#!/usr/bin/env python
# coding: utf-8

# ## Diffie-Helman with an additive group
# 
# ### Public parameters

# In[1]:


get_ipython().run_line_magic('display', 'latex')


# In[2]:


N = 2^100
G = Zmod(N)
G


# In[3]:


g = G(12345)
g, g.order()


# ### Romeo

# In[4]:


a = randint(1, N)
A = a*g
A


# ### Juliet

# In[5]:


b = randint(1, N)
B = b*g
B


# ### Romeo

# In[6]:


S = a*B
S


# ### Juilet

# In[7]:


S = b*A
S


# ### Lady Capulet

# In[8]:


a == A / g and b == B / g


# ## Diffie-Hellman with $\mathbb{F}_q^*$
# 
# ### Public parameters

# In[9]:


q = next_prime(2^100)
(q-1).factor()


# In[10]:


log(6070659658921032842417,2).n()


# In[11]:


G = Zmod(q)
G


# In[12]:


g = G(2)
g.multiplicative_order() == q-1


# ### Romeo

# In[13]:


a = randint(1, N)
A = g^a
A


# ### Juliet

# In[14]:


b = randint(1, N)
B = g^b
B


# ### Romeo

# In[15]:


S = B^a
S


# ### Juilet

# In[16]:


S = A^b
S


# ### Lady Capulet

# In[17]:


print(1000 * chr(63))


# In[18]:


get_ipython().run_line_magic('time', 'A.log(g)')


# ## Elliptic curves

# In[19]:


E = EllipticCurve([2,3])
E


# In[20]:


E.plot()


# In[21]:


k = GF(101)
E = EllipticCurve(k, [2,3])
E


# In[22]:


E.plot()


# In[23]:


R.<x1,y1,x2,y2> = FractionField(QQ["x1", "y1", "x2", "y2"])
E = EllipticCurve(R, [2,3])
P = E.point((x1, y1, 1), check=False)
Q = E.point((x2, y2, 1), check=False)
P, Q


# In[24]:


P+Q


# In[25]:


2*P


# In[ ]: