## modifying curve setting up

/* CG - February 2010
the program is supposed to find y for any given x
[in progress]
the representation is in polynomial
*/

#include <stdio.h>
#include <stdlib.h>
#include "field2n.h"
#include "poly.h"
#include "eliptic.h"

unsigned long random_seed;
extern FIELD2N poly_prime;

void set_field(value, n)
FIELD2N *value;
INDEX n;
{
value->e[0] = n;
}

void set_curve(curv)
CURVE *curv;
{
curv->form = 1;
set_field(&curv->a6, 1L);
null(&curv->a2);
//   set_field(&curv->a2, 0L);
}

int main()
{
FIELD2N	*data, *data_juga;
CURVE *curve;
POINT *point;
INDEX error;
FIELD2N f;
POINT *pnt;

curve = malloc(sizeof(CURVE));
if (curve == NULL) {exit(-1);}

point = malloc(sizeof(POINT));
if (point == NULL) {exit(-1);}

if (!irreducible(&poly_prime)) return(0);
print_field("poly_prime = ", &poly_prime);

if (error = init_poly_math())
{
printf("Can't initialize S matrix, row = %d\n", error);
return(-1);
}

printf("setting up curves \n\n");

set_curve(curve);
print_curve("the curve after setting up: ", curve);

return 0;
}



 poly_prime = 8 0 0 0 0 c9 setting up curves 

 

the curve after setting up: form: 1 a2: 0 0 0 0 0 0 a6: 1 0 0 0 0 0 

• #### zakimath 7:28 am on February 24, 2010 Permalink | Reply

how to find an elliptic curve E with #E (the order of group) is prime…? 🙂

• #### CG 9:21 am on February 24, 2010 Permalink | Reply

hadoh! susah, harus tanya sama dukun math dulu nih 😀

• #### zakimath 11:10 am on February 24, 2010 Permalink | Reply

hehehe, emang susah bgt! masih belum terbayang… 🙂
Kalo gitu dibalik: Jika diberikan bilangan prima p, apakah terdapat suatu kurva elliptik E sedemikian hingga #E = p ?

• #### CG 11:27 am on February 24, 2010 Permalink | Reply

[berpikir keras] #E itu order kurva ya? berarti jumlah titik dalam kurva? bisa gitu jumlah titiknya ganjil (prima) kan titik2xnya sepasang? y^2?

• #### zakimath 12:59 pm on February 24, 2010 Permalink | Reply

Iya bu, #E itu banyaknya titik dalam suatu kurva E… bener gk sih simbolnya? koq jd lupa, he2… Contohnya kurva elliptik E: y^2 = x^3 + 12x + 20 atas field F_23 itu mempunyai jumlah titik 17, yaitu #E = 17 yg merupakan bilangan prima, akibatnya setiap titiknya merupakan generator dari E. CMIIW…

• #### zakimath 1:03 pm on February 24, 2010 Permalink | Reply

Hmmm, tp rasanya ada yg salah dg komentarku… #E = 17 itu sudah termasuk titik infinity blom ya? aduh, koq jadi lupa… 😦

• #### CG 9:23 pm on February 24, 2010 Permalink | Reply

eh baru aja saya mau bales, iya bner ganjil kan ditambah point of infinity 🙂

• #### HongKong 10:09 pm on February 27, 2010 Permalink | Reply

jawabannya sih.. bisa bilangan ganjil dan bisa juga bilangan genap ((x,0) kan gak punya pasangan)
untuk setiap bilangan prima p selalu dapat dicari kurva eliptik atas Fp dengan orde (banyaknya titik) p

• #### CG 6:26 am on February 28, 2010 Permalink | Reply

thank you, HongKong 🙂

• #### CG 5:33 pm on February 28, 2010 Permalink | Reply

baru inget kalau si HongKong = the geometer 😀

## Questions on polynomial and normal bases

1. How to convert from polynomial to normal bases?
2. For what conditions ONB representation is available?

## More on fields and polynomial

Have just learned that:

$F_{2}=Z_{2}=\left\{ \bar{0},\bar{1}\right\}$

$F_{4}=\left\{\left(a,b: a,b \epsilon F_{2}\right) \right\}$ means that the elements are

$\left(\bar{0},\bar{0} \right)$, $\left(\bar{0},\bar{1} \right)$, $\left(\bar{1},\bar{0} \right)$ and $\left(\bar{1},\bar{1} \right)$

$F_{8}=F_{2^{3}}\left\{\left(a, b, c: a, b, c \epsilon F_{2} \right) \right\}$

$F_{2^{n}}$ means that we’ll have $n$ tuple, thus we’ll have a polynomial with $n$ degree

(degree is the highest exponent of the polynomial).

Example:

$\left(\bar{1}, \bar{0}, \bar{1}, \bar{1} \right)$ means $x^{3}+x+1$

Important notes:

$F_{p}=Z_{p}$ is only if $p$ is prime. $F_{8}\neq Z_{8}$

## Polynomial

The phd student had just found out more about $F_{2^{n}}$ representation. For $2^{n}$, there are $n$ tuples. The tuples turn out to be the degree of a polynomial!

She is so happy to know the basic terms and concepts in math. Interesting!

## Math Polynomial

Move forward to Galois Field representation in C. Zooming in all the modules of math for polynomial representation with the prime number 2 as the modulus (means that the coefficient and only take values of 0 or 1).

The text converting code is halfway to go.

• #### yuti 9:53 pm on January 1, 2009 Permalink | Reply

Aaaaaa…. selalu pengen belajar Galois, waktu itu pernah mau belajar tentang Galois waktu sedang seneng baca tentang ehmm…mmm… kayanya tentang social network dan transformasi-transformasi gitu… aaaaaa… cerita-cerita

• #### CG 4:44 am on January 2, 2009 Permalink | Reply

yuti: hah, gue sampe skarang juga belum ngerti2x yut 😀

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