Stream Ciphers

class sage.crypto.stream_cipher.LFSRCipher(parent, poly, IS)

Bases: SymmetricKeyCipher

Create a linear feedback shift register (LFSR) cipher.

INPUT:

• parent – parent

• poly – connection polynomial

• IS – initial state

EXAMPLES:

Sage

sage: FF = FiniteField(2)
sage: P.<x> = PolynomialRing(FF)
sage: E = LFSRCryptosystem(FF)
sage: E
LFSR cryptosystem over Finite Field of size 2
sage: IS = [ FF(a) for a in [0,1,1,1,0,1,1] ]
sage: g = x^7 + x + 1
sage: e = E((g,IS))
sage: B = BinaryStrings()
sage: m = B.encoding("THECATINTHEHAT")
sage: e(m)
0010001101111010111010101010001100000000110100010101011100001011110010010000011111100100100011001101101000001111
sage: FF = FiniteField(2)
sage: P.<x> = PolynomialRing(FF)
sage: LFSR = LFSRCryptosystem(FF)
sage: e = LFSR((x^2+x+1,[FF(0),FF(1)]))
sage: B = e.domain()
sage: m = B.encoding("The cat in the hat.")
sage: e(m)
00111001110111101011111001001101110101011011101000011001100101101011001000000011100101101010111100000101110100111111101100000101110101111010111101000011
sage: m == e(e(m))
True

Python

>>> from sage.all import *
>>> FF = FiniteField(Integer(2))
>>> P = PolynomialRing(FF, names=('x',)); (x,) = P._first_ngens(1)
>>> E = LFSRCryptosystem(FF)
>>> E
LFSR cryptosystem over Finite Field of size 2
>>> IS = [ FF(a) for a in [Integer(0),Integer(1),Integer(1),Integer(1),Integer(0),Integer(1),Integer(1)] ]
>>> g = x**Integer(7) + x + Integer(1)
>>> e = E((g,IS))
>>> B = BinaryStrings()
>>> m = B.encoding("THECATINTHEHAT")
>>> e(m)
0010001101111010111010101010001100000000110100010101011100001011110010010000011111100100100011001101101000001111
>>> FF = FiniteField(Integer(2))
>>> P = PolynomialRing(FF, names=('x',)); (x,) = P._first_ngens(1)
>>> LFSR = LFSRCryptosystem(FF)
>>> e = LFSR((x**Integer(2)+x+Integer(1),[FF(Integer(0)),FF(Integer(1))]))
>>> B = e.domain()
>>> m = B.encoding("The cat in the hat.")
>>> e(m)
00111001110111101011111001001101110101011011101000011001100101101011001000000011100101101010111100000101110100111111101100000101110101111010111101000011
>>> m == e(e(m))
True

Sage Live

FF = FiniteField(2)
P.<x> = PolynomialRing(FF)
E = LFSRCryptosystem(FF)
E
IS = [ FF(a) for a in [0,1,1,1,0,1,1] ]
g = x^7 + x + 1
e = E((g,IS))
B = BinaryStrings()
m = B.encoding("THECATINTHEHAT")
e(m)
FF = FiniteField(2)
P.<x> = PolynomialRing(FF)
LFSR = LFSRCryptosystem(FF)
e = LFSR((x^2+x+1,[FF(0),FF(1)]))
B = e.domain()
m = B.encoding("The cat in the hat.")
e(m)
m == e(e(m))

connection_polynomial()

The connection polynomial defining the LFSR of the cipher.

EXAMPLES:

Sage

sage: k = GF(2)
sage: P.<x> = PolynomialRing( k )
sage: LFSR = LFSRCryptosystem( k )
sage: e = LFSR((x^2+x+1,[k(0), k(1)]))
sage: e.connection_polynomial()
x^2 + x + 1

Python

>>> from sage.all import *
>>> k = GF(Integer(2))
>>> P = PolynomialRing( k , names=('x',)); (x,) = P._first_ngens(1)
>>> LFSR = LFSRCryptosystem( k )
>>> e = LFSR((x**Integer(2)+x+Integer(1),[k(Integer(0)), k(Integer(1))]))
>>> e.connection_polynomial()
x^2 + x + 1

Sage Live

k = GF(2)
P.<x> = PolynomialRing( k )
LFSR = LFSRCryptosystem( k )
e = LFSR((x^2+x+1,[k(0), k(1)]))
e.connection_polynomial()

initial_state()

The initial state of the LFSR cipher.

