Nonlinear Feedback Shift Registers Research Articles

Design and Implementation of 32 Bit Linear Feedback Shift Register using FPGA

Abstract: Shift register is a common device for generating signals and sequences. Usually register is the combination of flipflops.The shift register is of two types which are: linear and nonlinear. Various sequences including pseudorandom codes are generated by linear and nonlinear feedback shift registers, respectively. Linear feedback shift register (LFSR) is composed of dynamic or static master-slave flip-flop. Its characteristics are usually characterized by a characteristic polynomial. The twoinput XOR gate is used to calculate the characteristic polynomial of the maximum or near maximum length of the feedback function without rectifying the register.

The equivalence between Galois and Fibonacci NFSRs

Nonlinear feedback shift registers (NFSRs) are used in many stream ciphers as their main building blocks. In particular, Galois NFSRs with terminal bits are used in typical stream ciphers Grain and Trivium. Seven types of Galois NFSRs have been found equivalent to Fibonacci ones, among which three types are particular cases of another type of lower triangular Galois NFSRs. This paper continues the research of equivalence between Galois NFSRs and Fibonacci ones. It first enumerates the Galois NFSRs with terminal bits that are equivalent to a Fibonacci NFSR. It then discloses n-stage (n−2)-terminal-bit Galois NFSRs equivalent to Fibonacci ones, must be lower triangular Galois NFSRs. Finally, it presents two new types of Galois NFSRs that are equivalent to Fibonacci NFSRs. Some examples show that, compared to a Fibonacci NFSR, its equivalent Galois NFSR from our new types may decrease the area and increase the throughput of the circuits implementing feedback functions.

Bisimulations of nonlinear feedback shift registers

This work investigates how to reduce nonlinear feedback shift registers (NFSRs) in combination with bisimulation method via semi-tensor product (STP). By considering a large-scale NFSR as a smaller-scale Boolean system utilising bisimulation relations, the computing complexity of the system can be decreased. On the one hand, the properties of the bisimulation relation between two different NFSRs are examined, while on the other hand, a necessary and sufficient condition is discovered, as well as a reduction strategy utilising bisimulation. Besides, it is talked about how NFSR properties spread throughout the reduction process. The nonsingularity, driven stability and other aspects of the original NFSR are confirmed to be preserved by the reduced Boolean system.

Efficient image encryption scheme using a nonlinear shift register and chaos

Powerful cryptographic systems require a qualified random number generator. This research purposes to provide a comprehensive comparative analysis done on several of the well-known pseudo-random number generators (PRNGs) regarding their efficiency and resilience against crypto-analytical threats. These generators consist of the basic 8-bit Non-Linear Feedback Shift Register (NLFSR), the logistic map (LM), and our proposed hybrid random number generator named NLFSR-LM, which combines through XOR operation the sequences of the NLFSR with the LM to achieve a high quality of randomness. The performance of the created generator is examined and subsequently compared according to statistical tests of randomness alongside cryptographic features in terms of key space, key sensitivity and resistance to numerous attacks. The proposed generator produced good results and exhibited several interesting properties, such as a high degree of security, a sufficiently large key space, and it provided better randomness than other frequently used PRNGs.

Lightweight encryption with data and device integrity using NLFSR and PUF for the Internet of Medical Things

Nowadays the Internet of Medical Things (IoMT) is in top demand; they communicate with each other in an IoT network, but there are various security issues including privacy preservation, data and device integrity, and authentication. The IoMT devices have challenges like limited computational power and resources, so conventional techniques are not suitable for them. This paper presents a lightweight stream cipher to ensure privacy protection, data integrity, and device integrity using a message authentication code (MAC). The stream cipher is designed using a nonlinear feedback shift register (NLFSR). The cipher is associated with a MAC generated by concatenating a physical unclonable function (PUF) response and the hash of the plaintext to protect the device and data integrity. The use of NLFSR and PUF makes the proposed scheme robust and lightweight for resource-constrained devices like MIoT. To evaluate the quality of the cipher the NIST 800-22 statistical test suite (NIST STS) is used, and results are compared with existing stream ciphers like Rivest cipher 4 (RC4), RC4 based on chaotic mapping (RCCM), and product algebra RCCM (PARCCM). All the tests and comparisons were performed using a Heart Disease Dataset. The scheme ensures the confidentiality and integrity of sensitive medical data which are critical for protecting patient privacy and regulatory compliance. The proposed scheme can also resist cloning attacks on the IoMT devices.

