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= Sage Interactions - Number Theory =
goto [[interact|interact main page]]
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= Integer Factorization =

== Divisibility Poset ==
by William Stein
{{{#!sagecell
@interact
def _(n=(5..100)):
    Poset(([1..n], lambda x, y: y%x == 0) ).show()
}}}

{{attachment:divposet.png}}
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{{{ {{{#!sagecell
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== Continued Fraction Plotter ==
by William Stein
{{{
@interact
def _(number=e, ymax=selector([None,5,20,..,400],nrows=2), clr=Color('purple'), prec=[500,1000,..,5000]):
    c = list(continued_fraction(RealField(prec)(number))); print c
    show(line([(i,z) for i, z in enumerate(c)],rgbcolor=clr),ymax=ymax,figsize=[10,2])
}}}
{{attachment:contfracplot.png}}		 More complicated demonstration using Mathematica: http://demonstrations.wolfram.com/FactorTrees/

== Factoring an Integer ==
by Timothy Clemans

Sage implementation of the Mathematica demonstration of the same name. http://demonstrations.wolfram.com/FactoringAnInteger/

{{{#!sagecell
@interact
def _(r=selector(range(0,10000,1000), label='range', buttons=True), n=slider(0,1000,1,2,'n',False)):
    if not r and n in (0, 1):
        n = 2
    s = '$%d = %s$' % (r + n, factor(r + n))
    s = s.replace('*', '\\times')
    html(s)
}}}

= Prime Numbers =
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{{{ {{{#!sagecell
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== Computing Generalized Bernoulli Numbers ==
by William Stein (Sage-2.10.3)
{{{
@interact
def _(m=selector([1..15],nrows=2), n=(7,(3..10))):
    G = DirichletGroup(m)
    s = "<h3>First n=%s Bernoulli numbers attached to characters with modulus m=%s</h3>"%(n,m)
    s += '<table border=1>'
    s += '<tr bgcolor="#edcc9c"><td align=center>$\\chi$</td><td>Conductor</td>' + \
           ''.join('<td>$B_{%s,\chi}$</td>'%k for k in [1..n]) + '</tr>'
    for eps in G.list():
        v = ''.join(['<td align=center bgcolor="#efe5cd">$%s$</td>'%latex(eps.bernoulli(k)) for k in [1..n]])
        s += '<tr><td bgcolor="#edcc9c">%s</td><td bgcolor="#efe5cd" align=center>%s</td>%s</tr>\n'%(
             eps, eps.conductor(), v)
    s += '</table>'
    html(s)
}}}

{{attachment:bernoulli.png}}


== Fundamental Domains of SL_2(ZZ) ==
by Robert Miller
{{{
L = [[-0.5, 2.0^(x/100.0) - 1 + sqrt(3.0)/2] for x in xrange(1000, -1, -1)]
R = [[0.5, 2.0^(x/100.0) - 1 + sqrt(3.0)/2] for x in xrange(1000)]
xes = [x/1000.0 for x in xrange(-500,501,1)]
M = [[x,abs(sqrt(x^2-1))] for x in xes]
fundamental_domain = L+M+R
fundamental_domain = [[x-1,y] for x,y in fundamental_domain]
@interact
def _(gen = selector(['t+1', 't-1', '-1/t'], nrows=1)):
    global fundamental_domain
    if gen == 't+1':
        fundamental_domain = [[x+1,y] for x,y in fundamental_domain]
    elif gen == 't-1':
        fundamental_domain = [[x-1,y] for x,y in fundamental_domain]
    elif gen == '-1/t':
        new_dom = []
        for x,y in fundamental_domain:
            sq_mod = x^2 + y^2
            new_dom.append([(-1)*x/sq_mod, y/sq_mod])
        fundamental_domain = new_dom
    P = polygon(fundamental_domain)
    P.ymax(1.2); P.ymin(-0.1)
    P.show()
}}}

{{attachment:fund_domain.png}}

== Computing modular forms ==
by William Stein
{{{
j = 0
@interact
def _(N=[1..100], k=selector([2,4,..,12],nrows=1), prec=(3..40),
      group=[(Gamma0, 'Gamma0'), (Gamma1, 'Gamma1')]):
    M = CuspForms(group(N),k)
    print j; global j; j += 1
    print M; print '\n'*3
    print "Computing basis...\n\n"
    if M.dimension() == 0:
         print "Space has dimension 0"
    else:
        prec = max(prec, M.dimension()+1)
        for f in M.basis():
             view(f.q_expansion(prec))
    print "\n\n\nDone computing basis."
}}}

