Introduction

Today randomness is an important unit in many modern computer applications (especially games, simulations, cryptography). Computers use a form of randomness known as pseudo randomness, it means simulation of randomness. A pseudo random event looks random but is completely predictable or deterministic (result of completely predictable mathematical algorithm). Pseudo-random number generator (PRNG) can be used as key generator in stream ciphers [1] to form large key sequence with small input data of PRNG. This generator creates bit sequence similar to random sequence by statistical parameters. In practice, these sequences are not random and can be reproduced -key sequence should be as longer as possible and variable. PRNG output must be the function of encryption key and it's very important for privacy ensuring. To generate random numbers by computer the hardware should be used (for example, noise from semiconductor devices; bits of digitized sound from the microphone; intervals between interruptions of external or internal devices; intervals between keystrokes; air temperature on hardware components). New communication standard 5G has improved requirements for confidentiality (privacy) as well as novel secure encryption algorithms and PRNG should be developed (Fig. 1) [2].

Related papers analysis

Analysis of related papers in this direction [3][4][5][6] shows different approaches in PRNG creation that influences on their characteristics and properties. There are two following categories of PRNGs: Most up-to-date cryptographic applications require random numbers, for example key generation, cryptographic nonces, salts in certain signature schemes, including ECDSA, RSASSA-PSS. The security and privacy of 5G is better than in 4G (Fig. 2) [7] by using many secure algorithms.

The main part of the research study

Theoretical principles of the model construction

On the basis of analysis, as a prototype of computational model well-known PRNG Trivium [8] was chosen. Trivium is a synchronous stream cipher designed by cryptographers C. De Cannière and B. Preneel to provide a flexible trade-off between speed and gate count in hardware, and reasonably efficient software implementation. Trivium was one of the eSTREAM competition winners and its recommended for using in modern communication networks as hardware unit. To improve PRNG Trivium following modifications were proposed:

1. Parameters n , t , e , k were specified and after their fixing new PRNG structure is forming (capacity of operations is changing). All operations are performing not on the bits but on the vectors of some size (bytes).

2. To improve non-linearity parameters substitution operation ( ) S x was specified. For any new formed PRNG new unique ( ) S x can be specified.

3. To generate pseudo-random sequences PRNG internal status vector i E , key vector for sequence generation K and index of the current iteration of forming (generation) i are used. 4. For generation function gen F stage of variables initialization was modified, the dynamic carry shift and substitution operations were specified.

S x . Computational model description Let , n t Z + ∈ , then for generation pseudo-random sequence M , M V N ∈ , { } 0,1 N V N ∈ with length N n t

= ⋅ bits it is necessary to form t sequences with length n bit: ( )

1 2 1

, , ..., ,

t t M m m m m − = , i n m V ∈ , 1, i t = .

Process of generation everyi m , i n m V ∈ , 1, i t =

performs in the following manner:

( ) 1 , , , i i gen i m E F E K i − = , 1, i t = ,

where i E is internal status vector of PRNG after generation i

-th i m , i e E V ∈ , e Z + ∈ , 0 E IV = , IV is initialization vector, e IV V ∈ , K is key vector for sequence generation, k K V ∈ , k Z + ∈ , gen F is function of generation the sequence i m . Function ( ) , ,

gen F E K i is performing by 2 following stages:

1) variables initialization;

2) sequence forming.

Stage 1 of function ( )

, ,

gen F E K i performing. At the beginning the processing of internal status vector is performing E , e E V ∈ :

( )

E S E i = <<< ,

where x y <<< is operation of right dynamic carry shift of argument x on y bits, ( ) S x is some substitution operation. Next the internal status vector E of PRNG and key vector K are disintegrated on 4 components:

( ) , , , a b c d E E E E E = , e E V ∈ , e a b c d = + + + , a a E V ∈ , b b E V ∈ , c c E V ∈ , d d E V ∈ , , , , a b c d Z + ∈ , ( ) , ,,a b c d K K K K K = , k K V ∈ , k a b c d ′ ′ ′ ′ = + + + , a a K V ′ ∈ , b b K V ′ ∈ , c c K V ′ ∈ , d d K V ′ ∈ , , , , a b c d Z + ′ ′ ′ ′∈ . Vectors a E , b E , c E , d E and a K , b K , c K , d K will be used in the next stage of the function ( ) , , gen F E K i . Stage 2 of function ( ) , ,

gen F E K i . On this stage the forming of sequence m , n m V ∈ is performing. For this objective 4 additional functions ( )

, , A a a F E K i , ( ) , , B b b F E K i , ( ) , , C c c F E K i і ( ) , , D d d

F E K i are using, where A F , B F , C F and D F are functions, that have internal status vector and key vector as input data, and sequence with length n bits as output data. These functions can be constructed on the base of non-linear shift registers, block and stream symmetric ciphers, hash functions and others.

In this case, process of sequence m ,

vectors a E , b E , c E , d E : ( ) , , , a A a a A E F E K i = , n A V ∈ , a a E V ∈ , ( ) , , , b B b b B E F E K i = , n B V ∈ , b b E V ∈ , ( ) , , , c C c c C E F E K i = , n C V ∈ , c c E V ∈ , ( ) , ,,d D d d D E F E K i = , n D V ∈ , d d E V ∈ .

