Statistical Assessment of two Rekeying Mechanisms applied to the Generation of Random Numbers

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Journal of Science and Technology on Information Security

Statistical Assessment of two Rekeying

Mechanisms applied to the Generation of

Random Numbers

Adrián Alfonso Peñate, Daymé Almeida Echevarria, Laura Castro Argudín

Abstract—The block ciphers modes of

operation with internal rekeying mechanisms,

used during the encryption of a message to

increase their security, have been a subject of

analysis in recent years. In this paper, we will

analyze the randomness of the sequences

generated by two of these modes of operation,

which also will be used in the generation of pseudo-

random numbers.

generated and used to process the following

blocks of the message used to process the

following blocks of the message. This new

approach,

which

well-studied

and

recommended in the last few years [4]-[12],

ensures that encryption remains secure even

when the message is very large, by changing the

key periodically. It can be applied on three levels:

on the block cipher level (fresh rekeying), on the

block cipher mode of operation level (internal

rekeying), and on the message processing level

(external rekeying).

Tóm tắt—Trong những năm qua, các chế độ

mã hóa khối hoạt động với cơ chế tạo lại khóa bên

trong, được sử dụng trong quá trình mã hóa tin

nhắn đã trở thành đối tượng nghiên cứu trong

những năm qua để tăng tính bảo mật. Trong bài

báo này, chúng tôi sẽ phân tích tính ngẫu nhiên của

chuỗi được tạo bởi hai trong số các chế độ này, mà

cũng sẽ được sử dụng trong việc tạo ra các số giả

ngẫu nhiên.

In our work, we will analyze the randomness

properties of the outputs produced by the pseudo-

random number generator Fortuna [13], which

uses the block cipher mode of operation CTR with

an internal rekeying mechanism proposed by their

own designers. Then we will perform a similar

analysis replacing the original proposal for the

recently standardized in the Russian Federation

mode of operation CTR-ACPKM [see 14 for

example]. For the analysis we intend to perform,

we will use the NIST statistical test suite [15].

Keywords—rekeying; randomness.

Từ khóa—tạo lại khóa; ngẫu nhiên.

I. INTRODUCTION

During the encryption process of a message

with a mode of operation, the values of the

pseudo-random permutation determined by the

secret key are tours. Then, if the length of the

message exceeds the lifetime of the key, the

encryption process can be seriously compromised

[1]. This determines for each mode of operation

the maximum number of blocks that can be

processed with the same key, and therefore, a

message of length greater than this amount

cannot be encrypted without changing the key

first. Bounds for the life time of a key in some

modes of operation can be seen in [2], [3].

II. HOW FORTUNA WORKS

A. The original Fortuna

Generation of keys, initialization vectors,

seeds, and other kinds of sequences with random

behavior play

fundamental role in

cryptography. For this reason, a pseudo-random

number generator that can be used in real

cryptographic applications has become an

important tool. Fortuna [13] is that kind of tool

due to the relevance of its design and to the

strength of the underlying cryptographic

primitives chosen by their designers [16]-[18].

Rekeying mechanisms are used to encrypt a

large amount of data with a specific mode of

operation, with the peculiarity that for every

certain amount of processed blocks, a new key is

This particular generator has three specific

components: the accumulator (to extract and pool

entropy from some external sources), the

generator (to produce the requested sequences),

and the seed file control (to produce random data

This manuscript is received on May 22, 2020. It is

commented on May 28, 2020 and is accepted on July 14, 2020

by the first reviewer. It is commented on December 8, 2020

and is accepted on December 26, 2020 by the second reviewer.

38 No 2.CS (12) 2020

Khoa học và Công nghệ trong lĩnh vực An toàn thông tin

at any time). In this paper, we only focus on the

The blocks of the first section are encrypted

generator due to the character of our investigation. using the mode of operation CTR with the section

key 퐾¹= 퐾. And consequently, the blocks of the

The mechanism that produces the sequences

of interest consist of the block cipher standard

AES with 256 bits of key length working in the

mode of operation CTR. The input is one random

counter of 128 bits and one random key of 256

bits, with which we can only produce 1MB of

pseudo-random data.

푖^푡ℎsection are encrypted using the same mode of

operation with the section key

퐾^ꢀ+1= 퐸 (퐷 ) ⋯ 퐸 (퐷 ).

