Sz\303\241m\303\255t\303\263g\303\251pes sz\303\241melm\303\251let J\303\241rai Antal Ezek a programok csak szeml\303\251ltet\303\251sre szolg\303\241lnak

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

restart;

# # This routine randomly choose an elliptic "curve" modulo n, # where gcd(n,6)=1. The coordinates x,y are choosen # randomly, the parameter a too, and b is calculated. # The list [x,y,a,b] is given back or a divisor d of n. # ellrand:=proc(n) local x,y,a,b,r,d,f; r:=rand(n); d:=0; while d=0 do x:=r(n); y:=r(n); a:=r(n); b:=y^2-x^3-a*x mod n; d:=4*a^3+27*b^2 mod n; gcd(d,n); od; if %<n and %>1 then return % fi; [x,y,a,b]; end;

Zio2I0kibkc2IjYpSSJ4R0YlSSJ5R0YlSSJhR0YlSSJiR0YlSSJyR0YlSSJkR0YlSSJmR0YlRiVGJUMnPkYrLUklcmFuZEdGJUYjPkYsIiIhPyhGJSIiIkY1RiUvRixGM0MoPkYnLUYrRiM+RihGOT5GKUY5PkYqLUkkbW9kR0YlNiQsKCokKUYoIiIjRjVGNSokKUYnIiIkRjUhIiIqJkYpRjVGJ0Y1RkdGJD5GLC1GPjYkLCYqJiIiJUY1KUYpRkZGNUY1KiYiI0ZGNSlGKkZDRjVGNUYkLUkkZ2NkR0YlNiRGLEYkQCQzMkkiJUdGJUYkMkY1RllPRlk3JkYnRihGKUYqRiVGJUYl

# # This brute force procedure calculate the number of points # on an elliptic curve modulo p>3, a prime. The curve # parameters are a, b. # ellcount:=proc(a,b,p) local x,c; c:=1; for x from 0 to p-1 do c:=c+numtheory[jacobi](x^3+a*x+b,p)+1; od; c; end;

Zio2JUkiYUc2IkkiYkdGJUkicEdGJTYkSSJ4R0YlSSJjR0YlRiVGJUMlPkYqIiIiPyhGKSIiIUYtLCZGJ0YtRi0hIiJJJXRydWVHJSpwcm90ZWN0ZWRHPkYqLChGKkYtLSZJKm51bXRoZW9yeUdGJTYjSSdqYWNvYmlHRiU2JCwoKiQpRikiIiRGLUYtKiZGJEYtRilGLUYtRiZGLUYnRi1GLUYtRipGJUYlRiU=

ellrand(97); ellcount(%[3],%[4],97);

NyYiI2IiI1UiIyMpIiNa

IiQxIg==

with(plots);

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

implicitplot(y^2=x^3-10*x+10,x=-5..10,y=-20..20,numpoints=10000);

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# # Addition on an elliptic "curve" modulo n, where gcd(n,6)=1. # P and Q are the points to add and a,b are the parameters. # The return value is the sum of the points or a divisor d of n. # elladd:=proc(P,Q,a,b,n) local l,d; if P[3]=0 then return Q fi; if Q[3]=0 then return P fi; if P=Q then return elldou(P,a,b,n) fi; if P[1]=Q[1] then return [0,1,0] fi; d:=igcdex(P[1]-Q[1],n,'l'); if 1<d and d<n then return d fi; l:=(P[2]-Q[2])*l mod n; l^2-P[1]-Q[1] mod n; [%,l*(P[1]-%)-P[2] mod n,1]; end;

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# # Doubling on an elliptic "curve" modulo n, where gcd(n,6)=1. # P is the point to double and a, b are the parameters. # The return value is the double of the point P or # a divisor d of n. # elldou:=proc(P,a,b,n) local l,d; if P[3]=0 then return P fi; if P[2]=0 then return [0,1,0] fi; d:=igcdex(2*P[2],n,'l'); if 1<d and d<n then return d fi; l:=(3*P[1]^2+a)*l mod n; l^2-2*P[1] mod n; [%,l*(P[1]-%)-P[2] mod n,1]; end;

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# # This program compute k*P, k>=0 or a divisor of n # on an elliptic "curve" modulo n, where gcd(n,6)=1. # It use the left-to-right binary method. # ellmul:=proc(P,k,a,b,n) local L,i,Q; if k=0 then return [0,1,0] fi; if P[3]=0 then return P fi; L:=convert(k,base,2); Q:=P; for i from nops(L)-1 to 1 by -1 do Q:=elldou(Q,a,b,n); if type(Q,integer) then return Q fi; if L[i]=1 then Q:=elladd(P,Q,a,b,n); if type(Q,integer) then return Q fi; fi; od; Q; end;

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