Finite Fields 07/01/1999 and Zech's Logarithms 1 Pierre Douillet - PowerPoint PPT Presentation

Université Lille-1 Lab ANO P.L. Douillet Finite Fields 07/01/1999 and Zech's Logarithms 1 Pierre Douillet douillet@cnam.fr

� roots set of the Fermat's polynomial : � m x � x Z / p � F Z F p � F F x � � p q � � quotient field modulo an irreducible , i.e. P � Lab ANO set of polynomials into a proper element : P.L. Douillet 07/01/1999 F F q � F F p X P � X � powers of a primitive element : � a � 2 F F 0 0 � a � q � 2 q � three characterizations of the finite field GF(q) douillet@cnam.fr

X 256 � X � X � 1 � 3 � 5 � 15 � 17 � 51 � 85 � 255 GF(256)* 51 GF(4)* Lab ANO 255 GF(16)* P.L. Douillet 3 07/01/1999 15 17 GF(2)* 85 3 1 5 Subgroups of GF(256)* and subfields of GF(256) Substructures of GF(256) douillet@cnam.fr

��������������������������������������������������������������������� s � s � R � Z � s � Z � s R 2 R � s 0 0 � � 0 0 � 1 1 0 0 1 � � 2 � 1 1 � 4 5 3 � � 1 4 2 � 2 8 10 Lab ANO 4 � 2 2 3 � 3 14 2 5 � 2 � 1 8 P.L. Douillet 4 � � 1 1 5 6 � 2 � � 5 5 � 2 � � 10 0 07/01/1999 7 � 2 � � � 1 10 6 � 3 � � 2 13 4 8 � 3 3 7 � 3 � x � 1 9 1 9 � 3 � 1 14 8 � 2 � 1 2 10 10 � 3 � � 9 9 � 3 � � 7 1 11 � 3 � � � 1 7 10 � 2 � � � 1 5 0 12 � 3 � � 2 6 11 � 3 � � 2 � � 12 8 4 13 � 3 � � 2 � 1 13 12 � 3 � � 2 � � � 1 11 8 14 � 3 � � 2 � � 11 13 � 3 � � 2 � 1 6 4 15 � 3 � � 2 � � � � 12 14 � 3 � 1 3 2 the Imamura algorithm douillet@cnam.fr

put : ax 2 � bx � c � 0 z � ax / b inside obtain : Lab ANO z 2 � z � ac / b 2 � z � � z � � � 1 � 0 P.L. Douillet 07/01/1999 and observe that : L ac / b 2 � L � � L � � 1 � s � Z s x 2 � � 3 x � � 10 � 0 exercise : 5 answer : x � � , � 9 solving quadratic equations douillet@cnam.fr

P � s � ps (Frobenius) ; S � s � � s (inverse) ; Z (Zech's logarithm) Lab ANO P.L. Douillet 07/01/1999 6 � is another 0 � q � 2 F F � q � � douillet@cnam.fr q

groups acting over can be used for a � q 2 m space/time transaction sp k mod q � 1 x p k � 0 � m � 1 C s � Lab ANO s � min C s P.L. Douillet � 07/01/1999 x p , 1/ x s � min � q ± k k � C s � 7 Z ps � p Z s ; Z � s � Z s � s SPS : space saving algorithms douillet@cnam.fr

groups acting over can be used to � q compress and re-expand a Zech's table x � 1, 1/ x � PGL 2 p if p � 2 or p � 4 n � 3 Lab ANO � PSL 2 p if p � 2 or p � 4 n � 1 P.L. Douillet 07/01/1999 p 2 � 1 p p 2 � 1 p / 2 therefore or sized when p � 3 PGL 2 3 iso S 4 8 and s,z,s,z,s,z,s,z, generate a skew octagon REB : rebuilding algorithms douillet@cnam.fr

j � � � ; � � define q � 1 / 3 1 � j � j 2 � 0 0, 1, j , j � 1 thus and F F 4 � � s � j � � Y s � s � 1 � � Z s consider Y : like Lab ANO P.L. Douillet 07/01/1999 x p , 1/ x , jx 6 m sized s � min � q ± k , ± k ± � k � C s � Z s � � � Y s � � 9 Y ps � p ZY s ; Y � s � Y s � � � s SPS2 : space saving, , m even p � 2 douillet@cnam.fr

again, groups acting over can be used � q to compress and re-expand a Zech's table x � 1, x � j , 1/ x � PGL 2 4 is 60 sized Lab ANO P0z P0 P13y P.L. Douillet P13x P7z P9z P9 07/01/1999 PGL 2 4 P14x P1z � A 5 � P10z P12x P12y P6 P5 P11y P3z P3 PSL 2 5 P2y P8x P4y P4 P2z P2 P3y generating a P8 P5y 10 P5x P11z P6x P12z P10y dodecahedron P14z P1y P14 P9x P7y P7x P13z P0x P0y REB2 : rebuilding, , m even p � 2 douillet@cnam.fr

