Feedback With Carry Shift Registers Research Articles

Hybrid FCSR Based Stream Cipher for Secure Communications in IoT

<p>Linear Feedback Shift Register (LFSR) is the basic hardware of stream cipher, and Feedback with Carry Shift Register (FCSR) is the nonlinear analogues of LFSR. FCSR is a feedback architecture to generate long pseudorandom sequence. In this paper, we study the characteristics of FCSRs combined with nonlinear circuits such as Dawson’s Summation Generator (DSG), lp-Geffe generator and etc. Then we proposed a hybrid FCSR applying DSG and lp-Geffe generator as nonlinear combining elements to increase the period and the linear complexity of the output sequence. In addition, we further investigate the period, linear complexity, randomness, and use known attacks to verify the security strength of the proposed keystream generator. The pass rates of the proposed scheme are 100% for FIPS PUB 140-1 random tests, and at least 98% for SP800-22 random test, respectively.</p> <p> </p>

Rational complexity of binary sequences, Fmathbb {Q}SRs, and pseudo-ultrametric continued fractions in mathbb {R}

We introduce rational complexity, a new complexity measure for binary sequences. The sequence s ∈ Bω is considered as binary expansion of a real fraction s equiv {sum }_{kin mathbb {N}}s_{k}2^{-k}in [0,1] subset mathbb {R}. We compute its continued fraction expansion (CFE) by the Binary CFE Algorithm, a bitwise approximation of s by binary search in the encoding space of partial denominators, obtaining rational approximations r of s with r → s. We introduce Feedback inmathbb {Q}Shift Registers (Fmathbb {Q}SRs) as the analogue of Linear Feedback Shift Registers (LFSRs) for the linear complexity L, and Feedback with Carry Shift Registers (FCSRs) for the 2-adic complexity A. We show that there is a substantial subset of prefixes with “typical” linear and 2-adic complexities, around n/2, but low rational complexity. Thus the three complexities sort out different sequences as non-random.

FCSR connection integers: efficient generation, testing and counting

The period of a binary sequence generated by feedback with carry shift register (FCSR) depends on the connection integer (q) of the register. We define FCSR primes as primes generating maximum possible period sequences when used as connection integer of an FCSR. Hence, to design good FCSR pseudorandom generators, one needs suitable feedback primes. The authors of this article study some properties about these primes and give algorithms to generate some of them. The first algorithm is somewhat straightforward and is based on the criterion given in Arnault and Berger. We propose two new algorithms for the efficient generation of FCSR primes. Comparison between these algorithms, supported by experimental and analytical results is presented. Further, we address the question of quantification of such primes and derive some explicit bounds.

Autocorrelation and Lower Bound on the 2-Adic Complexity of LSB Sequence of p-Ary m-Sequence

LSB (Least Significant Bit) sequences are widely used as the initial inputs in some modern stream ciphers, such as the ZUC algorithm-the core of the 3GPP LTE International Encryption Standard. Therefore, analyzing the statistical properties (for example, autocorrelation, linear complexity and 2-adic complexity) of these sequences becomes an important research topic. In this article, we first reduce the autocorrelation distribution of the LSB sequence of a p-ary m-sequence with period p n -1 for any order n≥2 to the autocorrelation distribution of a corresponding Costas sequence with period p-1, and from the computing of which by computer, we obtain the explicit autocorrelation distribution of the LSB sequence for each prime p<; 100. In addition, we give a lower bound on the 2-adic complexity of each of these LSB sequences for all primes p<; 20, which proves to be large enough to resist the analysis of RAA (Rational Approximation Algorithm) for FCSRs (Feedback with Carry Shift Registers). In particular, for a Mersenne prime p=2 k -1 (i.e., k is a prime such that p is also a prime), our results hold for all its bit-component sequences since they are shift equivalent to the LSB sequence.

Solving the FCSR synthesis problem for multi-sequences by lattice basis reduction

Register synthesis for multi-sequences has significance for the security of word-oriented stream ciphers. Feedback with carry shift registers (FCSRs) are promising alternatives to linear feedback shift registers for the design of stream ciphers. In this paper, we solve the FCSR synthesis problem for multi-sequences by two rational approximation algorithms using lattice theory. One is based on the lattice reduction greedy algorithm proposed by Nguyen and Stehle (ACM Trans Algorithms (TALG) 5(4):46, 2009). The other is based on the LLL algorithm which is a polynomial time lattice reduction algorithm. Both of these rational approximation algorithms can find the smallest common FCSR for a given multi-sequence but with different numbers of known terms. When the number of sequences within the multi-sequence is less than or equal to 3, the former is suggested because it has better time complexity and fewer terms are needed. Otherwise, the latter will have better time complexity.

Fast construction of binary ring FCSRs for hardware stream ciphers

Stream ciphers based on linear feedback shift registers have been subject to algebraic attacks. To avoid these kinds of attacks, feedback with carry shift registers (FCSRs) have been proposed as an alternative. They are suitable for hardware implementations. FCSRs have been implemented using ring representation, in order to circumvent some weaknesses in the traditional representations. In this paper, we explore the simplest case of FCSRs, called binary FCSRs, which are common in applications. We give a fast algorithm to construct binary ring FCSRs for hardware stream ciphers.

