Random Sequence Ⅳ
Maximum Length Sequence
Before, we studied LFSR, and now let's take a look at the m-sequence of LFSR.
The m-sequence (Maximum Length Sequence) of LFSR is a pseudo-random sequence with the longest possible period. This sequence is generated in the LFSR by appropriately selecting the initial state and feedback polynomial, allowing the LFSR to achieve its maximum possible period. Such m-sequences are highly useful in various applications, especially in the fields of communication, cryptography, and test pattern generation.
Some basic concepts and characteristics of the m-sequence of LFSR are as follows:
Maximum Periodicity: The m-sequence of LFSR has the longest possible period, equal to 2^n - 1, where n is the number of bits in the LFSR. This is because the m-sequence will go through all 2^n - 1 different states before reaching its initial state.
Initial State: Selecting an appropriate initial state is crucial for generating an m-sequence with the maximum period. Generally, the initial state cannot be all zeros.
Feedback Polynomial: Choosing an appropriate feedback polynomial is also key. The feedback polynomial determines which bits participate in the feedback operation. A common feedback polynomial is a primitive polynomial, such as x^n + 1, where n is the number of bits in the LFSR.
Period Detection: To verify whether the generated sequence is an m-sequence, period detection methods can be used. A simple method is to observe whether the sequence returns to the initial state after reaching 2^n - 1 bits.
Applications: M-sequences have important applications in many areas, such as in pseudo-random number generation, pseudo-random test pattern generation, and sequence spreading in spread spectrum communication systems (e.g., CDMA).
This code implements a Linear Feedback Shift Register (LFSR) class for generating m-sequences.
Initialization Method: The constructor takes two parameters -
initial_state
represents the initial state of the LFSR, andfeedback_mask
is the mask for the feedback polynomial.feedback_mask
is a list where 1s and 0s indicate whether the corresponding bit participates in the feedback.Shift Method: This method simulates the shift operation of the LFSR. First, it calculates the feedback bit
feedback_bit
using list comprehension and bitwise operations. Then, it shifts the state to the right by one position and updates the leftmost bit using bitwise operations, simulating the LFSR shift process.generate_m_sequence
Method: This method generates the m-sequence. In each iteration, it appends the rightmost bit of the current state tom_sequence
and then calls theshift
method for state shifting.An instance of a 4-bit LFSR is created, with an initial state of
0b1011
and a feedback polynomial corresponding to the mask[1, 0, 0, 1]
, representingx^4 + x^3 + 1
. Then, a list containing a 15-bit m-sequence is generated, and the result is printed.
This example demonstrates how an LFSR generates m-sequences based on the feedback polynomial and initial state. In practical applications, different LFSR bit lengths and feedback polynomials can be chosen to generate m-sequences with varying periodicity and properties as needed.
Using m-sequences directly generated by an LFSR may not be secure, primarily due to the inherent properties of LFSRs and the fixed initial state, leading to some security weaknesses that make the generated sequences vulnerable to attacks. Specifically, these weaknesses include:
Predictability: If an attacker can deduce or guess the LFSR's initial state and feedback polynomial, they can predict the generated m-sequence. For shorter LFSRs, brute force might be an effective attack method since the state space of an LFSR is finite.
Periodicity: The period of an LFSR is finite and depends on its bit length and the choice of feedback polynomial. After a certain number of steps, the generated sequence will repeat. If an attacker knows the LFSR's period, they can predict subsequent outputs by observing part of the sequence.
Fixed Initial State: If the initial state is fixed and known to an attacker, they can repeatedly generate the same sequence. This can be insecure in cryptographic applications where passwords should exhibit unpredictability.
Linearity: Sequences generated by LFSRs exhibit linearity, meaning each bit in the sequence can be expressed as a linear combination of previous bits. This linearity can make the generated sequences more susceptible to statistical attacks.
Due to these security vulnerabilities, using m-sequences directly from an LFSR as cryptographic keys or in security applications is not recommended. In practical cryptography applications, more complex Pseudo-Random Number Generators (PRNGs) or stream cipher algorithms are widely used to enhance security and prevent potential attacks. These algorithms are typically designed with greater complexity, employing non-linear operations and larger state spaces to enhance cryptographic security.
Filter
We study three typical methods, with the first one related to filter generators.