{{attachment:modformbasis.png}}


== Computing the cuspidal subgroup ==
by William Stein
{{{
html('<h1>Cuspidal Subgroups of Modular Jacobians J0(N)</h1>')
@interact
def _(N=selector([1..8*13], ncols=8, width=10, default=10)):
    A = J0(N)
    print A.cuspidal_subgroup()
}}}

{{attachment:cuspgroup.png}}

== A Charpoly and Hecke Operator Graph ==
by William Stein

{{{
# Note -- in Sage-2.10.3; multiedges are missing in plots; loops are missing in 3d plots
@interact
def f(N = prime_range(11,400),
      p = selector(prime_range(2,12),nrows=1),
      three_d = ("Three Dimensional", False)):
    S = SupersingularModule(N)
    T = S.hecke_matrix(p)
    G = Graph(T, multiedges=True, loops=not three_d)
    html("<h1>Charpoly and Hecke Graph: Level %s, T_%s</h1>"%(N,p))
    show(T.charpoly().factor())
    if three_d:
        show(G.plot3d(), aspect_ratio=[1,1,1])
    else:
        show(G.plot(),figsize=7)
}}}

{{attachment:heckegraph.png}}

== Demonstrating the Diffie-Hellman Key Exchange Protocol ==
by Timothy Clemans (refereed by William Stein)
{{{
@interact
def diffie_hellman(button=selector(["New example"],label='',buttons=True),
    bits=("Number of bits of prime", (8,12,..512))):
    maxp = 2^bits
    p = random_prime(maxp)
    k = GF(p)
    if bits>100:
        g = k(2)
    else:
        g = k.multiplicative_generator()
    a = ZZ.random_element(10, maxp)
    b = ZZ.random_element(10, maxp)

    print """
<html>
<style>
.gamodp {
background:yellow
}
.gbmodp {
background:orange
}
.dhsame {
color:green;
font-weight:bold
}
</style>
<h2>%s-Bit Diffie-Hellman Key Exchange</h2>
<ol style="color:#000;font:12px Arial, Helvetica, sans-serif">
<li>Alice and Bob agree to use the prime number p=%s and base g=%s.</li>
<li>Alice chooses the secret integer a=%s, then sends Bob (<span class="gamodp">g<sup>a</sup> mod p</span>):<br/>%s<sup>%s</sup> mod %s = <span class="gamodp">%s</span>.</li>
<li>Bob chooses the secret integer b=%s, then sends Alice (<span class="gbmodp">g<sup>b</sup> mod p</span>):<br/>%s<sup>%s</sup> mod %s = <span class="gbmodp">%s</span>.</li>
<li>Alice computes (<span class="gbmodp">g<sup>b</sup> mod p</span>)<sup>a</sup> mod p:<br/>%s<sup>%s</sup> mod %s = <span class="dhsame">%s</span>.</li>
<li>Bob computes (<span class="gamodp">g<sup>a</sup> mod p</span>)<sup>b</sup> mod p:<br/>%s<sup>%s</sup> mod %s = <span class="dhsame">%s</span>.</li>
</ol></html>
    """ % (bits, p, g, a, g, a, p, (g^a), b, g, b, p, (g^b), (g^b), a, p,
       (g^ b)^a, g^a, b, p, (g^a)^b)
}}}

{{attachment:dh.png}}

== Plotting an elliptic curve over a finite field ==
{{{
E = EllipticCurve('37a')
@interact
def _(p=slider(prime_range(1000), default=389)):
    show(E)
    print "p = %s"%p
    show(E.change_ring(GF(p)).plot(),xmin=0,ymin=0)
}}}

{{attachment:ellffplot.png}}
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{{{ {{{#!sagecell
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{{{ {{{#!sagecell
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= Modular Forms =

== Computing modular forms ==
by William Stein
{{{#!sagecell
j = 0
@interact
def _(N=[1..100], k=selector([2,4,..,12],nrows=1), prec=(3..40),
      group=[(Gamma0, 'Gamma0'), (Gamma1, 'Gamma1')]):
    M = CuspForms(group(N),k)
    print j; global j; j += 1
    print M; print '\n'*3
    print "Computing basis...\n\n"
    if M.dimension() == 0:
         print "Space has dimension 0"
    else:
        prec = max(prec, M.dimension()+1)
        for f in M.basis():
             view(f.q_expansion(prec))
    print "\n\n\nDone computing basis."
}}}

{{attachment:modformbasis.png}}


== Computing the cuspidal subgroup ==
by William Stein
{{{#!sagecell
html('<h1>Cuspidal Subgroups of Modular Jacobians J0(N)</h1>')
@interact
def _(N=selector([1..8*13], ncols=8, width=10, default=10)):
    A = J0(N)
    print A.cuspidal_subgroup()
}}}

