Chapter 2. Finite Fields (Chapter 3 in the text)

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Chater 2. Fiite Fields (Chater 3 i the tet 1. Grou Structures 2. Costructios of Fiite Fields GF(2 ad GF( 3. Basic Theory of Fiite Fields 4. The Miimal Polyomials 5. Trace Fuctios 6. Subfields

1. Grou Structures - Grous ad Cyclic Grous - Rigs ad Fields Defiitio 1 A grou is a set G together with a biary oeratio o G such that the followig three roerties hold: (i is associative; that is, for ay a, b, c G, a(bc=(abc. (iii For each a G, there eists a iverse elemet a -1 G such that a a -1 = a -1 a = e. (ii There is a idetity or (uity elemet e i G such that for all a G, ae = ea = a. Sometimes, we deote the grou as a trile (G,, e. If the grou also satisfies (iv For all a, b G, ab = ba, the the grou is called abelia or commutative. @G. Gog 2

Eamle 1 Let Z, the set cosistig of all itegers Q, the set of all ratioal umbers + ad are ordiary additio ad multilicatio. The (Z, +, 0 (Q, +, 0 (Q*,, 1 are all grous where Q* is the all ozero ratioal umbers. Furthermore, they are abelia. How about (Z*,, 1? @G. Gog 3

Oe of the most imortat structures i cryto: Residues modulo. Let be a ositive iteger (>1 ad Z rereset the set of remaider of all itegers o divisio, i.e., Z = {0, 1, 2,..., -1}. We defie a + b ad ab the ordiary sum ad roduct of a ad b reduced by modulo, resectively. Let Proositio 1 (a (Z, +, 0 forms a grou, (b (Z *,, 1 forms a grou for ay rime. Z * = {a Z a! 0}. @G. Gog 4

b = a i Defiitio 2 A multilicative grou G is said to be cyclic if there is a elemet a G such that for ay b G there is some iteger i with b = a i. Such a elemet a is called a geerator of the cyclic grou, ad we write G = <a>. G 1 a a 3 a 2 @G. Gog 5

Eamles (Z 6, +, 0, cyclic grou with geerators 1 ad 5. (Z 3 *,, 1, cyclic grou with geerator 2. Z 3 * ={1, 2} = <2> ={2 0 = 1, 2}, 2 2 = 1 mod 3. (Z 7 *,, 1, cyclic grou, 3 is a geerator: 3 1 = 3, 3 2 = 2, 3 3 = 6, 3 4 = 4, 3 5 = 5, 3 6 = 1 mod 7 However, 2 3 =1 mod 7. Thus 2 is ot a geerator of Z 7 *. (Z 5 *,, 1, cyclic grou, 2 is a geerator. 2 1 = 2, 2 2 = 4, 2 3 = 3, 2 4 = 1 mod 5, thus Z 5 * = <2>. i.e., every elemet i Z 5 * ca be writte ito a ower of 2. @G. Gog 6

Fiite Grou Defiitio 3 A grou is called fiite if it cotais fiite may elemets. The umber of elemets i G is called the order of G, deoted as G. @G. Gog 7

E.g. - (Z, +,, (Q, +, are rigs. - (Z, +, forms a rig, called the residue class rig modulo. - (Z 4, +, is a rig. Defiitio 4 A rig (R, +, is a set R, together with two biary oeratios, deoted by + ad, such that: (i R is a abelia grou with resect to +. (ii is associative, that is, (a b c=a (b c for all a, b, c R. (iii The distributive laws hold; that is, for all a, b, c R, we have a (b + c = a b + a c ad (b + c a = b a + c a. Rigs ad Fields @G. Gog 8

Let (F, +, be a rig, ad let F* = {a F a! 0}, the set of elemets of F that are o zero. Defiitio 6 A fiite field is a field that cotais a fiite umber of elemets, this umber is called the order of the field. Fiite fields are called Galois fields after their discoverer. E. g. (Q, +,, (R, +, ad (C, +, are fields where R is the set of all real umbers, ad C, the set of all comle umbers. Defiitio 5 A field is a rig (F, +, such that F* together the multilicatio forms a commutative grou. (Z 2, +, forms a fiite field + 0 1 0 1 0 1 1 0 Proositio 2 Let be a rime, the (Z, +, is a fiite field with order. This field is deoted as GF(. 0 1 0 1 0 0 0 1!!! @G. Gog 9

