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

SLIDE 1

Lab ANO

P.L. Douillet

07/01/1999

1

douillet@cnam.fr

Université Lille-1

Finite Fields

and

Zech's Logarithms

Pierre Douillet

SLIDE 2

Lab ANO

P.L. Douillet

07/01/1999

2

douillet@cnam.fr

Z Z /p F F

p F

F

q

x m x x p P F F

q F

F

p X

P X F F

q

a

0 a q 2

three characterizations of the finite field GF(q)

roots set of the Fermat's polynomial : quotient field modulo an irreducible , i.e. set of polynomials into a proper element : powers of a primitive element :

SLIDE 3

Lab ANO

P.L. Douillet

07/01/1999

3

douillet@cnam.fr

GF(2)* GF(4)* GF(16)* GF(256)* 85 17 51 5 1 3 15 255

X 256 X X 1 3 5 15 17 51 85 255

Subgroups of GF(256)* and subfields of GF(256)

Substructures of GF(256)

SLIDE 4

Lab ANO

P.L. Douillet

07/01/1999

4

douillet@cnam.fr

s s R

Z s Z s

0 1
1

4 5 2 2 8 10 3 3 14 2 4 1 1 5 5 2 10 6 3 2 13 4 7 3 x 1 9 1 8 2 1 2 10 9 3 7 1 10 2 1 5 11 3 2 12 8 12 3 2 1 11 8 13 3 2 1 6 4 14 3 1 3 2 R 2 R s 0 0

1 1

2 1 3 1 4 4 2 2 5 2 1 8 6 2 5 7 2 1 10 8 3 3 9 3 1 14 10 3 9 11 3 1 7 12 3 2 6 13 3 2 1 13 14 3 2 11 15 3 2 12

the Imamura algorithm

SLIDE 5

Lab ANO

P.L. Douillet

07/01/1999

5

douillet@cnam.fr

z ax/b ax 2 bx c 0 z 2 z ac/b 2 z z 1 0 L ac/b 2 L L 1 s Z s x 2 3 x 10 0

x , 9

solving quadratic equations

put : inside

btain :

and observe that : exercise :

answer :

SLIDE 6

Lab ANO

P.L. Douillet

07/01/1999

6

douillet@cnam.fr

q

0 q 2

F F

q

P s ps S s s Z

is another

(Frobenius) ; (inverse) ; (Zech's logarithm)

SLIDE 7

Lab ANO

P.L. Douillet

07/01/1999

7

douillet@cnam.fr

q 2m x p

C s sp k mod q 1 k 0m1

s min C s

x p, 1/x

s minq

± k k C s

Z ps pZ s ; Z s Z s s

SPS : space saving algorithms

groups acting over can be used for a space/time transaction

SLIDE 8

Lab ANO

P.L. Douillet

07/01/1999

8

douillet@cnam.fr

q

x 1, 1/x PGL2 p if p 2 or p 4n 3 PSL2 p if p 2 or p 4n 1

p 2 1 p p 2 1 p / 2 p 3 PGL2 3 iso S4

REB : rebuilding algorithms

groups acting over can be used to compress and re-expand a Zech's table therefore

r

sized when and s,z,s,z,s,z,s,z, generate a skew octagon

SLIDE 9

Lab ANO

P.L. Douillet

07/01/1999

9

douillet@cnam.fr

p 2 j ; q 1 / 3 1 j j 2 0 F F

4

0, 1, j, j 1 Y s j Y s s 1 Z s x p, 1/x, jx 6m

s minq

± k, ±k ± k C s

Z s Y s Y ps pZY s ; Y s Y s s

SPS2 : space saving, , m even

define thus and consider : like sized

SLIDE 10

Lab ANO

P.L. Douillet

07/01/1999

10

douillet@cnam.fr

P13z P0x P0y P9x P7y P7z P5x P14 P8 P3z P12y P2z P7x P8x P5y P3 P2 P12x P14x P0z P3y P11y P6x P12z P1z P9 P13x P11z P2y P0 P10y P6 P10z P5 P4 P14z P9z P1y P4y P13y

p 2 q x 1, x j, 1/x PGL2 4 60 PGL2 4 A5 PSL2 5

REB2 : rebuilding, , m even

again, groups acting over can be used to compress and re-expand a Zech's table is sized generating a dodecahedron

SLIDE 11

Lab ANO

P.L. Douillet

07/01/1999

11

douillet@cnam.fr

Z Y Y Z Y Z Z Y

A four cosets wheel

SLIDE 12

Lab ANO

P.L. Douillet

07/01/1999

12

douillet@cnam.fr

[zsy, zsy, zs, zsy, Qzsy, Qs, zsy, Qzs, zsy, zsy, zsy, Qs, zsy, zsy]

14 13 12 11 10 9 8 7 6 5 4 3 2 1

A5

walking inside

SLIDE 13

Lab ANO

P.L. Douillet

07/01/1999

13

douillet@cnam.fr

m imam meth1 meth2 meth3 2 .015 .010 3 .005 .005 4 .005 .009 .020 .020 5 .010 .015 .005 .005 6 .025 .010 .005 .005 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 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

q1

P

P | q1

P P | d

ptimizing Imamura's algorithm

1: factorize , 2: proper and , 3: proper and not

SLIDE 14

Lab ANO

P.L. Douillet

07/01/1999

14

douillet@cnam.fr

a F F

q, primitive : a 1

k Z s Z ks ÷ k q 1 Admq Z k ÷ k gcd k, q 1 1 # Admq q 1 m a P X

admissible exponents

define " admissible" as if then mod the set of all admissible exponents is :

therefore roughly characterizes

SLIDE 15

Lab ANO

P.L. Douillet

07/01/1999

15

douillet@cnam.fr

s 1 ; z s a a

x 1, 1/x,

k t Z s k k Z a k k s t ; z s

JMP algorithm

start from , knowing that is an admissible exponent use , i.e. the REB algorithm, to walk through orbit(1) try until are already known and restart from : until the table is completed

