A set of modules and a simple chat in Python to illustrate key exchange, cryptography and discrete logarithm problems applied to rational points of elliptic curves over a finite field.

All the modules are well documented, here there is just a quick example of the main functionalities.

The ec module provides support to define and manipulate rational points on elliptic curves. Two classes are defined in it, EC and ECPt, representing respectively an elliptic curve over a finite field (of equation y^2=x^3+ax^2+bx+c) and a point on it.

Just call the constructor EC(a,b,c,p), providing the three coefficient of the equation and p, a prime number that defines the finite field:

>>> from modules.ec import *
>>> curve = EC(0, 5, 2, 967)
>>> str(curve)
'y^2==x^3+5x+2 over F_967'

We can generate a point on the curve by calling the constructor ECPt(curve,x,y), or using the method pickPoint() of the curve we just defined.

>>> P = ECPt(curve, 8, 39)
>>> Q = curve.pickPoint()
>>> str(P), str(Q)
('[8, 39]', '[40, 185]')

The operation of addition between points (as well as the multiplication by a scalar value) is implemented into the two builtin operators + and *. Let's compute P+Q, P+P and the first multiples of P:

>>> str( P+Q )
'[309, 703]'
>>> str( P+P )
'[756, 105]'
>>> for i in range(2, 5):       # i = 2, 3, 4
...     str( i*P )
...
'[756, 105]'
'[157, 602]'
'[783, 349]'
>>> ( 2*P ) == ( P+P )
True
>>> str( 345*P )
'[697, 843]'

The neutral element O can be obtained calling the static method ECPt.identity(). To verify if a given point is the identity, you can check the return value of isIdentity(), a method of ECPt. Let's see an example where we perform basic checks on the identity:

>>> O = ECPt.identity()
>>> P+O == O+P == P
True
>>> O == 2*O == O+O == -O == 50*O
True
>>> str( P-P )
'O'
>>> R = 41*P
>>> S = -41*P
>>> ( R+S ).isIdentity()
True
>>> 0*P == O
True

The method that computes the cardinality of the rational points group is based on three auxiliary algorithms:

• minOrderWithConstraints(...) (instance method of ECPt);
• orderInFactorGroup(...) (static method ECPt);
• cardinality() (method of EC).

Let's try the third algorithm, as the result can be appreciated immediately:

>>> curve.cardinality()
976
>>> othercurve = EC(1, 2, 300, 25169)
>>> othercurve.cardinality()
25136
>>> P = othercurve.pickPoint()
>>> str(P), str( 25136*P )
('[20982, 1348]', 'O')

Another method is provided: enumerateAllPoints(); it returns a list of all the rational points (relying on cardinality() to stop the scanning loop).

>>> points = othercurve.enumerateAllPoints()
>>> len(points)
25136

This module implements Shanks's and Pohlig-Hellman's algorithms, optimizing the execution time using sorted lists. The methods provided here are generic: they work perfectly on any object provided with the operations + and -, as well as scalar multiplication (* between an object and a int); because of that, they can be used in any group.

This algorithm is implemented in two versions, shanks(a,b,bs,gs) and autoshanks(a,b,n). The first allows the user to specify the number of baby and giant steps, the seconds instead requires the knowledge of the cardinality n of the group used, and sets the number of steps both to sqrt(n).

>>> from modules.dlog import autoshanks, pohlighellman
>>> Q = 3343*P
>>> autoshanks(P, Q, 25136)
3343L

The method pohlighellman(a, b, n) computes the discrete logarithm using the factorization of n. 25136 is small enough for the factorization to be computed quickly:

>>> pohlighellman(P, Q, 25136)
3343L

Also the method computeOrder() of the class ECPt relies on Shanks's algorithm. computeOrder() returns the order of a point in the rational points group. The method pickGenerator() of EC takes advantage of both this method and cardinality() to find a point with order equal to #C_k:

>>> P, Q = curve.pickPoint(), othercurve.pickPoint()
>>> P.computeOrder(), Q.computeOrder()
(122L, 12568L)
>>> G = othercurve.pickGenerator()
>>> G.computeOrder()  # we expect 25136
25136L

The cryptohelp and ecdh modules provide methods, such as ecdh_init(...) and ecdh_reply(...), that allow to implement Diffie-Hellman protocol on elliptic curves. More specifically, the class ECDHSession manages key exchange. Let's see an example of these methods altogether.

The script can be launched by running python main.py; it opens a connection between two machines via TCP/IP, then proceeds with a key exchange as in Diffie-Hellman protocol.

With rational points as group, the choice of the parameters implies the choice of a particular finite field and an elliptic curve on it. The machine that begins key exchange (let's call it M) performs the following operations:

1. Chooses a random prime p.
2. Randomly chooses paramteres to define an elliptic curve.
3. Chooses a random integer A and a generator g of the rational points group.

Then it sends to the other machine, D, the parameters a, b, c, the number p, the points g and Ag, and waits for a reply.

The machine D receives the parameters and chooses a random integer B; then sends Bg to M.

M receives Bg and computes ABg. From this last point, it derives a key and uses it to initialize a symmetric cypher (Salsa20 in this case). Now M sends a first encrypted message m_1.

D receives m_1 encrypted; it computes only now ABg with the known data, derives from ABg the same key that M has, initializing its own symmetric cypher. It uses then the cypher to decrypt and check m_1. It it matches the expected value, it prepares another confirmation message m_2, encrypts it and sends it to M. At this point D has finished its role in the key exchange.

M receives m_2, decrypts it and checks its correctness; the key exchange is concluded.

The software allows now the users to type and exchange messages: the inserted text is encrypted using the key derived from ABg.

Follows a sample run of main.py.

