\dplinecog0\dplinecob0}}}{\shp{\*\shpinst\shpleft2400\shptop171\shpright2913\shpbottom171\shpfhdr0\shpbxcolumn\shpbxignore\shpbypara\shpbyignore\shpwr3\shpwrk0\shpfblwtxt0\shpz8\shplid1031 {\sp{\sn shapeType}{\sv 20}}{\sp{\sn fFlipH}{\sv 0}}{\sp{\sn fFlipV}{\sv 0}}{\sp{\sn shapePath}{\sv 4}}{\sp{\sn fFillOK}{\sv 0}}{\sp{\sn fFilled}{\sv 0}}{\sp{\sn lineEndArrowhead}{\sv 1}}{\sp{\sn fArrowheadsOK}{\sv 1}} {\sp{\sn fLayoutInCell}{\sv 0}}{\sp{\sn fLayoutInCell}{\sv 0}}}{\shprslt{\*\do\dobxcolumn\dobypara\dodhgt8200\dpline\dpptx0\dppty0\dpptx513\dppty0\dpx2400\dpy171\dpxsize513\dpysize0\dplinew15\dplinecor0\dplinecog0\dplinecob0}}} {\shp{\*\shpinst\shpleft6504\shptop171\shpright7131\shpbottom171\shpfhdr0\shpbxcolumn\shpbxignore\shpbypara\shpbyignore\shpwr3\shpwrk0\shpfblwtxt0\shpz11\shplid1032{\sp{\sn shapeType}{\sv 20}}{\sp{\sn fFlipH}{\sv 0}}{\sp{\sn fFlipV}{\sv 0}} {\sp{\sn shapePath}{\sv 4}}{\sp{\sn fFillOK}{\sv 0}}{\sp{\sn fFilled}{\sv 0}}{\sp{\sn lineEndArrowhead}{\sv 1}}{\sp{\sn fArrowheadsOK}{\sv 1}}{\sp{\sn fLayoutInCell}{\sv 0}}{\sp{\sn fLayoutInCell}{\sv 0}}}{\shprslt{\*\do\dobxcolumn\dobypara\dodhgt8203 \dpline\dpptx0\dppty0\dpptx627\dppty0\dpx6504\dpy171\dpxsize627\dpysize0\dplinew15\dplinecor0\dplinecog0\dplinecob0}}}}{\fs20\lang2057\langfe1033\langnp2057 \par }{\fs20\lang1024\langfe1024\noproof\langnp1033 {\shp{\*\shpinst\shpleft5535\shptop123\shpright6162\shpbottom750\shpfhdr0\shpbxcolumn\shpbxignore\shpbypara\shpbyignore\shpwr3\shpwrk0\shpfblwtxt0\shpz7\shplid1033 {\sp{\sn shapeType}{\sv 20}}{\sp{\sn fFlipH}{\sv 1}}{\sp{\sn fFlipV}{\sv 0}}{\sp{\sn shapePath}{\sv 4}}{\sp{\sn fFillOK}{\sv 0}}{\sp{\sn fFilled}{\sv 0}}{\sp{\sn lineEndArrowhead}{\sv 1}}{\sp{\sn fArrowheadsOK}{\sv 1}} {\sp{\sn fLayoutInCell}{\sv 0}}{\sp{\sn fLayoutInCell}{\sv 0}}}{\shprslt{\*\do\dobxcolumn\dobypara\dodhgt8199\dpline\dpptx627\dppty0\dpptx0\dppty627\dpx5535\dpy123\dpxsize627\dpysize627\dplinew15\dplinecor0\dplinecog0\dplinecob0}}} {\shp{\*\shpinst\shpleft3939\shptop123\shpright4281\shpbottom693\shpfhdr0\shpbxcolumn\shpbxignore\shpbypara\shpbyignore\shpwr3\shpwrk0\shpfblwtxt0\shpz6\shplid1034{\sp{\sn shapeType}{\sv 20}}{\sp{\sn fFlipH}{\sv 0}}{\sp{\sn fFlipV}{\sv 0}} {\sp{\sn shapePath}{\sv 4}}{\sp{\sn fFillOK}{\sv 0}}{\sp{\sn fFilled}{\sv 0}}{\sp{\sn lineEndArrowhead}{\sv 1}}{\sp{\sn fArrowheadsOK}{\sv 1}}{\sp{\sn fLayoutInCell}{\sv 0}}{\sp{\sn fLayoutInCell}{\sv 0}}}{\shprslt{\*\do\dobxcolumn\dobypara\dodhgt8198 \dpline\dpptx0\dppty0\dpptx342\dppty570\dpx3939\dpy123\dpxsize342\dpysize570\dplinew15\dplinecor0\dplinecog0\dplinecob0}}} {\shp{\*\shpinst\shpleft3255\shptop123\shpright4053\shpbottom693\shpfhdr0\shpbxcolumn\shpbxignore\shpbypara\shpbyignore\shpwr3\shpwrk0\shpfblwtxt0\shpz5\shplid1035{\sp{\sn shapeType}{\sv 20}}{\sp{\sn fFlipH}{\sv 0}}{\sp{\sn fFlipV}{\sv 0}} {\sp{\sn shapePath}{\sv 4}}{\sp{\sn fFillOK}{\sv 0}}{\sp{\sn fFilled}{\sv 0}}{\sp{\sn lineEndArrowhead}{\sv 1}}{\sp{\sn fArrowheadsOK}{\sv 1}}{\sp{\sn fLayoutInCell}{\sv 0}}{\sp{\sn fLayoutInCell}{\sv 0}}}{\shprslt{\*\do\dobxcolumn\dobypara\dodhgt8197 \dpline\dpptx0\dppty0\dpptx798\dppty570\dpx3255\dpy123\dpxsize798\dpysize570\dplinew15\dplinecor0\dplinecog0\dplinecob0}}}}{\fs20\lang2057\langfe1033\langnp2057 \par \par }{\fs20\lang1024\langfe1024\noproof\langnp1033 {\shp{\*\shpinst\shpleft3426\shptop141\shpright6105\shpbottom768\shpfhdr0\shpbxcolumn\shpbxignore\shpbypara\shpbyignore\shpwr3\shpwrk0\shpfblwtxt0\shpz4\shplid1036 {\sp{\sn shapeType}{\sv 3}}{\sp{\sn fFlipH}{\sv 0}}{\sp{\sn fFlipV}{\sv 0}}{\sp{\sn lTxid}{\sv 327680}}{\sp{\sn fShadow}{\sv 1}}{\sp{\sn f3D}{\sv 0}}{\sp{\sn fLayoutInCell}{\sv 0}}{\sp{\sn fLayoutInCell}{\sv 0}}{\shptxt \pard\plain \qc \li0\ri0\widctlpar\faauto\adjustright\rin0\lin0\itap0 \fs24\lang3082\langfe1033\cgrid\langnp3082\langfenp1033 {\fs20 f(x}{\fs20\sub 0}{\fs20 ,x}{\fs20\sub 1}{\fs20 ,...,x}{\fs20\sub L-1}{\fs20 ) \par }}}{\shprslt{\*\do\dobxcolumn\dobypara\dodhgt8196\dptxbx\dptxlrtb{\dptxbxtext\pard\plain \qc \li0\ri0\widctlpar\faauto\adjustright\rin0\lin0\itap0 \fs24\lang3082\langfe1033\cgrid\langnp3082\langfenp1033 {\fs20 f(x}{\fs20\sub 0}{\fs20 ,x}{\fs20\sub 1}{ \fs20 ,...,x}{\fs20\sub L-1}{\fs20 ) \par }}\dpx3426\dpy141\dpxsize2679\dpysize627\dpfillfgcr255\dpfillfgcg255\dpfillfgcb255\dpfillbgcr255\dpfillbgcg255\dpfillbgcb255\dpfillpat1\dplinew15\dplinecor0\dplinecog0\dplinecob0}}}}{\fs20\lang2057\langfe1033\langnp2057 \par }{\fs20\lang1024\langfe1024\noproof\langnp1033 {\shp{\*\shpinst\shpleft2400\shptop207\shpright3426\shpbottom207\shpfhdr0\shpbxcolumn\shpbxignore\shpbypara\shpbyignore\shpwr3\shpwrk0\shpfblwtxt0\shpz9\shplid1037 {\sp{\sn shapeType}{\sv 20}}{\sp{\sn fFlipH}{\sv 1}}{\sp{\sn fFlipV}{\sv 0}}{\sp{\sn shapePath}{\sv 4}}{\sp{\sn fFillOK}{\sv 0}}{\sp{\sn fFilled}{\sv 0}} {\sp{\sn fArrowheadsOK}{\sv 1}}{\sp{\sn fLayoutInCell}{\sv 0}}{\sp{\sn fLayoutInCell}{\sv 0}}}{\shprslt{\*\do\dobxcolumn\dobypara\dodhgt8201\dpline\dpptx1026\dppty0\dpptx0\dppty0\dpx2400\dpy207\dpxsize1026\dpysize0 \dplinew15\dplinecor0\dplinecog0\dplinecob0}}}}{\fs20\lang2057\langfe1033\langnp2057 \par \par \par \tab \tab By representing the feedback functions in algebraic normal form \par }\pard \qc \li0\ri0\widctlpar\faauto\adjustright\rin0\lin0\itap0 {\pard\plain \qc \li0\ri0\widctlpar\faauto\adjustright\rin0\lin0\itap0 \fs20\lang2057\langfe1033\cgrid\langnp2057\langfenp1033 {\object\objemb\objw9481\objh380{\*\objclass Equation.3} {\*\objdata 01050000020000000b0000004571756174696f6e2e3300000000000000000000160000 d0cf11e0a1b11ae1000000000000000000000000000000003e000300feff0900060000000000000000000000010000000100000000000000001000000200000001000000feffffff0000000000000000ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff 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(in this case, both of them are). Strictly speaking, given a maximum-period function, if we swap the indexes of the coefficients: x}{ \fs20\lang2057\langfe1033\sub\langnp2057 i}{\fs20\lang2057\langfe1033\langnp2057 }{\fs20\lang2057\langfe1033\langnp2057 {\field{\*\fldinst SYMBOL 171 \\f "Symbol" \\s 10}{\fldrslt\f3\fs20}}}{\fs20\lang2057\langfe1033\langnp2057 x}{ \fs20\lang2057\langfe1033\sub\langnp2057 L-i}{\fs20\lang2057\langfe1033\langnp2057 we obtain another function having the same property. \par \par There are many different ways for showing this property. The most intuitive proof is arguing over the DeBruijn graph associated to the FSR (we will later define this graph), but it can be a lso demonstrated algebraically. We know that the function must have the (c) form: \par }\pard \qc \li284\ri0\widctlpar{\*\pn \pnlvlcont\ilvl12\ls0\pnrnot0\pndec }\faauto\ilvl12\adjustright\rin0\lin284\itap0 {\pard\plain \qc \li284\ri0\widctlpar{\*\pn \pnlvlcont\ilvl12\ls0\pnrnot0\pndec }\faauto\ilvl12\adjustright\rin0\lin284\itap0 \fs20\lang2057\langfe1033\cgrid\langnp2057\langfenp1033 {\object\objemb\objw3581\objh360{\*\objclass Equation.3}{\*\objdata 01050000020000000b0000004571756174696f6e2e3300000000000000000000100000 d0cf11e0a1b11ae1000000000000000000000000000000003e000300feff0900060000000000000000000000010000000100000000000000001000000200000001000000feffffff0000000000000000ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff 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Suppose it is a DeBruijn sequence, if we invert the time axis we obtain an another differe nt DeBruijn sequence (for example, given 11100010, the inversion would derive to 01000111). 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\fs24\lang3082\langfe1033\cgrid\langnp3082\langfenp1033 {\fs20 01000 \par }}\dpx5016\dpy0\dpxsize969\dpysize396\dpfillfgcr255\dpfillfgcg255\dpfillfgcb255\dpfillbgcr255\dpfillbgcg255\dpfillbgcb255\dpfillpat0\dplinehollow\dptxbx\dptxlrtb{\dptxbxtext\pard\plain \ql \li0\ri0\widctlpar\faauto\adjustright\rin0\lin0\itap0 \fs24\lang3082\langfe1033\cgrid\langnp3082\langfenp1033 {\fs20 10000 \par }}\dpx6213\dpy0\dpxsize969\dpysize396\dpfillfgcr255\dpfillfgcg255\dpfillfgcb255\dpfillbgcr255\dpfillbgcg255\dpfillbgcb255\dpfillpat0\dplinehollow\dpendgroup\dpx0\dpy0\dpxsize0\dpysize0}}}}{\fs20\lang2057\langfe1033\langnp2057 We call linear terms to those terms in the algebraic normal form that only contain one variable: x}{\fs20\lang2057\langfe1033\sub\langnp2057 0}{\fs20\lang2057\langfe1033\langnp2057 , x}{\fs20\lang2057\langfe1033\sub\langnp2057 1}{ \fs20\lang2057\langfe1033\langnp2057 , ..., x}{\fs20\lang2057\langfe1033\sub\langnp2057 L-1}{\fs20\lang2057\langfe1033\langnp2057 . 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