2 Costructios of Fiite Fields GF(2 ad GF( Ste 1 Select, a ositive iteger ad a rime. Ste 2 Choose that f( is a irreducible olyomial over GF( of degree. Ste 3 We agree that is a elemet that satisfies f( = 0. Let For two elemets g(, h( i GF(, we write g( = a 0 + a 1 +... + a -1-1 ad h( = b 0 + b 1 +... + b -1-1. Additio: g( + h( = (a 0 + b 0 + (a 1 +b 1 +... + (a -1 + b -1-1 Multilicatio: g(h( = r( where r( is the remaider of g(h( divided by f(. GF ( = 1 { a0 + a1 + L + a 1 a GF ( } i @G. Gog 10

Eamle 7. Let = 2 ad f( = 3 + + 1. The f( is irreducible over GF(2. Let be a root of f(, i.e., f( = 0. The fiite field GF(2 3 is defied by GF(2 3 = {a 0 + a 1 + a 2 2 a i GF(2}. Table 1. GF(2 3, defied by f( = 3 + + 1 ad f( = 0. As a 3-tule As a olyomial As a ower of 000 = 001 = 0 1 = 0 = 1 010 = = 100 = 2 = 2 011 = 1 + = 3 110 = + 2 = 4 111 = 1 + + 2 = 5 101 = 1 + 2 = 6 7 = 1 @G. Gog 11

Additio: Comutatio i GF(2 3 : g( + h( = (1 + + ( + 2 Multilicatio: We take g( = 1+ ad h( = + 2. = 1 + 2 GF(2 3 g(h( = (1 + ( + 2 = + 2 + 2 + 3 = + 2 + 3 = + 3 = 1 (sice 3 + + 1 = 0. O the other had, g( = 1+ = 3 ad h( = + 2 = 4 We may comute the roduct of g( ad h( as follows g(h( = 3 4 = 7 = 1 @G. Gog 12

3 Basic Theory of Fiite Fields A. Primitive Elemets ad Primitive Polyomials!!! Fact 1 For ay fiite field F, its multilicative grou F *, the set of o zero elemet of F, is cyclic. GF(2 3 * 6 7 5 = 1 4 Defiitio. A geerator of the cyclic grou GF( * is called a rimitive elemet of GF(. A olyomial havig a rimitive elemet as zero is called a rimitive olyomial. E.g Sice 3 + + 1 = 0, the GF(2 * 3 3 + + 1 is a rimitive olyomial over GF(2. 1 2 4 + + 1 is a rimitive olyomial over GF(2. @G. Gog 13

How about f( = 4 + 3 + 2 + + 1? Is it rimitive? f( is irreducible over GF(2. So we ca use f( to defie GF(2 4. Let be a root of f(. 4 5 2 3 = (1 + + + 2 3 4 = + + + = 1 2 = 1+ + + 3 Thus, is ot a rimitive elemet of GF(2 4, so f( is ot rimitive. Remark. The tables of rimitive olyomials over GF(2 of degree 229 ca be foud at Alfred J. Meezes, Paul C. Va Vorschot ad S. A. Vastoe, Hadbook of Alied Crytograhy, CRC Press, 1996, 161. @G. Gog 14

B. Structure of GF( Let GF( be defied by f( = 0 where f( is rimitive, the the elemets i the field have the followig two reresetatios. GF( 1 = { a0 + a1 + L+ a 1 a GF( } i (vector reresetatio i = { 0 i < 1, i = } (eoetial reresetatio (we defie 0 = Remark. Vector reresetatio is efficiet for comutatio of additio ad the eoetial reresetatio is efficiet for comutatio of multilicatio. For small field, for eamle, = 2 ad < 40, it is much more efficiet tha to use the eoetial reresetatio for comutatio of multilicatio ad the vector reresetatio for that of additio where it stores the add 1 table for coversio from the vector reresetatio to the eoetial reresetatio. @G. Gog 15

