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3.4 STREAM CIPHER:
Recall the discussion in Sections 3.1, 3.2 and 3.3 concerning the One-Time-Pad
system, the development of stream cipher systems was greatly influenced by the fact
that Shanoon had proved the OTP to be unconditionally secure. Stream cipher systems
are developed to emulate the OTP system in the sense that they produced long
sequence of displacement which were applied character by character to the plaintext
message as illustrated in figure 3-1. However, the fundamental difference is that; the
sequences generated by the stream system is not random, and it is completely
determined by a key.
Thus, a stream cipher is a system in which the key is fed to an algorithm, which
uses the key to generate finite sequence. The algorithm is usually referred to as the
sequence generator or key stream generator. Figure 3-2 illustrates stream cipher
system. Stream Ciphers lies between simple XOR and One-Time Pad. Encryption and
decryption processes are shown in figure 3-3.
K M=C
C K=M K K=M
K 0 =k
and K K K =0
Key Generator
Key stream
Ciphertext
Mixer
Plaintext
Figure (3-2)Stream cipher System.
Ki
Mi
Ki
Ci
Mi
Stream ciphers should posses the following characteristics:
Figure (3-3) Stream Cipher encryption and decryption.
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1. Easy to implement.
2. High speed in generating key stream.
3. Computationally secure.
3.1. It should be protected against known-plaintext attack.
3.2. Having a large key space.
3.3. The period of the key sequence should be greater than the message length.
3.4. Key sequence has random characteristics.
In the following sections, fundamental background and techniques of stream cipher
systems is discussed. A survey of previous stream ciphers cryptanalysis also
discussed.
3.5 KEYSTREAM GENERATOR:
The basic element in stream ciphers is the key stream generator, which will
generate the key stream (sequence) to be combined with the plaintext stream and
produce the cipher text.
3.5.1 LINEAR SHIFT REGISTERES:
A feed back shift register is an implementation of the key stream generator. It is
made up of two parts; a shift register and a feedback function. The shift register is a
sequence of bits. Each time a bit is needed, all the bits in the register are shifted 1 bit
to the right. The new left-most bit is computed as a function of the other bits in the
register. The output of the shift register is one bit. The simplest kind of feedback shift
register is a linear feedback shift register (LFSR). The feedback function is simply the
XOR function. Figure 3-4 shows a 4-bit LFSR. If it is initialized with the value
(1111), it produces the following output sequence:
111101011001000 ......
b4
b3
b2
b1
Figure 3-4 Linear Feedback Shift Register (4-bit LFSR)
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Three parameters; initial state, primitive polynomial, and the length of the
register affect the output stream of the linear shift register. For each linear shift
register there exist a linear equivalence, which defined as; the length of the smallest
linear shift register which can be used to generate the sequence.
3.5.2 REGISTER STAGES:
Shift registers consist of finite length of binary memory, called stages,
for n-binary memory, called m-stages shift register, and in any given time the contents
of the register, called state. The register could be in one of 2 m states. Zero state is
ignored because; it causes endless sequence of zeros. Thus, we left with 2m -1 states.
Next states depends on the feedback function (the mixer).
Example:
Given a LFSN of figure 3-4 with an initial state (0001) the register will cycle
with the following states and produce the corresponding outputs.
Feedback
R states
Out put
1
0001
1
1
1000
0
1
1100
0
1
1110
0
1
1111
1
0
1
0
1
1
0
0
1
0
0
0111
1011
0101
1010
1101
0110
0011
1001
0100
0010
1
1
1
0
1
0
1
1
0
0
To achieve maximal length of 2m -1 stages of LFSR; the tap sequence must
cause the register to cycle through 2m -1 non zero bit sequence before repeating; this
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will happen if the polynomial formed from the elements in the tap sequence is
primitive.
3.5.3 PRIMITIVE POLYNOMIAL
When talking about the polynomials, the term prime is replaced by
irreducible. Primitive polynomial of degree
n
is defined as an irreducible
polynomial that divides x2d-1 + 1, but not xd +1 for any d that divides 2n -1. A
polynomial is irreducible if it can not be expressed as the product of two other
polynomials (except 1 and itself). In another meaning maximal length sequence
with period 2n -1 are generated only in the case when the characteristic (generating)
polynomial ¢(x) is primitive, irreducible, and the initial state of the register must be
other than zero. Table 3-1 illustrates primitive polynomial of m =2 - 6.
Degree
2
3
Primitive polynomial
X2+x+l
X3+x+l
Degree Primitive polynomial
6
X6+x+1
X6+x5+l
x3+x2+l
X6+x5+x4+x+l
4
X4+x+l
x6+x5+x4+x+l
x6+x5+x3+x2+ x+l
5
x4+x3+l
X5+x2+1 .
X5+x2+1
x6+x4+x3+x+l
X5+x4+x3+x2+l
X5+x3+x2+ x+1
X5+x4+x3+x+l
Table 3-1 Primitive polynomials
x5+x4+x3+x+l