In pseudo-random sequence generators based on Linear Feedback Shift Registers (LFSRs), filter generators are an improved structure designed to enhance the statistical properties and security of the generated pseudo-random sequences. Filter generators combine the concepts of Linear Feedback Shift Registers (LFSRs) and linear complexity filters by processing the output of the LFSR through non-linear filtering operations.
Basic principles of filter generators include:
LFSR: Like traditional LFSRs, filter generators use a register to store states and generate pseudo-random sequences through shifting and linear feedback operations.
Linear Complexity Filter: Filter generators introduce a linear complexity filter, which is a non-linear operation typically composed of one or more non-linear components, such as Boolean functions. This filter enhances the complexity of the sequence by applying non-linear transformations to the output of the LFSR.
Feedback Path: The feedback path in filter generators includes not only feedback from the LFSR but also feedback from the linear complexity filter. This integrated feedback path increases the non-linearity of the generated sequence, improving its statistical properties.
Selection of Appropriate Boolean Functions: The security and performance of filter generators largely depend on the choice of Boolean functions for the non-linear filter. These Boolean functions should possess cryptographic properties, such as uniformity and resistance to analysis.
Initial State: The selection of the initial state is equally crucial for the generated sequence. Avoiding the use of an all-zero or all-one state is a good practice.
This code implements a filter generator based on both an LFSR and a non-linear filter function.
LFSR class: Represents a Linear Feedback Shift Register. The constructor takes the initial state (
initial_state
) and the feedback mask (feedback_mask
). Theshift
method simulates the LFSR shift operation, calculates the feedback bit, and updates the state accordingly.FilterFunctionGenerator class: Represents the filter generator. The constructor takes the length of the LFSR (
lfsr_length
), the length of the filter function (filter_length
), and the feedback polynomial (feedback_mask
). It initializes the LFSR and the filter function.The
generate_key_stream
method generates a key stream, producing one bit at a time. At each step, it generates a bit (lfsr_output
) using the LFSR and calculates the output of the non-linear filter function (filter_output
). The final key bit is obtained by XORing these two outputs.An example is created using a 4-bit LFSR and a 3-element non-linear filter function. The
feedback_mask
defines the feedback polynomial as x^3 + x^2 + 1. The filter generator is then used to generate 10 key stream bits, and the result is printed.
In summary, this filter generator combines an LFSR with a non-linear filter function, enhancing the complexity and statistical properties of the generated key stream by introducing non-linear operations. Such a design contributes to improving the security of the generated key. In practical applications, the choice of feedback polynomial and non-linear filter function should be adjusted based on specific security requirements.
Combining generator
A combiner generator is an extended form of a pseudo-random sequence generator based on LFSR, and it generates more complex key streams by combining multiple LFSRs and non-linear filter functions. Such a design helps enhance the non-linearity, complexity, and security of the generated sequences.
Basic principles of a combiner generator:
LFSR Combination: The combiner generator includes multiple LFSRs, each with its own initial state and feedback polynomial. These LFSRs can run in parallel, generating their respective portions of the key stream.
Non-linear Filter Functions: Each LFSR is followed by a non-linear filter function, and these functions can be different. These functions perform non-linear transformations on the sequences generated by their respective LFSRs, introducing additional complexity.
Combination Operation: The generator combines the individual key stream portions in some way, such as through XOR operations or other combination operations. This combination operation can further increase the overall complexity of the generated sequence.
Initial States: The initial states of each LFSR should be different to ensure diversity among the various portions of the sequence.
Let's analyze the provided Python code:
LFSR
Class:
LFSR
Class:Initialization (
__init__
method):Takes
initial_state
andfeedback_mask
as parameters.Initializes the LFSR instance with the provided initial state and feedback mask.
Shift (
shift
method):Simulates the LFSR shift operation.
Computes the feedback bit based on the feedback mask.
Updates the LFSR state by shifting and incorporating the feedback bit.
Returns the computed feedback bit.
CombinatorialGenerator
Class:
CombinatorialGenerator
Class:Initialization (
__init__
method):Takes
lfsr_configs
andcombiner_function
as parameters.Initializes multiple LFSR instances based on the provided configurations.
Stores the combiner function that will be used to combine the outputs of the LFSRs.
Generate Key Stream (
generate_key_stream
method):Takes
num_bits
as a parameter.Iterates for the specified number of bits.
Shifts each LFSR to generate outputs.
Uses the specified combiner function to combine the LFSR outputs.
Appends the combined output to the key stream.
Returns the generated key stream.