Stage 2. Calculation of the new value internal status vector E ,

e E V ∈ : ( ) , , , b d a c E E E E E = . Stage 3. Formation of the sequence m , n m V ∈ : AB A B = <<< , n AB V ∈ , CD C D = <<< , n CD V ∈ , BC B CD = + , n BC V ∈ , AD AB D = ⊕ , n AD V ∈ , ()m AD S BC = ⊕ , n m V ∈ ,

where ⊕ і + are operations modulo addition 2 and 2 n respectively, ( )

S( ) ( ) , , , gen m E F E K i = .

In this Section computational model of PRNG was described. Based on PRNG Trivium, this model includes internal status vector and key vector processing, dynamic carry shift and 4 nonlinear functions. It allows to construct effective PRNGs for privacy ensuring in modern communication networks (5G networks and others).

Software PRNG development

On the basis of developed computational model 3 PRNGs 5Gen-1, 5Gen-2 and 5Gen-3 was constructed:

5Gen (by the order 8 different substitution tables are using).

Substitution S i , 0,7 i = is constructed with parameters, presented in Tables 2-9.

Experimental study and discussion

Statistical parameters investigation

Statistical parameters of developed PRNGs were investigated using traditional techniques NIST STS [9] and DIEHARD [13]. Given results were compared with testing results of generator BBS as well as stream ciphers SNOW [10] and Trivium (used in 5G networks). NIST STS are used to determine the qualitative and quantitative features of the sequences randomness. Three basic criteria are used to draw conclusions about passing random sequences of statistical tests are following:

 Criterion for decision-making based on the establishment of some threshold level.  Criterion based on establishing a fixed confidence interval.  Criterion for some appropriate statistical test probability value P-value.

The statistical test is based on the verification of null hypothesis H0 -that the sequence under study is random. Alternative hypothesis H1 is also provided -the sequence under study is not random. Therefore, the generated sequence is examined by a set of tests, each of which concludes whether the hypothesis H0 is rejected or accepted. For each test, adequate randomness statistics are selected based on which the hypothesis H0 is further rejected or accepted. Theoretically, the distribution of statistics for the null hypothesis is calculated using mathematical methods. The critical value is then determined from such an exemplary distribution. When performing the test, the value of the test statistic is calculated, which is compared to the critical value. In case of exceeding the test critical value over the reference, hypothesis H1 is accepted, otherwise -hypothesis H0.

For the experiments, the following input parameters were selected for the use of NIST STS:

1. The length of the test sequence n=10 6 bit.

2. The number of sequences being tested m=100. The confidence interval rule was applied, the lower bound being 0.96015. For every algorithms 10 files with sequences (100 Mbit) and investigated by NIST STS technique.

The results of testing are presented on Fig. 3-5. Results presented on Fig. 3-5 as well as detail experimental data (Table 11) show that developed algorithms 5Gen-1, 5Gen-2 and 5Gen-3 have passed complex control by NIST STS technique and show better results (in some cases) than existed and well-known algorithms [11][12]. Results presented on Fig. 6 show that developed algorithms have passed complex security control by DIEHARD technique as well as verified good quality of PRNG and statistical security of potential systems based on these algorithms [14][15][16].

Speed parameters investigation

For investigation developed PRNGs 5Gen-1, 5Gen-2, 5Gen-3 were realized as software tools using programming language С++. Given results were compared with results of well-known SNOW generator. Experimental study was carried out using work station Intel Core i3-3220 3.3GHz and files with different size. Average results are presented on the Table 12. As we can see from Table 12 speed of the developed algorithms are higher in comparison with SNOW till 21% outside two results of 5Gen-3 algorithm.

Conclusions

In this paper analysis of different approaches in PRNG creation and implementation was carried out. Two categories of PRNGs were defined (cryptographically secure and cryptographically insecure) as well cryptographic applications that require random numbers were defined (key generation, cryptographic nonces, salts in certain signature schemes). But there are many relevant tasks related with data privacy risk mitigation and confidentiality ensuring in 4G/5G that can be solved by advanced PRNG development and implementation. Computational model of PRNG was developed. Based on PRNG Trivium, this model includes internal status vector and key vector processing, dynamic carry shift and 4 non-linear functions. It allows to construct effective PRNGs for data privacy risk mitigation in modern communication networks (5G networks and others).

On the basis of developed computational model 3 PRNGs 5Gen-1, 5Gen-2 and 5Gen-3 was constructed and realized as software tools. These PRNGs have passed complex statistical testing by NIST STS technique as well as speed parameters investigation (developed algorithms are higher in comparison with SNOW till 21%).

Additionally, developed algorithms were investigated by DIEHARD technique (one of the best practice and traditional approach in cryptography for measuring quality of PRNG and statistical security). These algorithms have passed complex security control by DIEHARD technique as well as verified good quality of PRNG and statistical security of potential systems based on these algorithms.