CC BY
16
Alexander V. Babash, Valery A. Sizov, Elena K. Baranova, Andrey A. Mikrukov
Тheoretica[ questions of computer science, computational mathematics, computer science and cognitive information technologies
УДК 004.056.55
DOI: 10.25559/SITITO.14.201803.573-577
THEORETICALLY UNBREAKABLE CIPHERS AS THEY SHOULD BE UNDERSTOOD
Alexander V. Babash1, Valery A. Sizov1, Elena K. Baranova2, Andrey A. Mikrukov1
1 Plekhanov Russian University of Economics, Moscow, Russia
2 National Research University - Higher School of Economics, Moscow, Russia
ТЕОРЕТИЧЕСКИ СТОЙКИЕ ШИФРЫ И КАК ИХ ПОНИМАТЬ
А.В. Бабаш1, В.А. Сизов1, Е.К. Баранова2, А.А. Микрюков1
1 Российский экономический университет имени Г.В. Плеханова, г. Москва, Россия
2 Национальный исследовательский университет «Высшая школа экономики», г. Москва, Россия
Babash A.V., Sizov V.A., Baranova E.K., Mikrukov A.A., 2018
Keywords
Cipher; perfectly secret encryption scheme; unbounded computing power
Ключевые слова
Шифр; совершенно секретные схемы шифрования; неограниченная вычислительная мощность.
Abstract
Perfectly-secret ciphers according to the Claude Shannon's theory, which are considered as unbreakable, and more specifically random keystream ciphers, are discussed. An analysis of the sources mentioned in the reference list showed that all of them come to the point that the perfect ciphers according to Claude Shannon's theory are unbreakable.
The article introduces some concepts, such as: the probabilistic model of cipher; the perfect cipher, which is secure against a plaintext recovery ciphertext-only attack; the perfect cipher, which is secure against a key recovery ciphertext-only attack; effective plaintext or key recovery attack; ineffective plaintext or key recovery attack; decipherable model of cipher; undecipherable model cipher. The introduced concepts were used to clarify Shannon's mathematical model and to prove that a statement about unbreakability of the perfect ciphers according to the Claude Shannon's theory, including random keystream cipher, were wrong. The purpose of the article is to attract the attention of specialists to the problem of developing methods for decrypting Vizhener cipher and using them in solving the problem of determining the cipher key of a random gamming according to a ciphertext, as well as developing methods for estimating the complexity and reliability of deciphering the cipher class in question.
Аннотация
В статье рассмотрены совершенные шифры на основе модели К. Шеннона, которые считаются не дешифруемыми шифрами, в частности, шифры случайного гаммирования. Анализ представленных в статье источников показал, что в них делается вывод о недешифруемости совершенных шифров по К. Шеннону. В статье введен ряд понятий, таких как: вероятностной модели шифра; шифра, совершенного по нападению на открытый текст при перехвате шифрованного текста; шифра, совершенного по нападению на ключ при перехвате шифрованного текста; эффективной атаки на открытый текст или ключ; неэффективной атаки на открытый текст или ключ; дешифруемой модели шифра; не дешифруемой модели шифра. С использованием введенных понятий уточнена математическая модель К. Шеннона и доказана ошибочность утверждения о недешифруемости совершенных шифров по К. Шеннону, в частности шифров случайного гаммирования. Целью статьи является привлечение внимания специалистов к проблеме развития методов дешифрования шифра Виженера и их использования в решении задачи определения ключа шифра случайного гаммирования по шифрованному тексту, а также разработки методов оценки трудоемкости и надежности дешифрования рассматриваемого класса шифров.
Alexander V. Babash, Dr. Phys.-Math. Sci., Full Professor, Professor of the Academic Department of Applied Information Technology and Information Security, Institute of Digital Economics, Plekhanov Russian University of Economics (36 Stremyanny Lane, Moscow 115093, Russia), ORCID: http://orcid. org/0000-0001-7578-6923, [email protected]
Valery A. Sizov, Dr of Technical Sci., Full Professor, Professor of the Academic Department of Applied Information Technology and Information Security, Institute of Digital Economics, Plekhanov Russian University of Economics (36 Stremyanny Lane, Moscow 115093, Russia), ORCID: http://orcid. org/0000-0002-4844-4714, [email protected]
Elena K. Baranova, Senior Lecturer, Senior Lecturer at Department of Information Security, National Research University - Higher School of Economics (20 Myasnitskaya Str., Moscow 101000, Russia), ORCID: http://orcid.org/0000-0003-4650-2623, [email protected]
Andrey A. Mikrukov, PhD in Technical Sci., Senior Lecturer, Senior Lecturer of the Academic Department of Applied Information Technology and Information Security, Institute of Digital Economics, Plekhanov Russian University of Economics (36 Stremyanny Lane, Moscow 115093, Russia), ORCID: https://orcid.org/0000-0002-8206-677X, [email protected]
Vol. 14, no 3. 2018 ISSN 2411-1473 sitito.cs.msu.ru
Modern Information Technologies and IT-Education
Теоретические вопросы информатики, прикладной математики, А.В. Бабаш, В.А. Си30в,
компьютерных наук и когнитивно-информационных технологий Е.К. Баранова, А.А. Микрюков
Preface
Searching of the links through the Yandex search system gives 22 mln. results on the request "perfect ciphers" and 43 mln. results on the request «schemes perfectly secret». Some of them related to the subject domain of information security were chosen for detailed analysis [1-29].