Y Z Z Y Lab ANO P.L. Douillet 07/01/1999 Z Y Y 11 Z A four cosets wheel douillet@cnam.fr

[zsy, zsy, zs, zsy, Qzsy, Qs, zsy, Qzs, zsy, zsy, zsy, Qs, zsy, zsy] 0 1 2 3 Lab ANO 4 P.L. Douillet 07/01/1999 5 6 14 13 12 7 12 11 10 9 8 A 5 walking inside douillet@cnam.fr

1: factorize , 2: proper and , P � P � | � q � 1 � q � 1 3: proper and not P � | � d P � m imam meth1 meth2 meth3 2 0 .015 .010 0 Lab ANO 3 0 .005 0 .005 4 .005 .009 .020 .020 P.L. Douillet 5 .010 .015 .005 .005 6 .025 .010 .005 .005 07/01/1999 7 .095 .040 .020 .005 8 .105 .040 .035 .015 9 .255 .170 .024 .175 10 .530 .275 .048 .020 11 1.065 1.457 .225 .209 12 2.720 1.579 .405 .099 13 6.505 18.235 .535 .120 13 14 11.861 33.325 1.314 .630 15 43.425 420.060 435.882 1.120 16 70.925 448.382 9.655 3.255 17 189.645 6.540 18 582.844 35.623 19 1627.618 193.291 optimizing Imamura's algorithm douillet@cnam.fr

� define " admissible" as a q , � primitive : � a � � � 1 � � � F F � � � k Lab ANO � if then Z � s � Z � ks ÷ k mod q � 1 P.L. Douillet 07/01/1999 � the set of all admissible exponents is : Z � k ÷ k gcd k , q � 1 � 1 Adm q � � # Adm q � � q � 1 m 14 � therefore roughly characterizes a P � X admissible exponents douillet@cnam.fr

� start from s � 1 ; z s � a , knowing that is an admissible exponent a � use , i.e. the REB algorithm, x � 1, 1/ x , � Lab ANO P.L. Douillet to walk through orbit(1) 07/01/1999 t � Z s � k � k � try until are already k � � Z a � k � k 15 known and restart from : s � t ; z s � � until the table is completed JMP algorithm douillet@cnam.fr

� run JMP, starting from a random a � � q � if a jumping offset can't be found, k discard (dubious value) a Lab ANO P.L. Douillet � if an offset leads to a s � t ; z s � � k 07/01/1999 that contradicts a previously obtained s ; z s , can't be admissible a � otherwise, is called "efficient exponent" 16 a SEEK algorithm douillet@cnam.fr

If JMP, started with a random , ends a � � q without contradiction, that was indeed an a Lab ANO admissible exponent, and the obtained function P.L. Douillet is actually the Zech's logarithm associated 07/01/1999 with the primitive polynomial P � X � � m � 1 X � � k � k � 0 17 as computed from the Zech function. THEOREM : efficient exponent douillet@cnam.fr

m � deg gcd X a � X � 1 ; X q � 1 not proved : deg gcd X a � X � 1 ; X q � 1 � m Lab ANO P.L. Douillet 07/01/1999 d|m mod discard 2 3 2 3 7 3, 5 18 4 15 2, 4, 5, 8, 11, 12, 14 2, 3, 5, 8, 10, 11, 12, 14, 17, 19, 20 6 21 several sieves douillet@cnam.fr

a _ eff laps imam m poly � 2 � � � 1 2 2 0 0 � 3 � � � 1 3 3 0 0 � 4 � � � 1 4 4 .005 .005 � 5 � � 3 � � 2 � � � 1 5 12 .005 .010 � 6 � � � 1 6 6 .010 .025 Lab ANO � 7 � � � 1 7 7 .010 .095 P.L. Douillet � 8 � � 7 � � 5 � � 3 � 1 8 13 .055 .105 07/01/1999 � 9 � � 8 � � 6 � � 5 � � 3 � � 2 � 1 9 11 .049 .255 � 10 � � 9 � � 7 � � 3 � 1 10 57 .170 .530 � 11 � � 10 � � 9 � � 8 � � 7 � � 4 � 1 11 25 .190 1.065 � 12 � � 8 � � 7 � � 6 � � 4 � � 3 � 1 12 448 1.070 2.720 � 13 � � 10 � � 8 � � 7 � � 4 � � 3 � � 2 � � � 1 13 18 1.225 6.505 � 14 � � 13 � � 11 � � 9 � � 8 � � 5 � � 4 � � 3 � 1 14 40 5.580 11.861 19 � 15 � � � 1 15 15 8.505 43.425 � 16 � � 13 � � 10 � � 8 � � 6 � � 5 � � 4 � � 3 � 1 29.750 16 15981 70.925 � 17 � � 14 � � 11 � � 9 � � 6 � � 3 � 1 17 385 48.160 189.645 some results douillet@cnam.fr