Example Usage:
Initializes an instance of
CombinatorialGenerator
with two 4-bit LFSRs and an XOR combiner function.Defines configurations for each LFSR, specifying initial state and feedback mask.
Defines an XOR combiner function (
xor_combiner
) that XORs the outputs of the LFSRs.Creates an instance of
CombinatorialGenerator
with the specified configurations and combiner function.Generates and prints a key stream of 10 bits using the combinatorial generator.
Notes:
The
LFSR
class represents a basic Linear Feedback Shift Register.The
CombinatorialGenerator
class demonstrates the concept of combining multiple LFSRs with a non-linear combiner function.The example uses a simple XOR combiner, but more complex combiner functions could be implemented.
Overall, this code illustrates the construction of a combinatorial generator using LFSRs, providing a foundation for generating complex key streams. The specific configurations and combiner functions can be customized based on the desired cryptographic requirements.
Clock-Controlled Generator
Clock-Controlled Generator (CCG) is a type of pseudo-random sequence generator based on LFSR (Linear Feedback Shift Register), and its operation relies on a clock signal. The clock signal controls the shifting operations of the LFSR during the sequence generation process. This generator is commonly used in synchronous systems where the periodic variation of the clock signal is utilized to influence the generation of the pseudo-random sequence.
The basic principles of the Clock-Controlled Generator are as follows:
LFSR Module: The CCG consists of one or more LFSR modules, each of which is an independent LFSR. These LFSRs may have different initial states and feedback polynomials.
Clock Signal: The clock signal is a periodically changing signal that controls the shifting operations of the LFSR. During each clock cycle, the LFSR undergoes a shifting operation, generating a bit.
Clock Control: Changes in the clock signal directly impact the state update of the LFSR. When the clock signal changes, the LFSR performs shifting operations based on its feedback polynomial.
Output Sequence: The output sequence of the generator is a combination of outputs from all LFSR modules. Typically, a combination function (such as XOR operation) is used to combine the outputs of individual LFSRs to form the final pseudo-random sequence.
Now, let's take a look at the code.
Legendre sequence
In addition to leveraging the superior structure of Linear Feedback Shift Registers (LFSR) for designing pseudo-random sequence generators, there are also other generators that rely on knowledge of number theory or finite fields. These generators utilize mathematical tools, making theoretical analysis, including periodicity, random statistical properties, and linear complexity, more accessible. We will explore two common types of pseudo-random sequence generators, namely Legendre sequences and elliptic curve sequences.
Let's begin by studying Legendre sequences.
Legendre sequence is a pseudo-random sequence based on number theory, commonly employed in constructing pseudo-random number generators. The construction of such sequences is based on the properties of the Legendre symbol, which involves some important concepts in number theory, such as quadratic residues and the law of quadratic reciprocity.
Here are the basic steps for constructing a Legendre sequence:
Choose a modulus ( p ): The construction of Legendre sequences depends on a prime number ( p ). Choosing a sufficiently large prime number helps ensure the randomness and uniformity of the sequence.
Compute the Legendre symbol: The Legendre symbol is a mathematical notation in number theory, typically represented as ( \left(\frac{a}{p}\right) ), where ( a ) is an integer and ( p ) is a prime number. The computation of the Legendre symbol involves determining whether ( a ) is a quadratic residue modulo ( p ). The Legendre symbol is often used in conjunction with the law of quadratic reciprocity.
onstructing a Legendre sequence, we focus on quadratic residues and non-residues modulo p. The computation of Legendre symbols involves the properties of quadratic residues modulo p.
Constructing the sequence: By continuously selecting specific numbers and computing their Legendre symbols, a sequence can be obtained. Each term in the Legendre sequence is derived from a specific set of numbers, usually chosen based on their Legendre symbol properties.
Applying appropriate rules: Based on the values of Legendre symbols, rules can be defined. For example, if the symbol is 1, the corresponding position in the sequence takes the value 1; if the symbol is -1, it takes 0. This results in the final pseudo-random sequence.
The construction of Legendre sequences exhibits good mathematical properties, especially when the choice of modulo p is appropriate. These sequences find applications in cryptography and random number generation, among other fields, but they are not as widely used as LFSRs. In practical applications, analyzing the periodicity and statistical properties of Legendre sequences is crucial.
Now, let's look at the code.