{{attachment:cuspgroup.png}}

== A Charpoly and Hecke Operator Graph ==
by William Stein

{{{#!sagecell
# Note -- in Sage-2.10.3; multiedges are missing in plots; loops are missing in 3d plots
@interact
def f(N = prime_range(11,400),
      p = selector(prime_range(2,12),nrows=1),
      three_d = ("Three Dimensional", False)):
    S = SupersingularModule(N)
    T = S.hecke_matrix(p)
    G = Graph(T, multiedges=True, loops=not three_d)
    html("<h1>Charpoly and Hecke Graph: Level %s, T_%s</h1>"%(N,p))
    show(T.charpoly().factor())
    if three_d:
        show(G.plot3d(), aspect_ratio=[1,1,1])
    else:
        show(G.plot(),figsize=7)
}}}

{{attachment:heckegraph.png}}

= Modular Arithmetic =
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{{{ {{{#!sagecell
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{{{ {{{#!sagecell
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    if Mod(a,3)!=0 and Mod(b,3)==0:
        return True
    else:
        return False		     return Mod(a,3)!=0 and Mod(b,3)==0
Line 581: Line 496:
= Cyclotomic Fields =
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{{{ {{{#!sagecell
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{{{ {{{#!sagecell
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= Elliptic Curves =

== Adding points on an elliptic curve ==
by David Møller Hansen
{{{#!sagecell
def point_txt(P,name,rgbcolor):
    if (P.xy()[1]) < 0:
        r = text(name,[float(P.xy()[0]),float(P.xy()[1])-1],rgbcolor=rgbcolor)
    elif P.xy()[1] == 0:
        r = text(name,[float(P.xy()[0]),float(P.xy()[1])+1],rgbcolor=rgbcolor)
    else:
        r = text(name,[float(P.xy()[0]),float(P.xy()[1])+1],rgbcolor=rgbcolor)
    return r

E = EllipticCurve('37a')
list_of_points = E.integral_points()
html("Graphical addition of two points $P$ and $Q$ on the curve $ E: %s $"%latex(E))
@interact
def _(P=selector(list_of_points,label='Point P'),Q=selector(list_of_points,label='Point Q'), marked_points = checkbox(default=True,label = 'Points'), Lines = selector([0..2],nrows=1), Axes=True):
 curve = E.plot(rgbcolor = (0,0,1),xmin=25,xmax=25,plot_points=300)
 R = P + Q
 Rneg = -R
 l1 = line_from_curve_points(E,P,Q)
 l2 = line_from_curve_points(E,R,Rneg,style='--')
 p1 = plot(P,rgbcolor=(1,0,0),pointsize=40)
 p2 = plot(Q,rgbcolor=(1,0,0),pointsize=40)
 p3 = plot(R,rgbcolor=(1,0,0),pointsize=40)
 p4 = plot(Rneg,rgbcolor=(1,0,0),pointsize=40)
 textp1 = point_txt(P,"$P$",rgbcolor=(0,0,0))
 textp2 = point_txt(Q,"$Q$",rgbcolor=(0,0,0))
 textp3 = point_txt(R,"$P+Q$",rgbcolor=(0,0,0))
 if Lines==0:
  g=curve
 elif Lines ==1:
  g=curve+l1
 elif Lines == 2:
  g=curve+l1+l2
 if marked_points:
  g=g+p1+p2+p3+p4
 if P != Q:
  g=g+textp1+textp2+textp3
 else:
  g=g+textp1+textp3
 g.axes_range(xmin=-5,xmax=5,ymin=-13,ymax=13)
 show(g,axes = Axes)

def line_from_curve_points(E,P,Q,style='-',rgb=(1,0,0),length=25):
 """
 P,Q two points on an elliptic curve.
 Output is a graphic representation of the straight line intersecting with P,Q.
 """
 # The function tangent to P=Q on E
 if P == Q:
  if P[2]==0:
   return line([(1,-length),(1,length)],linestyle=style,rgbcolor=rgb)
  else:
   # Compute slope of the curve E in P
   l=-(3*P[0]^2 + 2*E.a2()*P[0] + E.a4() - E.a1()*P[1])/((-2)*P[1] - E.a1()*P[0] - E.a3())
   f(x) = l * (x - P[0]) + P[1]
   return plot(f(x),-length,length,linestyle=style,rgbcolor=rgb)
 # Trivial case of P != R where P=O or R=O then we get the vertical line from the other point
 elif P[2] == 0:
  return line([(Q[0],-length),(Q[0],length)],linestyle=style,rgbcolor=rgb)
 elif Q[2] == 0:
  return line([(P[0],-length),(P[0],length)],linestyle=style,rgbcolor=rgb)
 # Non trivial case where P != R
 else:
  # Case where x_1 = x_2 return vertical line evaluated in Q
  if P[0] == Q[0]:
   return line([(P[0],-length),(P[0],length)],linestyle=style,rgbcolor=rgb)
  