Discrete Logarithm i GF(!!! Let be a rimitive elemet i GF( ad, a arbitrary ozero elemet i GF(. Fid k such that k = is called a discrete logarithm of uder the base. Zech s logarithm: For ay 0< k < -1, fid k such that ' k = k + 1 @G. Gog 16

4 Miimal Polyomials Defiitio. Let GF( ad m( be a moic olyomial over GF(, m( r 1 = = i 0 c i i + r, c i GF ( m( is called a miimal olyomial (MP of if m( is the olyomial with the lowest degree such that m( = 0. @G. Gog 17

Proerties of Miimal Polyomials Suose that m( is the MP of GF(, the (1 m( is irreducible over GF(. (2 m ( (3 the degree of m( is a factor of, i.e. deg m(. (4 the miimal olyomial of a rimitive elemet of GF( has degree. @G. Gog 18

Algorithm for Fidig Miimal Polyomial Iut: - f(, a rimitive olyomial over GF( of degree ; -, a root of f(; - = k a elemet i GF(. Outut: The miimal olyomial of over GF(. Procedure_MP( Ste 1. Geerate the fiite field GF( by f( Ste 2. Comute s such that s is the smallest umber satisfyig k s k mod 1. Ste 3. Comute m ( = ( ( L( Retur m ( igf( @G. Gog 19 s1.

Eamle 8. For GF(2 4, defied by 4 + + 1 = 0, comute the miimal olyomial of 7. Solutio. Alyig rocedure_mp(, here we have = 7 where k = 7. Comute s such that s is the smallest iteger satisfyig 7 2 s 7 mod 15 ( trial ad error method, we have s = 4. m 7 7 27 2 7 ( = ( ( ( ( 7 14 13 11 = ( + ( + ( + ( + 2 14 7 6 = [ + ( + + ][ 2 6 2 4 9 = ( + + ( + + 4 3 9 5 6 = + + ( + + 2 2 + 1 3 2 7 4 4 3 9 5 6 2 10 10 = + ( + + ( + + + ( + + 1 4 3 = + + 1 2 11 13 9 + ( + + ] 4 3 Thus, m 7 ( = + + 1. @G. Gog 20

5 Trace Fuctios Proerty 1. (a The trace fuctio is a liear fuctio. (b Defiitio. A trace fuctio Tr ( of GF( /GF( is a fuctio from GF( to GF( defied as follows ( + Tr( = y = + + y + L+ 1 Tr( : GF( GF(,, ygf(. Eamle 9. Let GF(2 3 be defied by 3 + + 1 = 0. Solutio. Tr( = + + Tr 3 Comute Tr( ad Tr(. 2 2 = (1 + + (1 + + (1 + + = 1 2 2 4 = + + + 3 3 6 5 ( = + + 2 = 0 @G. Gog 21

6. Subfields A subfield of GF( N GF( N is a subset of which itself forms a field. E is a subfield of GF( where m N ( m is a factor of N, i. e., GF( m N E = GF( GF( m if ad oly if N. E.g. 2 GF(2 GF(2 4 GF(2 GF(2 12 4 GF(2 6 We write N = m ad deote the trace GF(2 3 GF(2 2 fuctio from GF( Tr N m (, i. e., N to GF( m as GF(2 Tr N m ( = + q + L+ q m1, where q = m. @G. Gog 22

Proerty 2. (Trasitivity For a field chai GF( we have m GF( GF( m Tr1 ( = Tr1 ( Tr ( m!!! Tr m GF( m 1 ( GF( GF(2 8 Tr m ( Tr 1 ( GF( 8 ( Tr 4 This is the uderlie structure of GMW sequeces. Tr 1 GF( @G. Gog 23 8 ( 4 ( Tr 1 GF(2 4

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