The function
legendre_sequence
takes two parameters:n
(length of the sequence) andp
(the modulo).It initializes a list called
sequence
withn
elements, all set to 0, and sets the first element to 1.It then enters a loop from 1 to
n-1
to generate the Legendre sequence. Inside the loop:It calculates
x
as the square ofi
modulop
.It checks if
x
is equal to 1 orp - 1
. If true, it sets the corresponding element in the sequence to 0; otherwise, it sets it to 1.
The function returns the generated Legendre sequence.
An example is provided using
n = 10
andp = 7
to generate a Legendre sequence. The result is printed.
The Legendre sequence is constructed based on the properties of quadratic residues and non-residues modulo p
. The output is a binary sequence indicating whether each element is a quadratic residue (0) or a non-residue (1) modulo p
. The example demonstrates the generation of the Legendre sequence with a specific length and modulo.
Elliptic Curve Sequence
Elliptic Curve Sequence is a pseudo-random sequence based on elliptic curves, and its construction utilizes the discrete logarithm problem of elliptic curves. Elliptic Curve Sequences are commonly used in cryptography to generate keystreams for encryption algorithms or pseudo-random number generators.
Basic steps for constructing an Elliptic Curve Sequence:
Choose an elliptic curve: Select an elliptic curve, usually represented as (y^2 = x^3 + ax + b) over a finite field (\mathbb{F}_p), where (a) and (b) are constants in the finite field, ensuring that the curve satisfies certain security properties.
Select a base point: Choose a base point (G) on the curve, and its order (the number of times the point must be added to itself to cover the entire curve) should be a large prime number.
Choose a private key: Select a private key (d), typically a random number.
Generate the sequence: By continuously calculating (d \cdot G), where (d) is the private key and (G) is the base point, a sequence can be obtained. Each term in the Elliptic Curve Sequence corresponds to a point on the curve.
Apply an appropriate hash function: To map points on the elliptic curve to a pseudo-random bit sequence, a hash function, such as SHA-256, is commonly used to map the coordinates on the elliptic curve to a binary sequence.
The security of Elliptic Curve Sequences relies on the difficulty of the elliptic curve discrete logarithm problem. This problem involves finding (d) such that (d \cdot G = Q), where (Q) is a known point on the elliptic curve. This problem is hard to solve with classical computational power, making the generation of Elliptic Curve Sequences reasonably secure.
Elliptic Curve Sequences find widespread applications in modern cryptography, especially in Elliptic Curve Cryptography (ECC). These sequences offer high security and efficiency, making them suitable for constructing secure encryption algorithms and key exchange protocols.
The
elliptic_curve_sequence
function generates an Elliptic Curve Sequence based on the specified elliptic curve and the number of points to generate.An ECC key pair is generated using the
ECC.generate
function with the specified elliptic curve.The public key is extracted from the generated key pair.
A list called
sequence
is initialized to store the x-coordinates of points in the Elliptic Curve Sequence.The base point (
pointQ
) from the public key is assigned to the variablepoint
.A loop iterates
num_points
times, appending the x-coordinate of the current point to the sequence and updating the point by adding the base point.The resulting Elliptic Curve Sequence is printed for demonstration purposes using the P-256 curve and generating 5 points.
Stream cipher
We have previously learned various methods for generating random sequences, and now let's delve into stream ciphers.
There exists a close relationship between random sequences and stream ciphers, especially in their applications within cryptography.
A stream cipher is a type of symmetric encryption algorithm that uses a randomly generated bit stream, known as a key stream, of equal length to the plaintext. Encryption or decryption is achieved by performing a bitwise XOR operation between the key stream and the plaintext.
In stream ciphers, the quality of the key stream is crucial for the security of encryption. Ideally, the key stream should exhibit characteristics of a completely random and unpredictable sequence. This introduces the concept of a random sequence, as it represents a sequence with randomness and unpredictability.
Stream cipher algorithms typically employ Pseudo-Random Number Generators (PRNGs) to generate the key stream. A PRNG is an algorithm capable of producing an approximately random sequence. It takes an initial value (seed) as input and generates a long sequence of pseudo-random bits, which serves as the key stream for the stream cipher.
The security of a stream cipher relies on the randomness of the key stream. If the key stream is predictable or follows a pattern, the encryption becomes vulnerable to attacks. Therefore, one of the design goals of stream ciphers is to use key streams with high randomness, often achieved by employing high-quality PRNGs.