Matrix parametrized shift registers

High speed pseudorandom sequence generators have played roles in an array of applications in communications, cryptography, and computing. At the heart of many of these generators are shift registers of various types. Researchers have studied linear feedback shift registers (LFSRs), feedback with carry shift registers (FCSRs), and various generalizations such as ring FCSRS and algebraic feedback shift registers. The analysis of these sequence generators typically proceeds by defining an algebraic structure on the set of infinite output sequences, based on a choice of uniformizing parameter (e.g., power series for LFSRs, N-adic numbers for FCSRs). In this paper we introduce a new generalization of FCSRs in which the uniformizing parameter \(N\in \mathbb {Z}\) is replaced by a square matrix, T, resulting in what we call T-generators. We describe an algebraic structure, the T-adic numbers, and use it to study the periodicity of T-generators.

AFSRs synthesis with the extended Euclidean rational approximation algorithm

Pseudo-random sequence generators are widely used in many areas, such as stream ciphers, radar systems, Monte-Carlo simulations and multiple access systems. Generalization of linear feedback shift registers (LFSRs) and feedback with carry shift registers (FCSRs), algebraic feedback shift registers (AFSRs) [ 7 ] can generate pseudo-random sequences over an arbitrary finite field. In this paper, we present an algorithm derived from the Extended Euclidean Algorithm that can efficiently find a smallest AFSR over a quadratic field for a given sequence. It is an analog of the Extended Euclidean Rational Approximation Algorithm [ 1 ] used in solving the FCSR synthesis problem. For a given sequence $\mathbf{a}$, $2\Lambda(\alpha)+1$ terms of sequence $\mathbf{a}$ are needed to find the smallest AFSR, where $\Lambda(\alpha)$ is a complexity measure that reflects the size of the smallest AFSR that outputs $\mathbf{a}$.

Practical construction of ring LFSRs and ring FCSRs with low diffusion delay for hardware cryptographic applications

Linear Feedback Shift Registers (LFSRs) and Feedback with Carry Shift Registers (FCSRs) are two pseudo-random generators which are widely used in many cryptographic applications. The Ring represent...

Construction of Transition Matrices for Ternary Ring Feedback With Carry Shift Registers

Since the linear structure of linear feedback shift registers (LFSRs) has a drawback that may lead to attacks against LFSR-based stream ciphers, feedback with carry shift registers (FCSRs) have been proposed as an alternative to LFSRs for the design of stream ciphers. However, some weaknesses in stream ciphers based on Fibonacci or Galois FCSRs have been exposed. Then, a new ring FCSRs has been proposed which is based on a matrix definition. This new approach generalizes Galois and Fibonacci FCSRs, and circumvents some of their severe weaknesses. In this paper, we give an affirmative answer to the following conjecture: for any given connection integer there exists a ternary transition matrix with a critical path of length 1 and fan-out of 2. We also give an algorithm to construct such transition matrices with given connection integer.

INVESTIGATIONS OF THE BASIC COMPONENT OF FCSR – GENERATOR

The analysis of the statistical parameters of feedback with carry shift register (FCSR) quasi-random generator has been done. For the investigation were chosen 8 - bits and 9-bits registers. Were investigated the period of the generator and distribution of “one” and “zero” values. Additionally, the generated sequences where tested by NIST STS package.

Clock-controlled generators with large period output sequences

Clock-controlled generators are a kind of pseudo-random number generators (PRNG). Recently, some clock-controlled generators based on jumping Linear Finite State Machines (LFSMs) have been proposed, such as Pomaranch and MICKEY. The period and the linear complexity of their output sequences need to be large enough to provide security against linear attacks. In this paper, a new condition for the period to reach its maximal value is presented. The condition is better than the previous one. Further, some clock-controlled generators are considered, including a new generator which uses a Feedback with Carry Shift Register (FCSR) as the control register. How to maximise the period of their output sequences is investigated.

A generalized birthday approach for efficiently finding linear relations in $$\ell $$ ℓ -sequences

Feedback with carry shift registers (FCSRs) have previously been available in two configurations, the Fibonacci and Galois architectures. Recently, a generalized and unifying FCSR structure and theory was presented. The new ring FCSR model repairs some weaknesses of the older architectures. Most notably, the carry cell bias property that was exploited for an attack on the eSTREAM final portfolio cipher F-FCSR-H v2 is no longer possible for the updated (and unbroken) F-FCSR-H v3 stream cipher. In this paper we show how to exploit a particular set of linear relations in ring FCSR sequences. We show what biases can be expected, and we also present a generalized birthday algorithm for actually realizing these relations. As all prerequisites of a distinguishing attack are present, we explicitly show a new such attack on F-FCSR-H v3 with an online time complexity of only $$2^{37.2}$$ 2 37.2 . The offline time complexity (for finding a linear relation) is $$2^{56.2}$$ 2 56.2 . This is the first successful attack on F-FCSR-H v3, the first attack to breach the exhaustive search complexity limit. Note that this attack is completely different from that of F-FCSR-H v2. We focus on this particular application in the paper, but the presented algorithm is actually very general. The algorithm can be applied to any FCSR automaton, so linearly filtered FCSRs and FCSR combiners may be particularly interesting targets for cryptanalysis.