EXAMPLES:

Sage

sage: k = GF(2)
sage: P.<x> = PolynomialRing( k )
sage: LFSR = LFSRCryptosystem( k )
sage: e = LFSR((x^2+x+1,[k(0), k(1)]))
sage: e.initial_state()
[0, 1]

Python

>>> from sage.all import *
>>> k = GF(Integer(2))
>>> P = PolynomialRing( k , names=('x',)); (x,) = P._first_ngens(1)
>>> LFSR = LFSRCryptosystem( k )
>>> e = LFSR((x**Integer(2)+x+Integer(1),[k(Integer(0)), k(Integer(1))]))
>>> e.initial_state()
[0, 1]

Sage Live

k = GF(2)
P.<x> = PolynomialRing( k )
LFSR = LFSRCryptosystem( k )
e = LFSR((x^2+x+1,[k(0), k(1)]))
e.initial_state()

class sage.crypto.stream_cipher.ShrinkingGeneratorCipher(parent, e1, e2)

Bases: SymmetricKeyCipher

Create a shrinking generator cipher.

INPUT:

• parent – parent

• poly – connection polynomial

• IS – initial state

EXAMPLES:

Sage

sage: FF = FiniteField(2)
sage: P.<x> = PolynomialRing(FF)
sage: LFSR = LFSRCryptosystem(FF)
sage: IS_1 = [ FF(a) for a in [0,1,0,1,0,0,0] ]
sage: e1 = LFSR((x^7 + x + 1,IS_1))
sage: IS_2 = [ FF(a) for a in [0,0,1,0,0,0,1,0,1] ]
sage: e2 = LFSR((x^9 + x^3 + 1,IS_2))
sage: E = ShrinkingGeneratorCryptosystem()
sage: e = E((e1,e2))
sage: e
Shrinking generator cipher on Free binary string monoid

Python

>>> from sage.all import *
>>> FF = FiniteField(Integer(2))
>>> P = PolynomialRing(FF, names=('x',)); (x,) = P._first_ngens(1)
>>> LFSR = LFSRCryptosystem(FF)
>>> IS_1 = [ FF(a) for a in [Integer(0),Integer(1),Integer(0),Integer(1),Integer(0),Integer(0),Integer(0)] ]
>>> e1 = LFSR((x**Integer(7) + x + Integer(1),IS_1))
>>> IS_2 = [ FF(a) for a in [Integer(0),Integer(0),Integer(1),Integer(0),Integer(0),Integer(0),Integer(1),Integer(0),Integer(1)] ]
>>> e2 = LFSR((x**Integer(9) + x**Integer(3) + Integer(1),IS_2))
>>> E = ShrinkingGeneratorCryptosystem()
>>> e = E((e1,e2))
>>> e
Shrinking generator cipher on Free binary string monoid

Sage Live

FF = FiniteField(2)
P.<x> = PolynomialRing(FF)
LFSR = LFSRCryptosystem(FF)
IS_1 = [ FF(a) for a in [0,1,0,1,0,0,0] ]
e1 = LFSR((x^7 + x + 1,IS_1))
IS_2 = [ FF(a) for a in [0,0,1,0,0,0,1,0,1] ]
e2 = LFSR((x^9 + x^3 + 1,IS_2))
E = ShrinkingGeneratorCryptosystem()
e = E((e1,e2))
e

decimating_cipher()

The LFSR cipher generating the decimating key stream.