  #Case where x_1 != x_2 return line trough P,R evaluated in Q"
  l=(Q[1]-P[1])/(Q[0]-P[0])
  f(x) = l * (x - P[0]) + P[1]
  return plot(f(x),-length,length,linestyle=style,rgbcolor=rgb)
}}}
{{attachment:PointAddEllipticCurve.png}}


== Plotting an elliptic curve over a finite field ==
{{{#!sagecell
E = EllipticCurve('37a')
@interact
def _(p=slider(prime_range(1000), default=389)):
    show(E)
    print "p = %s"%p
    show(E.change_ring(GF(p)).plot(),xmin=0,ymin=0)
}}}

{{attachment:ellffplot.png}}

= Cryptography =

== The Diffie-Hellman Key Exchange Protocol ==
by Timothy Clemans and William Stein
{{{#!sagecell
@interact
def diffie_hellman(bits=slider(8, 513, 4, 8, 'Number of bits', False),
    button=selector(["Show new example"],label='',buttons=True)):
    maxp = 2 ^ bits
    p = random_prime(maxp)
    k = GF(p)
    if bits > 100:
        g = k(2)
    else:
        g = k.multiplicative_generator()
    a = ZZ.random_element(10, maxp)
    b = ZZ.random_element(10, maxp)

    print """
<html>
<style>
.gamodp, .gbmodp {
color:#000;
padding:5px
}
.gamodp {
background:#846FD8
}
.gbmodp {
background:#FFFC73
}
.dhsame {
color:#000;
font-weight:bold
}
</style>
<h2 style="color:#000;font-family:Arial, Helvetica, sans-serif">%s-Bit Diffie-Hellman Key Exchange</h2>
<ol style="color:#000;font-family:Arial, Helvetica, sans-serif">
<li>Alice and Bob agree to use the prime number p = %s and base g = %s.</li>
<li>Alice chooses the secret integer a = %s, then sends Bob (<span class="gamodp">g<sup>a</sup> mod p</span>):<br/>%s<sup>%s</sup> mod %s = <span class="gamodp">%s</span>.</li>
<li>Bob chooses the secret integer b=%s, then sends Alice (<span class="gbmodp">g<sup>b</sup> mod p</span>):<br/>%s<sup>%s</sup> mod %s = <span class="gbmodp">%s</span>.</li>
<li>Alice computes (<span class="gbmodp">g<sup>b</sup> mod p</span>)<sup>a</sup> mod p:<br/>%s<sup>%s</sup> mod %s = <span class="dhsame">%s</span>.</li>
<li>Bob computes (<span class="gamodp">g<sup>a</sup> mod p</span>)<sup>b</sup> mod p:<br/>%s<sup>%s</sup> mod %s = <span class="dhsame">%s</span>.</li>
</ol></html>
    """ % (bits, p, g, a, g, a, p, (g^a), b, g, b, p, (g^b), (g^b), a, p,
       (g^ b)^a, g^a, b, p, (g^a)^b)
}}}


{{attachment:dh.png}}

= Other =

== Continued Fraction Plotter ==
by William Stein
{{{#!sagecell
@interact
def _(number=e, ymax=selector([None,5,20,..,400],nrows=2), clr=Color('purple'), prec=[500,1000,..,5000]):
    c = list(continued_fraction(RealField(prec)(number))); print c
    show(line([(i,z) for i, z in enumerate(c)],rgbcolor=clr),ymax=ymax,figsize=[10,2])
}}}
{{attachment:contfracplot.png}}

== Computing Generalized Bernoulli Numbers ==
by William Stein (Sage-2.10.3)
{{{#!sagecell
@interact
def _(m=selector([1..15],nrows=2), n=(7,(3..10))):
    G = DirichletGroup(m)
    s = "<h3>First n=%s Bernoulli numbers attached to characters with modulus m=%s</h3>"%(n,m)
    s += '<table border=1>'
    s += '<tr bgcolor="#edcc9c"><td align=center>$\\chi$</td><td>Conductor</td>' + \
           ''.join('<td>$B_{%s,\chi}$</td>'%k for k in [1..n]) + '</tr>'
    for eps in G.list():
        v = ''.join(['<td align=center bgcolor="#efe5cd">$%s$</td>'%latex(eps.bernoulli(k)) for k in [1..n]])
        s += '<tr><td bgcolor="#edcc9c">%s</td><td bgcolor="#efe5cd" align=center>%s</td>%s</tr>\n'%(
             eps, eps.conductor(), v)
    s += '</table>'
    html(s)
}}}