‖ ‖

푖

푁

퐾

푛

{

}

Here 퐷₁, 퐷₂, ⋯ , 퐷_푁∈ 0,1 are different

constants established in the standard, aimed to

prevent collisions between the cycles of the

permutations determined by the section keys, and

the counter block is never changed between

sections. A better description of the mode of

operation with internal rekeying mechanism

CTR-ACPKM can be seen in [10], [11].

If the request of a user exceeds this amount,

then the next two blocks are generated and used

as a new key for the generation of the following

pseudo-random data. This principle of Fortuna

constitutes an internal rekeying mechanism,

offering greater security than if we always use the

mode CTR with the initial key [13].

The only difference of the internal rekeying in

CTR-ACPKM with respect to the internal

rekeying in Fortuna lies in the fact that, in the first

case constants are always encrypted to produce

the next section key, while in the second case the

counter block is encrypted to produce the next

section key, then the mode CTR-ACPKM may

replace the mode CTR used in Fortuna. We

implemented Fortuna using CTR-ACPKM such

that the section keys are replaced once there are

1MB of pseudo-random data generated.

More detailed, the generator of Fortuna only

produces 푁 = 65536 blocks (AES encryptions)

with the same key 퐾^ꢀfor the requested sequence

퐸 (퐶 ), ⋯ , 퐸 (퐶 ).

푖

푁

퐾

After that, two new blocks are produced to

compute the new key for the next data

퐾^ꢀ+1= 퐸_퐾퐶_푁+1||퐸 (퐶

)

(

)

푖

푁+2

퐾

and the actual counter is not restarted (퐶_푁+3

퐶₁) to encrypt with the next key 퐾^ꢀ+1

III. ABOUT THE NIST TEST SUITE

Statistical assessments have been performed

for several cryptographic primitives, including

random and pseudo-random number generators

[20]-[25].

B. Fortuna working with CTR-ACPKM

Recently, the mode of operation with internal

rekeying mechanism CTR-ACPKM was adopted

in the Russian Standardization System. The main

characteristic of this mode is that, for every

certain amount of blocks processed during the

encryption of a same message, another master

key is derived to the previous key.

In [16], a statistical evaluation of random data

generated by Fortuna was already performed

using the Diehard test suite [19]. In this case, ﬁfty

files of 12MB each were generated and analyzed

in both software and hardware implementations,

showing a highly random behavior.

That message is split into sections, each one

of them composed of the same amount of blocks

(except for the last section), and with different

keys for the encryption process. The parameter

푁, the section length, is established

independently taking into account the specific

protocol where it will be used, depending on the

capacity of the system and the lifetime of the keys.

In this paper, we will analyze the statistical

behavior of random data generated by Fortuna,

but using the NIST statistical test suite [15]. It

contains 15 tests for randomness, each of which

establishes a comparison between the pseudo-

random sequences generated and a “truly”

random sequence.

Since the first publication of the NIST

statistical test suite in 2000, it has been well

studied through the years [26]-[32], and several

corrections have been made. For the evaluation of

the sequences generated in this paper, we will use

the latest version of this set of tests [15], proposed

Given 푃 one 푚 blocks message of 푛 bits

푚

each, it splits into 푙 = ⌈

⌉ sections denoted as

⁄

푁

푃¹, ⋯, 푃^푙, then we have 푃^ꢀ∈ {0,1}^푛푁for 푖 ∈

{1, ⋯ , 푙 − 1} and 푃^푙∈ {0,1}^푟, 푟 ≤ 푛푁.

Số 2.CS (12) 2020 39

Journal of Science and Technology on Information Security

in 2010, which continues to be a powerful tool for

randomization assessments.

longest-run

rank

Since 2010 until now, there have been a

number of publications [35]-[39] showing

inaccuracies in the NIST SP 800-22, but NIST

has not shown that they will update this test

suite. Furthermore, there have been many works

[40]-[44] showing the correlations of several

tests in this test suite. Therefore, the selection of

the most efficient tests is a research direction in

the future. We do not consider this issue in the

scope of this article.

fft

nonperiodic-templates

overlapping-templates

universal

148

apen

serial

linear-complexity

To make the statistical assessments, we took

into account:

We don't describe the internal tests or the

complete suite at all. A wise description of the

test, the suite, or even the interpretation of the

results can appear in any one of the papers cited

before, including [15].

[Aspect 1.] As indicated in [15], for a sample of

size 1000 and the significance level 훼 = 0.01,

the range for acceptable proportions (number of

p-values greater or equal than 0.01 divided by

1000) is (0.9805608 ; 0.9994392).

