Constructive Relationships Between Algebraic Thickness and Normality

We study the relationship between two measures of Boolean functions; \emph{algebraic thickness} and \emph{normality}. For a function $f$, the algebraic thickness is a variant of the \emph{sparsity}, the number of nonzero coefficients in the unique GF(2) polynomial representing $f$, and the normality is the largest dimension of an affine subspace on which $f$ is constant. We show that for $0 < \epsilon<2$, any function with algebraic thickness $n^{3-\epsilon}$ is constant on some affine subspace of dimension $\Omega\left(n^{\frac{\epsilon}{2}}\right)$. Furthermore, we give an algorithm for finding such a subspace. We show that this is at most a factor of $\Theta(\sqrt{n})$ from the best guaranteed, and when restricted to the technique used, is at most a factor of $\Theta(\sqrt{\log n})$ from the best guaranteed. We also show that a concrete function, majority, has algebraic thickness $\Omega\left(2^{n^{1/6}}\right)$.Comment: Final version published in FCT'201

On the normality of pp-ary bent functions

Depending on the parity of $n$ and the regularity of a bent function $f$ from $\mathbb F_p^n$ to $\mathbb F_p$, $f$ can be affine on a subspace of dimension at most $n/2$, $(n-1)/2$ or $n/2- 1$. We point out that many $p$-ary bent functions take on this bound, and it seems not easy to find examples for which one can show a different behaviour. This resembles the situation for Boolean bent functions of which many are (weakly) $n/2$-normal, i.e. affine on a $n/2$-dimensional subspace. However applying an algorithm by Canteaut et.al., some Boolean bent functions were shown to be not $n/2$- normal. We develop an algorithm for testing normality for functions from $\mathbb F_p^n$ to $\mathbb F_p$. Applying the algorithm, for some bent functions in small dimension we show that they do not take on the bound on normality. Applying direct sum of functions this yields bent functions with this property in infinitely many dimensions.Comment: 13 page

A New Distinguisher on Grain v1 for 106 rounds

In Asiacrypt 2010, Knellwolf, Meier and Naya-Plasencia proposed distinguishing attacks on Grain v1 when (i) Key Scheduling process is reduced to 97 rounds using $2^{27}$ chosen IVs and (ii) Key Scheduling process is reduced to 104 rounds using $2^{35}$ chosen IVs. Using similar idea, Banik obtained a new distinguisher for 105 rounds. In this paper, we show similar approach can work for 106 rounds. We present a new distinguisher on Grain v1 for 106 rounds with success probability 63\%

Grein. A New Non-Linear Cryptoprimitive

In this thesis, we will study a new stream cipher, Grein, and a new cryptoprimitive used in this cipher. The second chapter gives a brief introduction to cryptography in general. The third chapter looks at stream ciphers in general, and explains the advantages and disadvantages of stream ciphers compared to block ciphers. In the fourth chapter the most important building blocks used in stream ciphers are explained. The reader is excepted to know elementary abstract algebra, as much of the results in this chapter depend on it. In the fifth chapter, the stream cipher Grain is introduced. In chapter six, the new stream cipher, Grein, is introduced. Here, we look at the different components used in the cipher, and how they operate together. In chapter seven, we introduce an alteration to the Grein cryptosystem, which hopefully have some advantagesMaster i InformatikkMAMN-INFINF39

A Dynamic Cube Attack on 105105 round Grain v1

As far as the Differential Cryptanalysis of reduced round Grain v1 is concerned, the best results were those published by Knellwolf et al. in Asiacrypt $2011$. In an extended version of the paper, it was shown that it was possible to retrieve {\bf (i)} $5$ expressions in the Secret Key bits for a variant of Grain v1 that employs $97$ rounds (in place of $160$) in its Key Scheduling process using $2^{27}$ chosen IVs and {\bf (ii)} $1$ expression in Secret Key bits for a variant that employs $104$ rounds in its Key Scheduling using $2^{35}$ chosen IVs. However, the second attack on $104$ rounds, had a success probability of around $50$\%, which is to say that the attack worked for only around one half of the Secret Keys. In this paper we propose a dynamic cube attack on $105$ round Grain v1, that has a success probability of $100$\%, and thus we report an improvement of $8$ rounds over the previous best attack on Grain v1 that attacks the entire Keyspace. We take the help of the tool $\Delta${\sf Grain}$_{\sf KSA}$, proposed by Banik at ACISP 2014, to track the differential trails induced in the internal state of Grain v1 by any difference in the IV bits, and we prove that a suitably introduced difference in the IV leads to a distinguisher for the output bit produced in the $105^{th}$ round. This, in turn, helps determine the values of $6$ expressions in the Secret Key bits

Differential Fault Attack against Grain family with very few faults and minimal assumptions

The series of published works, related to Differential Fault Attack (DFA) against the Grain family, require (i) quite a large number (hundreds) of faults (around $n \ln n$, where $n = 80$ for Grain v1 and $n = 128$ for Grain-128, Grain-128a) and also (ii) several assumptions on location and timing of the fault injected. In this paper we present a significantly improved scenario from the adversarial point of view for DFA against the Grain family of stream ciphers. Our model is the most realistic one so far as it considers that the cipher to be re-keyed a very few times and fault can be injected at any random location and at any random point of time, i.e., no precise control is needed over the location and timing of fault injections. We construct equations based on the algebraic description of the cipher by introducing new variables so that the degrees of the equations do not increase. In line of algebraic cryptanalysis, we accumulate such equations based on the fault-free and faulty key-stream bits and solve them using the SAT Solver Cryptominisat-2.9.5 installed with SAGE 5.7. In a few minutes we can recover the state of Grain v1, Grain-128 and Grain-128a with as little as 10, 4 and 10 faults respectively (and may be improved further with more computational efforts)

Key recovery attacks on Grain family using BSW sampling and certain weaknesses of the filtering function

A novel internal state recovery attack on the whole Grain family of ciphers is proposed in this work. It basically uses the ideas of BSW sampling along with employing a weak placement of the tap positions of the driving LFSRs. The currently best known complexity trade-offs are obtained, and due to the structure of Grain family these attacks are also key recovery attacks. It is shown that the internal state of Grain-v1 can be recovered with the time complexity of about $2^{66}$ operations using a memory of about $2^{58.91}$ bits, assuming availability of $2^{45}$ keystream sequences each of length $2^{49}$ bits generated for different initial values. Moreover, for Grain-128 or Grain-128a, the attack requires about $2^{105}$ operations using a memory of about $2^{82.59}$ bits, assuming availability of $2^{75}$ keystream sequences each of length $2^{76}$ bits generated for different initial values. These results further show that the whole Grain family, due to the choice of tap positions mainly, does not provide enough security margins against internal state recovery attacks. A simple modification of the selection of the tap positions, as a countermeasure against the attacks described here, is given

Enhancing Data Security: A Study of Grain Cipher Encryption using Deep Learning Techniques

Data security has become a paramount concern in the age of data driven applications, necessitating the deployment of robust encryption techniques. This paper presents an in-depth investigation into the strength and randomness of the keystream generated by the Grain cipher, a widely employed stream cipher in secure communication systems. To achieve this objective, we propose the construction of sophisticated deep learning models for keystream prediction and evaluation. The implications of this research extend to the augmentation of our comprehension of the encryption robustness offered by the Grain cipher, accomplished by harnessing the power of deep learning models for cryptanalysis. The insights garnered from this study hold significant promise for guiding the development of more resilient encryption algorithms, thereby reinforcing the security of data transmission across diverse applications