{{attachment:bernoulli.png}}


== Fundamental Domains of SL_2(ZZ) ==
by Robert Miller
{{{#!sagecell
L = [[-0.5, 2.0^(x/100.0) - 1 + sqrt(3.0)/2] for x in xrange(1000, -1, -1)]
R = [[0.5, 2.0^(x/100.0) - 1 + sqrt(3.0)/2] for x in xrange(1000)]
xes = [x/1000.0 for x in xrange(-500,501,1)]
M = [[x,abs(sqrt(x^2-1))] for x in xes]
fundamental_domain = L+M+R
fundamental_domain = [[x-1,y] for x,y in fundamental_domain]
@interact
def _(gen = selector(['t+1', 't-1', '-1/t'], nrows=1)):
    global fundamental_domain
    if gen == 't+1':
        fundamental_domain = [[x+1,y] for x,y in fundamental_domain]
    elif gen == 't-1':
        fundamental_domain = [[x-1,y] for x,y in fundamental_domain]
    elif gen == '-1/t':
        new_dom = []
        for x,y in fundamental_domain:
            sq_mod = x^2 + y^2
            new_dom.append([(-1)*x/sq_mod, y/sq_mod])
        fundamental_domain = new_dom
    P = polygon(fundamental_domain)
    P.ymax(1.2); P.ymin(-0.1)
    P.show()
}}}

{{attachment:fund_domain.png}}

Contents

  1. Integer Factorization
    1. Divisibility Poset
    2. Factor Trees
    3. Factoring an Integer
  2. Prime Numbers
    1. Illustrating the prime number theorem
    2. Prime Spiral - Square
    3. Prime Spiral - Polar
  3. Modular Forms
    1. Computing modular forms
    2. Computing the cuspidal subgroup
    3. A Charpoly and Hecke Operator Graph
  4. Modular Arithmetic
    1. Quadratic Residue Table
    2. Cubic Residue Table
  5. Cyclotomic Fields
    1. Gauss and Jacobi Sums in Complex Plane
    2. Exhaustive Jacobi Plotter
  6. Elliptic Curves
    1. Adding points on an elliptic curve
    2. Plotting an elliptic curve over a finite field
  7. Cryptography
    1. The Diffie-Hellman Key Exchange Protocol
  8. Other
    1. Continued Fraction Plotter
    2. Computing Generalized Bernoulli Numbers
    3. Fundamental Domains of SL_2(ZZ)

Integer Factorization

Divisibility Poset

by William Stein

divposet.png

Factor Trees

by William Stein

factortree.png

More complicated demonstration using Mathematica: http://demonstrations.wolfram.com/FactorTrees/

Factoring an Integer

by Timothy Clemans

Sage implementation of the Mathematica demonstration of the same name. http://demonstrations.wolfram.com/FactoringAnInteger/

Prime Numbers

Illustrating the prime number theorem

by William Stein

primes.png

Prime Spiral - Square

by David Runde

SquareSpiral.PNG

Prime Spiral - Polar

by David Runde

PolarSpiral.PNG

Modular Forms

Computing modular forms

by William Stein

modformbasis.png

Computing the cuspidal subgroup

by William Stein

cuspgroup.png

A Charpoly and Hecke Operator Graph

by William Stein

heckegraph.png

Modular Arithmetic

Quadratic Residue Table

by Emily Kirkman

quadres.png

quadresbig.png

Cubic Residue Table

by Emily Kirkman

cubres.png

Cyclotomic Fields

Gauss and Jacobi Sums in Complex Plane

by Emily Kirkman

jacobising.png

Exhaustive Jacobi Plotter

by Emily Kirkman

jacobiexh.png

Elliptic Curves

Adding points on an elliptic curve

by David Møller Hansen

PointAddEllipticCurve.png

Plotting an elliptic curve over a finite field

ellffplot.png

Cryptography

The Diffie-Hellman Key Exchange Protocol

by Timothy Clemans and William Stein

dh.png

Other

Continued Fraction Plotter

by William Stein

contfracplot.png

Computing Generalized Bernoulli Numbers

by William Stein (Sage-2.10.3)

bernoulli.png

Fundamental Domains of SL_2(ZZ)

by Robert Miller

fund_domain.png

interact/number_theory (last edited 2020-06-14 09:10:48 by chapoton)