Computer efficient linear feedback shift register

Details Computer efficient linear feedback shift register Computer efficient linear feedback shift register

1999-12-02

2004-07-13

Chaki, Kakali (Department: 2124)

Electrical computers: arithmetic processing and calculating

Electrical digital calculating computer

Particular function performed

C708S253000

Reexamination Certificate

active

06763363

ABSTRACT:

THE FIELD OF THE INVENTION
The present invention generally relates to psuedo-random number generators (PRNGs), and more particularly relates to systems, such as private-key stream cipher cryptosystems, which employ linear feedback shift registers to produce pseudo-random bit keystreams, such as keystreams for combining with plaintext to encrypt the plaintext into ciphertext and keystreams for combining with the ciphertext to decipher the ciphertext into plaintext.
BACKGROUND OF THE INVENTION
Pseudo-random number generators (PRNGs) are used in a variety of systems such as cryptosystems, Monte Carlo simulation systems, games, and heuristic design systems (e.g., gate array placement and routing systems). In particular, cryptosystems perform cryptography to transform plaintext into ciphertext so that only an authorized receiver can transform the ciphertext back into the original plaintext. Encryption or enciphering is the process that transforms plaintext into ciphertext. Decryption or deciphering is the process that transforms ciphertext into plaintext.
A parameter called an encryption key is employed by a cryptosystem to prevent the plaintext from being easily revealed by an unauthorized person. A sender transforms a given plaintext into a large variety of possible ciphertexts selected by the specific encryption key. A receiver of the ciphertext deciphers the ciphertext by employing a parameter referred to as a decryption key. In a public-key cryptosystem, the encryption key is made public while the decryption key is kept secret. Therefore, in public key cryptosystems, the decryption key must be computationally infeasible to deduce from the encryption key. In a private-key cryptosystem, the sender and the receiver typically share a common key that is used for both enciphering and deciphering. In such a private-key cryptosystem, the common key is alterable and must be kept secret.
Private-key cryptosystems are typically implemented as block cipher cryptosystems or stream cipher cryptosystems. Block cipher cryptosystems divide the plaintext into blocks and encipher each block independently using a stateless transform. In block cipher cryptosystems if one fixed common private-key is employed to encipher different occurrences of a particular plaintext block, all of these occurrences are encrypted into identical corresponding ciphertext blocks. Therefore, the block size is preferably selected to be large enough to frustrate attacks from a cryptanalyst, which analyzes the occurrence frequencies of various patterns among the ciphertext blocks. Example block sizes are 64 bits and 128 bits.
In stream cipher cryptosystems, the plaintext is typically encrypted on a word-by-word basis using a stateful transform that evolves as the encryption progresses. In encrypting the plaintext binary data sequence for transmission as a ciphertext binary data sequence, the common private-key is a parameter that controls a pseudo-random bit generator to create a long sequence of binary data referred to as a keystream. The stream cipher cryptosystem includes a cryptographic combiner, which combines the keystream with the plaintext sequence. The cryptographic combiner is typically implemented with exclusive-or (XOR) bit-wise logic functions, which perform bit-wise modulo-2 addition. The cryptographic combiner produces the ciphertext. At the receiver, the common private-key controls a receiver pseudo-random bit generator to produce a decryption keystream. The decryption keystream is combined with a decryption combiner to decrypt the ciphertext to provide the plaintext to the receiver. The receiver decryption combiner operation must be the inverse of the sender encryption combiner operation. For this reason, the most common combiner operation is bit-wise XOR, which is its own inverse.
One problem with stream cipher cryptosystems is the difficulty of generating a long, statistically uniform, and unpredictable sequence of binary data in the keystream from a short and random key. Such sequences are desirable in the keystream in cryptography to make it impossible, given a reasonable segment of its data and sufficient computer resources, to find out more about the sequences.
There are four general requirements for cryptographically secure keystream PRNGs. First, the period of a keystream must be large enough to accommodate the length of the transmitted message. Second, the keystream output bits must have good statistical properties (e.g. values are uniformally distributed). Third, the keystream output bits must be easy to generate. Fourth, the keystream output bits must be hard to predict. For example, given the PRNG and the first N output bits, a(
0
), a(
1
), . . . , a(N−1), it should be computationally infeasible to predict the (N+1)
th
bit a(N) in a sequence with better than a 50—50 chance. In otherwords, a cryptanalyst should not be able to generate other forward bits or backward bits if presented with a given portion of the keystream output sequence.
The PRNG employed in stream cipher cryptosystems, often employs a feedback shift register (FSR) which includes N storage elements and a feedback function that expresses each new element a(t) of the sequence in terms of the previous generated elements a(t−N), a(t−N+1), . . . , a(t−1). Each individual storage element of the FSR is called a stage, and the binary signals a(
0
), a(
1
), a(
2
), . . . , a(N−1) are loaded into the stages as initial data to generate the keystream sequence. The period of the keystream sequence produced by the FSR depends both on the number of stages and on the details of the feedback function. The maximal period of a keystream sequence generated by an N-stage FSR with a non-singular feedback function is 2
N
, which represents the number of possible states of the N-stage FSR.
Depending on whether the feedback function is linear or is non-linear, the FSR is referred to respectively as a linear feedback shift register (LFSR) or a non-linear feedback shift register (NLFSR).
In particular, the LFSR is employed in many pseudo-random bit generators for stream cipher cryptosystems. LFSRs are preferred over most other PRNGs because mathematics are available to design LFSRs with guaranteed long sequence length and good statistics. The LFSR feedback function is of the form a(t)=c
1
a(t−1) XOR c
2
a(t−2) XOR . . . XOR c
N−1
a(t−N+1) XOR n
N
a(t−N), where c
i
is an element of the set {0,1}. Each stage that is associated with a non-zero c
i
is referred to as a tap. The feedback function of an LFSR can be represented formally by what is referred to as a feedback polynomial:
f
(
x
)=1
+c
1
x+c
2
x
N−2
+. . . +c
N−1
x
N−1
+c
N
x
N
where the intermediate x has no other meaning than as a mathematical symbol. This feedback polynomial decides the period and the statistical behavior of the keystream output sequence. To avoid trivial output, the zero-state should be excluded from the initial setting. This limits the largest possible period of an LFSR to 2
N
−1
In general, to generate the largest possible period 2
N
−1 for the output sequence, the feedback polynomial f(x) of the LFSR should be primitive. A sequence generated by an LFSR with a primitive feedback polynomial is referred to as a maximal-length LFSR sequence or simply an m-sequence. However, m-sequences cannot be used as keystreams without undergoing further cryptographic transformation. Without this further cryptographic transformation, the key of secrecy (i.e, the initial state of the LFSR and the feedback function of the LFSR) of an N-stage LFSR can be determined from just 2N successive bits of the output sequence.
Efficient synthesis procedures exist for finding feedback polynomials of the shortest LFSR that would generate a given output sequence. The length of such an LFSR is referred to as the linear complexity of the sequence. As a result, an LFSR suitable for employment in a cryptosystem, must guarantee a large enough key-independe

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