To emit a reasonable criteria about the

samples, each one must contain a sufficient

amount of data. For every sample, we are

generating a file with 1000 sequences of 1000000

bits each (≈120 MB), and something less than in

[16], we are analyzing twenty files for every one

of the rekeying mechanisms of interest. For every

file, we check the proportion and the uniformity

of the p-values, the two approaches

recommended in [15]; and also we use a useful

approach utilized in [16] to establish a

comparison, where the range of possible

proportions in this case is scored.

We carefully observe each of the 162

proportion values obtained in the 20 files

generated with every rekeying mechanism, and

conclude that only a few of those values are out

of range. More specifically, only 14 proportions

are out of range in case of Fortuna for an average

of 0.7 faults per file, and only 16 proportions are

out of range in case of Fortuna with CTR-

ACPKM for an average of 0.8 faults per file,

which we interpret as a well random behavior in

both cases.

A. Practical results

Two graphs for every variant of Fortuna,

original and with CTR-ACPKM, are shown in the

Appendix B and C.

We implement Fortuna using the language

C# in Visual Studio 2017, and run for random

data in a PC Intel(R) Core (TM) i3-4130 CPU

(3.40GHz, 4GB-RAM and OS-Windows 7x64).

A pseudo-code of the generator of Fortuna is

showing in Appendix A.

[Aspect 2.] The idea behind the analysis of [16]

is split the range of possible p-values resulting

from the Diehard statistical test suite, and score

each of the sub-range considering a meaning of

goodness. A similar assignment is done in our

case but scoring the possible range of proportions.

With respect to NIST statistical test suite, we

excluded in the analysis: Random Excursions

Test and Random Excursions Variant Test, this

way we get 162 testing for every file distributed

in the following tests:

The next table shows how we did this.

Meaning of

goodness

Assigned

score

Proportion range

Statistical test

frequency

Number of testing

[0.985 ; 0.995]

Desirably good

[0.980561; 0.985)

(0.995 ; 0.999439]

Almost not good

but still good

block-frequency

cumulative-sums

runs

[0 ; 0.980561)

(0.999439 ; 1]

Not good

40 No 2.CS (12) 2020

Khoa học và Công nghệ trong lĩnh vực An toàn thông tin

Only for uniformity at the check of that

We recommend the use of Fortuna, and so of

criteria, we excluded of the NIST suite: Random Fortuna with CTR-ACPKM, in real and practical

Excursions Test and Random Excursions Variant applications. Nevertheless, it should be taken into

Test, since both analyze only an arbitrary amount account that each generated sequence must be

of sequences which satisfy some conditions. A carefully analyzed for randomness before being

similar approach is adopted in [33], [34].

used although similar results are expected.

This way, with the remaining tests, we

evaluate 162 proportion values per file, and the

accumulated score gives us another idea of

goodness. The possible best score is 0 and the

possible worst score is 648 for every file.

IV. CONCLUSION

Generation of keys, initialization vectors,

seeds, and other kinds of sequences with random

behavior play

fundamental role in

cryptography. Thus, a pseudo-random number

generator cryptographically strong with proven

pseudo-random behavior is a powerful tool in

A comparative analysis between the twenty

files generated with Fortuna and the twenty files

generated with the proposed variant is shown this area.

below. All results have been tabulated and are

The used simple technique is the

shown in the Appendix D, E. These results do not

throw significant differences between both

variants, and we appreciate a random behavior

since the final score of every file is very low.

implementation of one block cipher algorithm

using one mode of operation. This way, good

random behavior is achieved if the underlying

encryption method has good diffusion properties.

Generator

Fortuna (original)

Score

The pseudo-random number generator

Fortuna is based on this principle, using the

standard AES and the mode of operation CTR as

underlying primitives, and one internal rekeying

mechanism proposed by their own designers.

12 – best score

38 – worst score

8 – best score

Fortuna-CTR-ACPKM

40 – worst score

In this paper, we have analyzed the random

behavior of the sequences generated by Fortuna

as well as a variant that uses another internal

rekeying mechanism that included in the

Russian Standardization System and named

CTR-ACPKM. In both cases, twenty files of 120

MB were generated, all of them show a good

random behavior.