Word-based FCSRs with fast software implementations

Feedback with carry shift registers (FCSRs) over 2-adic number would be suitable in hardware implementation, but they are not efficient in software implementation since their basic unit (the size of register cells) is 1-bit. In order to improve the efficiency we consider FCSRs over 2l -adic number (i.e., FCSRs with register cells of size l-bit) that produce l bits at every clocking where l will be taken as the size of normal words in modern CPUs (e.g., l = 32). But, it is difficult to deal with the carry that happens when the size of summation results exceeds that of normal words. We may use long variables (declared with 'unsigned _int64' or'unsigned long long') or conditional operators (such as 'if statement) to handle the carry, but both the arithmetic operators over long variables and the conditional operators are not efficient comparing with simple arithmetic operators (such as shifts, maskings, xors, modular additions, etc.) over variables of size l-bit. In this paper, we propose some conditions for FCSRs over 2l -adic number which admit fast software implementations using only simple operators. Moreover, we give two implementation examples for the FCSRs. Our simulation result shows that the proposed methods are twice more efficient than usual methods using conditional operators.

Security Provisions in Stream Ciphers Through Self Shrinking and Alternating Step Generator

in cryptography stream ciphers used to encrypt plain text data bits one by one. The security of stream ciphers depend upon randomness of key stream, good linear span and low probability of finding the initial states of pseudorandom generators. In this paper we propose a new stream cipher model use Feed back with carry shift registers (FCSRs) as building blocks which are considered as a source of long pseudorandom generator. Proposed model is the combined structure of alternating step generator and self shrinking generators which are commonly used stream ciphers generators. In this research we compare proposed generator against self shrinking generator (SSG), Alternating step generator (ASG) and alternating step self shrinking generator (ASSG) and we concludes that proposed generator architecture is best for encryption because it satisfies all necessary conditions of a good Stream cipher.

$2$-Adic Complexity of Binary $m$-Sequences

Although 2 -adic complexity was proposed more than ten years ago, even for <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> -sequences which are thought of as the most important linear recurring sequences, no theoretical results about their 2-adic complexity has been presented. In this paper, it is shown that for a binary <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> -sequence, its 2-adic complexity attains the maximum, which implies that no feedback with carry shift registers (FCSRs) with connection integer less than <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> - 1 can generate <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> -sequences of order <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> .

Some remarks on FCSRs and implications for stream ciphers

Feedback with carry shift registers (FCSRs) are extensively discussed in the context of pseudorandom number generation and as building blocks for stream ciphers. Similarly to linear feedback shift registers, FCSRs may be represented in Galois and in Fibonacci architecture. We describe the first formal characterization of periodic Galois states and show an efficient mapping between periodic Galois states and periodic Fibonacci states. Additionally we provide a method for explicitly computing the autocorrelation of maximum-period FCSR sequences and discuss the impact of our findings on the design of FCSR-based stream ciphers.

Some Results on FCSR Automata With Applications to the Security of FCSR-Based Pseudorandom Generators

This article describes new theoretical results concerning the general behavior of a feedback with carry shift register (FCSR) automaton. They help to better understand how the initial parameters must be chosen to use this automaton as a basic block of a filtered stream cipher. These results especially concern the structure of the transition graph of an FCSR automaton and the number of iterations of the FCSR transition function required to reach the main part of the graph. A potential linear weakness and a easy way to prevent the corresponding attack are also given.

Period and Complementarity Properties of FCSR Memory Sequences

In this correspondence, we investigate feedback with carry shift register (FCSR) memory sequences. For an -sequence generated by an FCSR with connection integer and initial memory , we prove that for , where and is the memory sequence. Generally speaking, the period of a memory sequence is a factor of that of the FCSR output binary sequence. We show there are a large number of connection integers, with which an FCSR can generate sequences that have the same period as their memory sequences, especially including all connection integers for -sequences.

Design and Properties of a New Pseudorandom Generator Based on a Filtered FCSR Automaton

Feedback with carry shift registers (FCSR) was introduced by Goresky and Klapper in 1993. It is similar to the classical linear feedback shift registers (LFSR) used in many pseudorandom generators. The main difference is that the elementary additions are not additions modulo 2 but with propagation of carries. The main problem for the use of an FCSR automaton is the fact that the generated sequences are predictable. In order to remove this weakness of FCSR-based generators, we propose filtering the state of the FCSR with a linear function. This method is efficient since the FCSR structure is not related to a linear property. This paper presents an extensive study of FCSR automata, a security analysis of our generator (concerning linear and 2-adic cryptanalysis, algebraic attack, correlation attack, etc.), and a practical example of parameters in order to design this generator. An important point concerning this generator is the fact that it is simple and efficient, both in hardware and software implementation.