The relationship between random sequences and stream ciphers finds widespread application in modern cryptography. Many stream cipher algorithms utilize specially designed PRNGs to generate high-quality key streams, ensuring the security of encryption.
In summary, the key stream in stream ciphers should exhibit characteristics of high randomness and unpredictability to ensure the security of cryptographic systems.
Below is a schematic representation of how stream ciphers operate:
Analysis:
Import Libraries:
The code imports the
get_random_bytes
function from theCrypto.Random
module.
Key Stream Generation:
The
generate_key_stream
function creates a key stream by repeating the provided key until it reaches the desired size.
Encryption and Decryption Functions:
The
encrypt
anddecrypt
functions perform encryption and decryption using bitwise XOR (^
) on corresponding elements of the plaintext/ciphertext and key stream.
Key and Message Initialization:
A random key of 16 bytes (
key
) is generated usingget_random_bytes
.The sample message (
message
) is defined.
Key Stream Generation and Usage:
The key stream (
key_stream
) is generated with the same length as the message using thegenerate_key_stream
function.
Encryption and Decryption:
The message is encrypted using the key stream, resulting in
encrypted_message
.The encrypted message is then decrypted using the same key stream, yielding
decrypted_message
.
Print Results:
The original message, encrypted message, and decrypted message are printed for verification.
Overall, the code demonstrates a basic stream cipher implementation using bitwise XOR for encryption and decryption. It showcases the generation of a key stream, encryption of a message, and subsequent decryption using the same key stream.
RC4
A typical representation of a stream cipher algorithm is RC4 (Rivest Cipher 4). Also known as Ron's Code 4 or Rivest Cipher, RC4 is a symmetric key stream cipher algorithm designed by American cryptographer Ron Rivest and published in 1987. RC4 is renowned for its simplicity and efficiency, and it is widely used in practical applications, including as a part of the SSL/TLS protocols.
The main steps of the RC4 algorithm are as follows:
Initialization: RC4 utilizes a state array called the S-box (Substitution box), which contains all possible values from 0 to 255. Additionally, there is a transformation array K used for generating the key stream. At the beginning of the algorithm, the S-box is initialized as an ordered sequence from 0 to 255, and then it is shuffled based on the key K.
Key Scheduling: The initialized S-box undergoes initial confusion with the key, involving a series of swap operations. The purpose of this stage is to complicate the initial state of the S-box using the key, thereby enhancing the strength of the cipher.
Key Stream Generation: RC4 utilizes the S-box and the key to generate a pseudo-random key stream. This key stream is employed for both encrypting plaintext and decrypting ciphertext. The process of generating the key stream involves continuous swapping of elements in the S-box, followed by addition or XOR operations, resulting in a key stream.
Encryption and Decryption: Using the generated key stream, the plaintext undergoes bitwise XOR operations or addition with the key stream to achieve encryption and decryption. This XOR operation constitutes the core operation of RC4.
Key Stream Update: After encrypting or decrypting each byte, elements in the key stream may be updated. This allows the continued use of the key stream for processing the next byte.
The workflow of RC4 is illustrated as follows:
Analysis:
Key-scheduling Algorithm:
The function begins by initializing the S-box (
S
) with values from 0 to 255 and settingj
to 0.The key-scheduling algorithm shuffles the S-box based on the provided key (
key
).
Pseudo-random Generation Algorithm:
The pseudo-random generation algorithm generates a key stream and encrypts the plaintext using the RC4 algorithm.
The S-box values are continuously swapped, and a keystream byte is calculated for each character in the plaintext.
Example Usage:
An example key (
b"Key"
) and plaintext (b"Hello, RC4!"
) are provided.The
rc4
function is used to encrypt the plaintext (cipher
).
Decryption:
The
rc4
function is again used to decrypt the ciphertext (decrypted_text
).
Print Results:
The original text, encrypted text, and decrypted text are printed for verification.
The RC4 algorithm is a stream cipher that uses a key-scheduling algorithm and a pseudo-random generation algorithm. The XOR operation with the key stream is the core of the encryption and decryption processes. The example usage demonstrates the functionality of the RC4 implementation.
Reference
1. https://www.cse.ust.hk/faculty/cding/JOURNALS/it982.pdf
2. https://www.researchgate.net/figure/Bit-distribution-of-the-Legendre-sequence-S-23_tbl2_332741870
3. https://cnj.atu.edu.iq/wp-content/uploads/2019/10/10.pdf
Last updated