[Aspect 3.] The other approach according to [15]

is to examine the distribution of the p-values in

the range [0 ; 1], so this interval is divided into

ten equal sub-intervals and then the number of p-

values that lie within each sub-interval are

counted. The optimal case for randomness is that

100 p-values fall in each sub-interval.

As reference we used the NIST test suite,

although we have removed from it two tests and

have included an extra analysis technique

performed previously, but using the Diehard

statistical test suite. In terms of this approach, the

sequences generated by Fortuna seem as random

as the sequences generated by the variant of

Fortuna, although the difference observed is

really small to make a comparison between the

two kinds of sequences.

The NIST test suite returns a p-value of p-

values in every testing, corresponding to the chi

square test for the same ones. If that p-value is

greater than 0.0001, then the sequence can be

considered to be uniformly distributed, in all the

checks carried out this was satisfactorily fulfilled.

B. Final considerations

Both the original Fortuna and Fortuna with the

internal rekeying mechanism CTR-ACPKM,

have shown that the generated sequences have a

random behavior. The performing tests do not

offer sufficient reasons to establish a significant

comparison between both types of generated

sequence actually, using the criteria established

in [16] the sequences generated by Fortuna have

similar randomness than the other kind of

generated sequences.

To conclude this paper, we propose the

application of the two analyzed pseudo-random

number generators, Fortuna and Fortuna with

internally rekeying mechanism CTR-ACPKM, in

real and practical applications, as long as the

generated sequences in both cases are previously

analyzed and show a random behavior similar to

the examples shown here.

Số 2.CS (12) 2020 41

Journal of Science and Technology on Information Security

V. APPENDIX

C. Proportion analysis of two samples generated

by Fortuna with CTR-ACPKM

A. Pseudo-code of Fortuna with both, original

and new rekeying mechanisms

Pseudo-random generator Fortuna

Counter 퐶 with 128 bits (random)

Key 퐾 with 256 bits (random)

Number of bytes 푛 to generate

1. Define an empty string 푟

Function:

Input:

Procedure:

⌈

⌉

2. Define 푘 = 푛/16

3. For 푖 from 1 to 푘 − 1 do

4. 푟 = 푟||퐸_퐾(퐶)

5. 퐶 = 퐶 + 1

6. 푛 = 푛 − 16

7. If 푖 = 2¹⁶then

8. Change the key 퐾 using the

internal rekeying method of the

original Fortuna or the ACPKM

method

9. 퐶 = 퐶 + 1

10. End If

D. Score analysis of the 20 samples generated by

Fortuna

11. 푖 = 푖 + 1

12. End For

24 14 32 12 26 26 38 30 22 20

22 32 34 26 24 30 24 32 26 22

13. 푟 = 푟||first-푛-bytes-(퐸_퐾(퐶))

Pseudo-random string 푟 of 푛 bytes

Return:

E. Score analysis of the 20 samples generated by

Fortuna with CTR-ACPKM

B. Proportion analysis of two samples generated

by the original Fortuna

18 40 34 18 36 28 34 12 34 30

26 24 24 32 30

18 28

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ABOUT THE AUTHOR

Adrián Alfonso Peñate

Workplace:

Cryptography. University of Havana.

Education: Graduated of

Mathematics in 2014; received his

Master's degree in 2018.

Institute

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Security, 2012. 7(2): pp. 491-505.

Current research direction: design

and analysis of block ciphers.

Daymé Almeida Echevarria

Workplace: Institute

Cryptography. University of Havana.

Education: Graduated of

[37] Takeda, Y., et al., Modified Non-overlapping

template matching test and proposal on setting

template. 2014. 27(1): pp. 49-60.

Mathematics in 2013; currently

aspires to pursue her Master's degree.

[38] Okada, H. and K. Umeno, Randomness evaluation

with the discrete Fourier transform test based on Current research direction: design and analysis of

rekeying mechanisms for block ciphers modes of

operation.

exact analysis of the reference distribution. IEEE

Transactions on Information Forensics Security,

2017. 12(5): pp. 1218-1226.

Laura Castro Argudín

[39] Iwasaki, A. and K. Umeno, A new randomness test

solving problems of Discrete Fourier Transform

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Workplace:

Institute

Cryptography. University of Havana.

The education process: Graduated of

Cryptology in 2019.

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of NIST statistical randomness tests and

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sequences." Turkish Journal of Electrical

Current research direction: design

and analysis of pseudo-random

number generators.

44 No 2.CS (12) 2020

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