A novel method for driven stability of NFSRs via STP

The nonlinear feedback shift register (NFSR) is the main component of the convolutional decoder. This paper provides a novel method for investigating the driven stability of NFSRs with input via semi-tensor product (STP). By using the STP method, a backward state transition matrix, as well as a backward NFSR, is constructed. Backward NFSRs can help to find the predecessors of any states so that an algorithm for the global stability of autonomous NFSRs is designed. On this basis, two algorithms are proposed for the driven stability of NFSRs with input, and they have lower computational complexity than the existing methods. Meanwhile, some numerical examples are presented to support the results of this paper. Finally, the relationship between stability and connectedness is discussed fully, which derives some interesting results.

Multi de Bruijn Sequences and the Cross-Join Method

We show a method to construct binary multi de Bruijn sequences using the cross-join method. We extend the proof given by Alhakim for ordinary de Bruijn sequences to the case of multi de Bruijn sequences. In particular, we establish that all multi de Bruijn sequences can be obtained by cross-joining an ordinary de Bruijn sequence concatenated with itself an appropriate number of times. We implemented the generation of all multi de Bruijn sequences of type C(2,2,2) and C(3,2,2). We experimentally confirm that some multi de Bruijn sequences can be generated by Galois Nonlinear Feedback Shift Registers (NLFSRs). It is supposed that all multi de Bruijn sequences can be generated using Galois NLFSRs.

A new idea in response to fast correlation attacks on small-state stream ciphers

In the conference “Fast Software Encryption 2015”, a new line of research was proposed by introducing the first small-state stream cipher (SSC). The goal was to design lightweight stream ciphers for hardware applications by going beyond the rule that the internal state size must be at least twice the intended security level. Fast correlation attack (FCA) was successfully applied to all proposed SSCs which can be implemented by less than 1000 gate equivalents in hardware. It is possible to increase the security of stream ciphers against FCA by exploiting more complicated functions for the nonlinear feedback shift register and the output function, but we use lightweight functions to design the lightest SSC in the world while providing more security against FCA. Our proposed cipher provides 80-bit security against all types of Time-memory-data trade-off (TMDTO) attacks, while Lizard and Plantlet provide only 60-bit and 58-bit security against TMDTO distinguishing attacks, respectively. Our main contribution is to propose a lightweight round key function with a very long period that increases the security of SSCs against FCA.

Weak-keys and key-recovery attack for text{ TinyJAMBU }

In this paper, we study NIST lightweight 3rd round candidate text{ TinyJAMBU }. The core component of text{ TinyJAMBU } is the keyed permutation mathcal {P}_n, which is based on a non-linear feedback shift register. By analysing this permutation carefully, we are able to find good cubes that are used to build distinguishers in the weak-key setting. In particular, we show that there are at least 2^{108} keys for which TinyJAMBU can be distinguished from a random source for up to 476 rounds. These distinguishers outperform the best-known distinguishers, which were proposed in ‘Scientific Reports - Nature’ by Teng et al. We are the first to study the exact degree of the feedback polynomial mathcal {P}_n in the nonce variables. This helped us in concluding that text{ TinyJAMBU } with more than 445 rounds is secure against distinguishers using 32 sized cubes in the normal setting. Finally, we give new key-recovery attacks against text{ TinyJAMBU } using the concepts of monomial trail presented by Hu et al. at ASIACRYPT 2020. Our attacks are unlikely to jeopardise the security of the entire 640 rounds text{ TinyJAMBU }, but we strongly anticipate that they will shed new lights on the cipher’s security.