SLIDE 16

Lab ANO

P.L. Douillet

07/01/1999

16

douillet@cnam.fr

a q k a k s t ; z s s ; z s a a

SEEK algorithm

run JMP, starting from a random if a jumping offset can't be found, discard (dubious value) if an offset leads to a that contradicts a previously obtained , can't be admissible

therwise, is called "efficient exponent"

SLIDE 17

Lab ANO

P.L. Douillet

07/01/1999

17

douillet@cnam.fr

a q a P X

m 1 k 0

X k

THEOREM : efficient exponent

If JMP, started with a random , ends without contradiction, that was indeed an admissible exponent, and the obtained function is actually the Zech's logarithm associated with the primitive polynomial as computed from the Zech function.

SLIDE 18

Lab ANO

P.L. Douillet

07/01/1999

18

douillet@cnam.fr

m deg gcd X a X 1 ; X q 1 deg gcd X a X 1 ; X q 1 m

several sieves

not proved : d|m mod discard 2 3 2 3 7 3, 5 4 15 2, 4, 5, 8, 11, 12, 14 6 21

2, 3, 5, 8, 10, 11, 12, 14, 17, 19, 20

SLIDE 19

Lab ANO

P.L. Douillet

07/01/1999

19

douillet@cnam.fr

m a_eff poly laps imam 2 2 2 1 3 3 3 1 4 4 4 1 .005 .005 5 12 5 3 2 1 .005 .010 6 6 6 1 .010 .025 7 7 7 1 .010 .095 8 13 8 7 5 3 1 .055 .105 9 11 9 8 6 5 3 2 1 .049 .255 10 57 10 9 7 3 1 .170 .530 11 25 11 10 9 8 7 4 1 .190 1.065 12 448 12 8 7 6 4 3 1 1.070 2.720 13 18 13 10 8 7 4 3 2 1 1.225 6.505 14 40 14 13 11 9 8 5 4 3 1 5.580 11.861 15 15 15 1 8.505 43.425 16 15981 16 13 10 8 6 5 4 3 1 29.750 70.925 17 385 17 14 11 9 6 3 1 48.160 189.645

Université Lille-1

Finite Fields

and

Zech's Logarithms

Pierre Douillet

Z Z /p F F

F

x m x x p P F F

F

P X F F

0 a q 2

three characterizations of the finite field GF(q)

roots set of the Fermat's polynomial : quotient field modulo an irreducible , i.e. set of polynomials into a proper element : powers of a primitive element :

X 256 X X 1 3 5 15 17 51 85 255

Substructures of GF(256)

the Imamura algorithm

z ax/b ax 2 bx c 0 z 2 z ac/b 2 z z 1 0 L ac/b 2 L L 1 s Z s x 2 3 x 10 0

solving quadratic equations

put : inside

and observe that : exercise :

q

F F

P s ps S s s Z

is another

(Frobenius) ; (inverse) ; (Zech's logarithm)

q 2m x p

x p, 1/x

Z ps pZ s ; Z s Z s s

SPS : space saving algorithms

groups acting over can be used for a space/time transaction

q

p 2 1 p p 2 1 p / 2 p 3 PGL2 3 iso S4

REB : rebuilding algorithms

groups acting over can be used to compress and re-expand a Zech's table therefore

sized when and s,z,s,z,s,z,s,z, generate a skew octagon

p 2 j ; q 1 / 3 1 j j 2 0 F F

0, 1, j, j 1 Y s j Y s s 1 Z s x p, 1/x, jx 6m

Z s Y s Y ps pZY s ; Y s Y s s

SPS2 : space saving, , m even

define thus and consider : like sized

p 2 q x 1, x j, 1/x PGL2 4 60 PGL2 4 A5 PSL2 5

REB2 : rebuilding, , m even

again, groups acting over can be used to compress and re-expand a Zech's table is sized generating a dodecahedron

A four cosets wheel

A5

walking inside

P

P P | d

1: factorize , 2: proper and , 3: proper and not

a F F

k Z s Z ks ÷ k q 1 Admq Z k ÷ k gcd k, q 1 1 # Admq q 1 m a P X

admissible exponents

define " admissible" as if then mod the set of all admissible exponents is :

s 1 ; z s a a

k t Z s k k Z a k k s t ; z s

JMP algorithm

start from , knowing that is an admissible exponent use , i.e. the REB algorithm, to walk through orbit(1) try until are already known and restart from : until the table is completed

a q k a k s t ; z s s ; z s a a

SEEK algorithm

run JMP, starting from a random if a jumping offset can't be found, discard (dubious value) if an offset leads to a that contradicts a previously obtained , can't be admissible

a q a P X

X k

THEOREM : efficient exponent

If JMP, started with a random , ends without contradiction, that was indeed an admissible exponent, and the obtained function is actually the Zech's logarithm associated with the primitive polynomial as computed from the Zech function.

m deg gcd X a X 1 ; X q 1 deg gcd X a X 1 ; X q 1 m

several sieves

not proved : d|m mod discard 2 3 2 3 7 3, 5 4 15 2, 4, 5, 8, 11, 12, 14 6 21

some results