EXAMPLES:

Sage

sage: FF = FiniteField(2)
sage: P.<x> = PolynomialRing(FF)
sage: LFSR = LFSRCryptosystem(FF)
sage: IS_1 = [ FF(a) for a in [0,1,0,1,0,0,0] ]
sage: e1 = LFSR((x^7 + x + 1,IS_1))
sage: IS_2 = [ FF(a) for a in [0,0,1,0,0,0,1,0,1] ]
sage: e2 = LFSR((x^9 + x^3 + 1,IS_2))
sage: E = ShrinkingGeneratorCryptosystem()
sage: e = E((e1,e2))
sage: e.decimating_cipher()
LFSR cipher on Free binary string monoid

Python

>>> from sage.all import *
>>> FF = FiniteField(Integer(2))
>>> P = PolynomialRing(FF, names=('x',)); (x,) = P._first_ngens(1)
>>> LFSR = LFSRCryptosystem(FF)
>>> IS_1 = [ FF(a) for a in [Integer(0),Integer(1),Integer(0),Integer(1),Integer(0),Integer(0),Integer(0)] ]
>>> e1 = LFSR((x**Integer(7) + x + Integer(1),IS_1))
>>> IS_2 = [ FF(a) for a in [Integer(0),Integer(0),Integer(1),Integer(0),Integer(0),Integer(0),Integer(1),Integer(0),Integer(1)] ]
>>> e2 = LFSR((x**Integer(9) + x**Integer(3) + Integer(1),IS_2))
>>> E = ShrinkingGeneratorCryptosystem()
>>> e = E((e1,e2))
>>> e.decimating_cipher()
LFSR cipher on Free binary string monoid

Sage Live

FF = FiniteField(2)
P.<x> = PolynomialRing(FF)
LFSR = LFSRCryptosystem(FF)
IS_1 = [ FF(a) for a in [0,1,0,1,0,0,0] ]
e1 = LFSR((x^7 + x + 1,IS_1))
IS_2 = [ FF(a) for a in [0,0,1,0,0,0,1,0,1] ]
e2 = LFSR((x^9 + x^3 + 1,IS_2))
E = ShrinkingGeneratorCryptosystem()
e = E((e1,e2))
e.decimating_cipher()

keystream_cipher()

The LFSR cipher generating the output key stream.

EXAMPLES:

Sage

sage: FF = FiniteField(2)
sage: P.<x> = PolynomialRing(FF)
sage: LFSR = LFSRCryptosystem(FF)
sage: IS_1 = [ FF(a) for a in [0,1,0,1,0,0,0] ]
sage: e1 = LFSR((x^7 + x + 1,IS_1))
sage: IS_2 = [ FF(a) for a in [0,0,1,0,0,0,1,0,1] ]
sage: e2 = LFSR((x^9 + x^3 + 1,IS_2))
sage: E = ShrinkingGeneratorCryptosystem()
sage: e = E((e1,e2))
sage: e.keystream_cipher()
LFSR cipher on Free binary string monoid

Python

>>> from sage.all import *
>>> FF = FiniteField(Integer(2))
>>> P = PolynomialRing(FF, names=('x',)); (x,) = P._first_ngens(1)
>>> LFSR = LFSRCryptosystem(FF)
>>> IS_1 = [ FF(a) for a in [Integer(0),Integer(1),Integer(0),Integer(1),Integer(0),Integer(0),Integer(0)] ]
>>> e1 = LFSR((x**Integer(7) + x + Integer(1),IS_1))
>>> IS_2 = [ FF(a) for a in [Integer(0),Integer(0),Integer(1),Integer(0),Integer(0),Integer(0),Integer(1),Integer(0),Integer(1)] ]
>>> e2 = LFSR((x**Integer(9) + x**Integer(3) + Integer(1),IS_2))
>>> E = ShrinkingGeneratorCryptosystem()
>>> e = E((e1,e2))
>>> e.keystream_cipher()
LFSR cipher on Free binary string monoid

Sage Live

FF = FiniteField(2)
P.<x> = PolynomialRing(FF)
LFSR = LFSRCryptosystem(FF)
IS_1 = [ FF(a) for a in [0,1,0,1,0,0,0] ]
e1 = LFSR((x^7 + x + 1,IS_1))
IS_2 = [ FF(a) for a in [0,0,1,0,0,0,1,0,1] ]
e2 = LFSR((x^9 + x^3 + 1,IS_2))
E = ShrinkingGeneratorCryptosystem()
e = E((e1,e2))
e.keystream_cipher()

Make live