Research Status of Nonlinear Feedback Shift Register Based on Semi-Tensor Product

Nonlinear feedback shift registers (NFSRs) are the main components of stream ciphers and convolutional decoders. Recent years have seen an increase in the requirement for information security, which has sparked NFSR research. However, the NFSR study is very imperfect as a result of the lack of appropriate mathematical tools. Many scholars have discovered in recent years that the introduction of semi-tensor products (STP) of matrices can overcome this issue because STP can convert the NFSR into a quasi-linear form. As a result of STP, new NFSR research has emerged from a different angle. In view of this, in order to generalize the latest achievements of NFSRs based on STP and provide some directions for future development, the research results are summarized and sorted out, broadly including the modeling of NFSRs, the analysis of the structure of NFSRs, and the study of the properties of NFSRs.

Improved conditional differential attacks on lightweight hash family QUARK

Nonlinear feedback shift register (NFSR) is one of the most important cryptographic primitives in lightweight cryptography. At ASIACRYPT 2010, Knellwolf et al. proposed conditional differential attack to perform a cryptanalysis on NFSR-based cryptosystems. The main idea of conditional differential attack is to restrain the propagation of the difference and obtain a detectable bias of the difference of the output bit. QUARK is a lightweight hash function family which is designed by Aumasson et al. at CHES 2010. Then the extended version of QUARK was published in Journal of Cryptology 2013. In this paper, we propose an improved conditional differential attack on QUARK. One improvement is that we propose a method to select the input difference. We could obtain a set of good input differences by this method. Another improvement is that we propose an automatic condition imposing algorithm to deal with the complicated conditions efficiently and easily. It is shown that with the improved conditional differential attack on QUARK, we can detect the bias of output difference at a higher round of QUARK. Compared to the current literature, we find a distinguisher of U-QUARK/D-QUARK/S-QUARK/C-QUARK up to 157/171/292/460 rounds with increasing 2/5/33/8 rounds respectively. We have performed the attacks on each instance of QUARK on a 3.30 GHz Intel Core i5 CPU, and all these attacks take practical complexities which have been fully verified by our experiments. As far as we know, all of these results have been the best thus far.

Finding Non-liner Register on Binary M-Sequence Generating Binary Multiplication Sequence

In the current time there is an important problem that is for a received linear or nonlinear binary sequence {zn} how we can find the nonlinear feedback shift register and its linear equivalent which generate this sequence. The linear orthogonal sequences, special M-Sequences, play a big role in these methods for solving this problem. In the current research trying give illuminations about the methods which are very useful for solving this problem under short sequences, and study these methods for finding the nonlinear feedback shift register of a multiplication sequence and its linear equivalent feedback shift register of a received multiplication binary sequence{zn} where the multiplication on h degrees of a binary linear sequence {an}, or finding the equivalent linear feedback shift register of {zn}, where the sequence {zn}of the form M-sequence, and these methods are very effectively. We can extend these methods for the large sequences using programming and modern computers with large memory.

A new result on irreducible NFSRs with respect to cascade connection

Nonlinear feedback shift registers (NFSRs) are widely used in stream cipher design as building blocks. The cascade connection of NFSRs, known as an important architecture, has been adopted in Grain family of stream ciphers. In this paper, a new sufficient condition under which an NFSR cannot be decomposed into the cascade connection of two smaller NFSRs is presented, which is easy to be verified from the algebraic normal form (ANF) of the characteristic function. In fact, our results are also applicable to nonsingular Boolean functions, which actually improve a previous research of Rhodes [6] where the characteristic functions of NFSRs cannot be contained.

On the Relationships between Different Methods for Degree Evaluation

In this paper, we compare several non-tight degree evaluation methods i.e., Boura and Canteaut’s formula, Carlet’s formula as well as Liu’s numeric mapping and division property proposed by Todo, and hope to find the best one from these methodsfor practical applications. Specifically, for the substitution-permutation-network (SPN) ciphers, we first deeply explore the relationships between division property of an Sbox and its algebraic properties (e.g., the algebraic degree of its inverse). Based on these findings, we can prove theoretically that division property is never worse than Boura and Canteaut’s and Carlet’s formulas, and we also experimentally verified that the division property can indeed give a better bound than the latter two methods. In addition, for the nonlinear feedback shift registers (NFSR) based ciphers, according to the propagation of division property and the core idea of numeric mapping, we give a strict proof that the estimated degree using division property is never greater than that of numeric mapping. Moreover, our experimental results on Trivium and Kreyvium indicate the division property actually derives a much better bound than the numeric mapping. To the best of our knowledge, this is the first time to give a formal discussion on the relationships between division property and other degree evaluation methods, and we present the first theoretical proof and give the experimental verification to illustrate that division property is the optimal one among these methods in terms of the accuracy of the upper bounds on algebraic degree.