>>> connect or listen? connect
>>> ip address (empty=>localhost)? 
... net: starting listening thread
... net: connecting to localhost:55755
... net: listening thread started
... crypto: generating a random pseudoprime
... crypto: chosen 48239
... crypto: picking random a, b, c to define an e.c.
... crypto: y^2==x^3+26531x^2+36476x+38073 over F_48239
... crypto: looking for a generator of the rational pts group
... crypto: using [20999, 24164] as generator
... crypto: choosing a random integer a
... crypto: chosen 11798761, a*g is [591, 44543]
... crypto: using bg=[44972, 26192]
... crypto: computed ab*g=[33338, 6910]
... crypto: deriving key from [33338, 6910]
... crypto: the key computed is e2fc1d9ef1630eac4e7b2fbc76a969b058b1917e29304181
...                             7103c0c980ab7d534a59cd5731c20e0ccefaeaedc80a8d5f
... ecdh: key exchange done. Parameters:
...           curve - y^2==x^3+26531x^2+36476x+38073 over F_48239
...     generator g - [20999, 24164]
...        secret a - 11798761
...             a*g - [591, 44543]
...             b*g - [44972, 26192]
...     secret ab*g - [33338, 6910]
...             key - e2fc1d9ef1630eac4e7b2fbc76a969b058b1917e29304181
...                   7103c0c980ab7d534a59cd5731c20e0ccefaeaedc80a8d5f
... ecdh: sending the hash of ec as reply
... ecdh: encrypting '.\xd9nK\x80\x18\xa3\x0c?\xc0\xfb(\xaa\x9dA\xfa'
... ecdh: sending '\xe6\x03\x07\xbf\x9e:Z\x9a\xceI\xee\x87rzxz&\xd7\xb1H'...
... ecdh: checking message against a*g, b*g hash
... ecdh: decrypting '`\xd3?\xd1+\\D{\xf0\x1c8\xb3\xf4&Gs\xea\x9d\x8a\xf5'...
... ecdh: encrypted session ready
... ecdh: received encrypted message
... ecdh: decrypting ' \x8e\x87\xfe.\xa2/x\x8b\xea\x87rw$\xc1\xaa6\xa9\x91\xb1'...
<<< FERMAT
EULERO
... ecdh: encrypting 'EULERO'
... ecdh: sending 't\xd9\xd7\x82?pL\xb3\xc6[Q\x18\xdb\x01q<\xf0\xeb8\xbe'...
... ecdh: received encrypted message
... ecdh: decrypting '\x0e\xcf\xfa0\xfd\xe1slZ)5/\xb0;\x01^\x88z8\x00'...
<<< Curve ellittiche
Moduli in Python
... ecdh: encrypting 'Moduli in Python'
... ecdh: sending '*\xe9\x8eY\x06\xb0)\xec\x96\xc7\xafA\x0eIl\xf2Sa\xce\xb6'...







>>> connect or listen? listen
>>> choose one of the network interfaces to bind (empty=>loopback lo0):
	en1	140.105.232.19
	lo0	127.0.0.1
>>> 
... ready to listen at lo0 on 127.0.0.1
... wait for an incoming connection before typing!
... net: incoming connection from ('127.0.0.1', 59456)
... crypto: using the e.c. y^2==x^3+26531x^2+36476x+38073 over F_48239
... crypto: using the generator [20999, 24164]
... crypto: a*g=[591, 44543]
... crypto: choosing a random b
... crypto: chosen 14429519, b*g is [44972, 26192]
... crypto: computed ab*g, that is [33338, 6910]
... crypto: deriving key from [33338, 6910]
... crypto: the key computed is e2fc1d9ef1630eac4e7b2fbc76a969b058b1917e29304181
...                             7103c0c980ab7d534a59cd5731c20e0ccefaeaedc80a8d5f
... ecdh: key exchange done. Parameters:
...           curve - y^2==x^3+26531x^2+36476x+38073 over F_48239
...     generator g - [20999, 24164]
...        secret b - 14429519
...             a*g - [591, 44543]
...             b*g - [44972, 26192]
...     secret ab*g - [33338, 6910]
...             key - e2fc1d9ef1630eac4e7b2fbc76a969b058b1917e29304181
...                   7103c0c980ab7d534a59cd5731c20e0ccefaeaedc80a8d5f
... ecdh: decrypting '\xe6\x03\x07\xbf\x9e:Z\x9a\xceI\xee\x87rzxz&\xd7\xb1H'...
... ecdh: replying with the hash of a*g, b*g
... ecdh: encrypting '\xf0y\xc5G\r{\xc5 \xf1\xb2s\xac\xdel\x031'
... ecdh: sending '`\xd3?\xd1+\\D{\xf0\x1c8\xb3\xf4&Gs\xea\x9d\x8a\xf5'...
... ecdh: encrypted session ready
FERMAT
... ecdh: encrypting 'FERMAT'
... ecdh: sending ' \x8e\x87\xfe.\xa2/x\x8b\xea\x87rw$\xc1\xaa6\xa9\x91\xb1'...
... ecdh: received encrypted message
... ecdh: decrypting 't\xd9\xd7\x82?pL\xb3\xc6[Q\x18\xdb\x01q<\xf0\xeb8\xbe'...
<<< EULERO
Curve ellittiche
... ecdh: encrypting 'Curve ellittiche'
... ecdh: sending '\x0e\xcf\xfa0\xfd\xe1slZ)5/\xb0;\x01^\x88z8\x00'...
... ecdh: received encrypted message
... ecdh: decrypting '*\xe9\x8eY\x06\xb0)\xec\x96\xc7\xafA\x0eIl\xf2Sa\xce\xb6'...
<<< Moduli in Python