A metamorphic cryptography approach towards securing medical data using chaotic sequences and Ramanujan conjecture

Medical Images has evolved from time to time and the need to protect sensitive information related to such images have also grown significantly. Nowadays data being the primary asset of an individual, protecting them has posed new challenges from time and again. With the advancement of the digital medium, the medical image has been at greater risk and there is an urgent need to protect the information and the raw pixel data of such images. This research focuses on ensuring the confidentiality of DICOM medical images and protecting the Electronic Patient Report (EPR) by applying both cryptographic and steganographic protection techniques. The encryption keys are generated using the Logistic Chaotic Map and Non-Linear Feedback Shift Register. the electronic patient report (EPR) is converted into a 9-space-format using Continued Fraction of Ramanujan conjecture and then using steganography techniques EPR is embedded into the Most Significant Bits of Medical Images. The final encrypted image contains the EPR as well as encrypted medical image. The key sequence used in order to encrypt the medical image is highly pseudo random and the Key Space Analysis depicted that the algorithm is very strong. This work has paved a better way of securely storing 16-bit DICOM medical images and EPR into the Picture Archive and Communication System repository. The proposed security model has shown promising results in comparison to other research and the various analysis did show significant improvements over past studies.

Grain-like structures with minimal and maximal period sequences

Nonlinear feedback shift registers (NFSRs) are important building blocks for stream ciphers. The cascade connection of an n-stage full-length linear feedback shift register (LFSR) into an m-stage NFSR is called a Grain-like structure. In this paper, we focus on Grain-like structures which can generate minimal and maximal possible period sequences. The existence of Grain-like structures which can generate minimal possible period sequences is proved for the cases $$m=n$$ and $$m>n$$ . The number of such Grain-like structures is estimated in both cases. Two necessary conditions are presented for Grain-like structures to generate maximal possible period sequences. Moreover, some interesting properties of such Grain-like structures are discussed.

ОБФУСКАЦИЯ ЛОГИЧЕСКИХ СХЕМ ГЕНЕРАТОРОВ ПСЕВДОСЛУЧАЙНЫХ ЧИСЕЛ НА РЕГИСТРАХ СДВИГА С ЛИНЕЙНЫМИ И НЕЛИНЕЙНЫМИ ОБРАТНЫМИ СВЯЗЯМИ

The paper describes methods of protection against reverse engineering of logic circuits of pseudo-random number generators (PRNG) on linear and non-linear feedback shift registers. These methods are based on the use of additional logic elements in the structure of the generator to hide its original functionality. Obfuscation of the generator logic circuit changes its design such that the device works correctly only if the signals at the additional key inputs of the generator take on the correct values. It is shown that even with a small bit capacity of generators, it is possible to provide a huge number of PRNG implementations with different numbers of states and different properties. The concept of a (М + 1)-sequence generator is introduced. The possibility of transforming (M – 1)- and (M – 3)-sequences generators into (M + 1)-sequences generators is demonstrated. The proposed methods can be used to ensure the security of IoT devices.

Analysis and Efficient Implementations of a Class of Composited de Bruijn Sequences

A binary de Bruijn sequence is a sequence of period 2n in which every binary n-tuple occurs exactly once in each period. A de Bruijn sequence has good randomness properties, such as long period, ideal tuple distribution, and high linear complexity, and can be generated by a nonlinear feedback shift register (NLFSR). Finding an efficient NLFSR that can generate a de Bruijn sequence with a long period is a significant challenge. “Composited construction” is a technique for constructing a de Bruijn sequence of period 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n+k</sup> by an NLFSR from a de Bruijn sequence of period 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> through a composition operation repeatedly applying k times. The goal of this article is to further investigate the composited construction of de Bruijn sequences with efficient hardware implementations, and determine randomness properties such as linear complexity. Our contributions in this article are as follows. First, we present a generalized construction of composited de Bruijn sequences that is constructed by adding a combination of conjugate pairs of different lengths in the feedback function of the composited construction, which results in generating a class of de Bruijn sequences of size 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sup> , whereas the original composited construction can generate only two sequences. Second, we investigate the linear complexity and the correlation property of the new class of de Bruijn sequences. We prove theoretically that the linear complexity of this class of de Bruijn sequences is optimal or close to optimal. Interestingly, we also prove that the linear complexities of all the sequences of this class are equal, which strengthens Etzion's conjecture (JCTA 1985, IEEE-IT 1999) about the number of de Bruijn sequences with equal linear complexity. This is the first known construction of de Bruijn sequences of an arbitrarily long period whose linear complexities are determined theoretically. Finally, we implement our construction in hardware to demonstrate its practicality. We synthesize our implementations for a 65 nm ASIC and a Xilinx Spartan FPGA and present hardware areas, and performances of de Bruijn sequences of periods in the range of 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">160</sup> to 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1056</sup> . For instance, a class of de Bruijn sequences of period 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">160</sup> (resp. 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">288</sup> ) can be implemented with an area of 3.43 (resp. 6.71) kGEs in 65 nm ASIC, and 83 (resp. 229) slices in Spartan6 FPGA.

Improved Conditional Differential Analysis on NLFSR-Based Block Cipher KATAN32 with MILP

In this paper, a new method for constructing a Mixed Integer Linear Programming (MILP) model on conditional differential cryptanalysis of the nonlinear feedback shift register- (NLFSR-) based block ciphers is proposed, and an approach to detecting the bit with a strongly biased difference is provided. The model is successfully applied to the block cipher KATAN32 in the single-key scenario, resulting in practical key-recovery attacks covering more rounds than the previous. In particular, we present two distinguishers for 79 and 81 out of 254 rounds of KATAN32. Based on the 81-round distinguisher, we recover 11 equivalent key bits of 98-round KATAN32 and 13 equivalent key bits of 99-round KATAN32. The time complexity is less than 2 31 encryptions of 98-round KATAN32 and less than 2 33 encryptions of 99-round KATAN32, respectively. Thus far, our results are the best known practical key-recovery attacks for the round-reduced variants of KATAN32 regarding the number of rounds and the time complexity. All the results are verified experimentally.

The MILP-aided conditional differential attack and its application to Trivium

Conditional differential attacks were proposed by Knellwolf et al. at ASIACRYPT 2010 which targeted at cryptographic primitives based on non-linear feedback shift registers. The main idea of conditional differential attacks lies in controlling the propagation of a difference through imposing some conditions on public/key variables. In this paper, we improve the conditional differential attack by introducing the mixed integer linear programming (MILP) method to it. Let $$J=\{f_i({\varvec{x}},{\varvec{v}})=\gamma _i| 1\le i\le N\}$$ be a set of conditions that we want to impose, where $${\varvec{x}}=(x_1,x_2,\ldots ,x_n)$$ (resp. $$ {\varvec{v}}=(v_1,v_2,\ldots ,v_n)$$ ) represents key (resp. public) variables and $$\gamma _i \in \{0,1\}$$ needs evaluating. Previous automatic conditional differential attacks evaluate $$\gamma _1,\gamma _2,\ldots ,\gamma _N$$ just in order with the preference to zero. Based on the MILP method, conditions in J could be automatically analysed together. In particular, to enhance the effect of conditional differential attacks, in our MILP models, we are concerned with minimizing the number of 1’s in $$\{\gamma _1,\gamma _2,\ldots ,\gamma _N\}$$ and maximizing the number of weak keys. We apply our method to analyse the security of Trivium. As a result, key-recovery attacks are preformed up to the 978-round Trivium and non-randomness is detected up to the 1108-round Trivium out of its 1152 rounds both in the weak-key setting. All the results are the best known so far considering the number of rounds and could be experimentally verified. Hopefully, the new method would provide insights on conditional differential attacks and the security evaluation of Trivium.