The book [1: 32] contains a definition (2.3) of the concept «perfectly secret cipher» as follows: "An encryption scheme over a message space M is perfectly secret if for every probability distribution over M, every message m £ M, and every ciphertext c £ C for which P(c) > 0:
p(m/c)=p(m).
(The requirement that P(c)>0 is a technical one needed to prevent conditioning on a zero-probability event.)". Such ciphers were introduced by Claude Shannon [13], who called them perfectly secret.
Goal of research
Our strategic aim is to draw attention of the information protection specialists to questions of cryptanalysis of the random keystream cipher (Vernam cipher) and particularly to the question of development of the Vigenere cipher decryption techniques and use them to determine a key of random keystream cipher from ciphertext.
Substantiation of the perfectly secret ciphers unbreakability in the works cited
The next sources [1-11] present the results of considerations of various encryption schemes that are secure even when the adversary has unbounded computational power. Such schemes are called perfectly secret. Qualitative descriptions of the ciphers under consideration are submitted below.
«In this chapter, we look at the other extreme and study encryption schemes that are provably secure even against an adversary who has unbounded computational power. Such schemes are called perfectly secret» [1: 35].
«The hotline between the United States and the former Soviet Union was (is it still active?) rumored to be encrypted with a one-time pad. Many Soviet spy messages to agents were encrypted using one-time pads. These messages are still secure today and will remain that way forever. It doesn't matter how long the supercomputers work on the problem. Even after the aliens from Andromeda land with their massive spaceships and undreamed-of computing power, they will not be able to read the Soviet spy messages encrypted with one-time pads (unless they can also go back in time and get the one-time pads» [2: 26 ].
«The one-time pad is the only available for us encryption technique, the absolute security of which can be proved» [3: 104].
«Theoretically, perfectly secret cipher (in other words, absolutely unbreakable cipher) does exist, but the only such cipher is only one of the forms of the so called "one time pad", in which the plaintext is encrypted by combining it with the key, which is truly random, and available for us algorithm of the same length» [4: 22]. It seems like to prevent any misconception the authors of this work [4] dropped a hint of doubt: 'keys produced by some truly random number generator will be high-quality with a probability that does not differ from unity more than a negligible quantity'» [4: 23].
"Considering the issue of theoretical unbreakability of ciphers, somebody can pay no attention to the real cost of complexity aspects and time needed for breaking a cipher (that determines an approach to the practical security of the cryptosystem). Pride of place goes to the
possibility in principle to get any information about plaintext or used key. For the first time ever this approach was investigated by Claude Shannon [13]. Examining that model of cipher with which we are already well acquainted, he considered a single ciphertext-only crypt attack. Let us follow his reasoning. As we already specified, the ultimate objective of any cryptanalyst is a message text or a cryptographic key. However, even some probabilistic information about the plaintext can be very useful. For example, even before looking at a ciphertext the cryptanalyst may have a priori information about message because of his assumption only that plaintext is written in English [5: 172].
Russian cryptanalysts consider these ciphers as theoretically unbreakable [9-11]. Authors of these sources insist that these ciphers are undecipherable by an adversary who has unbounded computational power. Following their approach, all other ciphers, i. e. imperfect ciphers, must be considered as decipherable ones. As a rule, they refer random keystream cipher as an example of perfectly secret cipher [1-13].
Clarification of the concept of perfect cipher according to the Claude Shannon's theory
Let us introduce following terms and symbols: M - a finite set, consisting of two or more elements, is called the plaintext space;
K - a finite set, consisting of two or more elements, is called the key space;
C - a finite set, consisting of two or more elements, is called the ciphertext space;
CA )keK - a set (family) of injective mappings;
(Л )кек: - inverse mappings for (fk ')kGK, if ^ k = C
fk'ic) = m; f: M x K ^ C
, then
- surjective mapping,
f (m k) = fk(m)
fk (m) = c f (m, k) = c,
k ) - encryption equation;
(fk\c) = m
k ^ ' - decryption equation;
P(M = (p(m), m <= M)
over space M;
P(K) = (p(k), k e K)
over space K.
- discrete probability distribution - discrete probability distribution
Definition 1. Let's consider an assembly of five introduced concepts ( f ) (f
(M,K,C, vJkJkeK, vJk Jk^K P(M), P(K))
as a probabilistic model of the Claude Shannon's cipher (encryption scheme) or, for the sake of brevity, a cipher model.
For ease of explanation of ensuing results, we will suppose, that for any m e M, k e K, c e C p(m)*0, p(k)*0, p(c)*0. Probability distributions P(M), P(K) induce: Probability distribution P(C) = (p(c),Ce C) over space C; Conditional probability distribution P(C / m) = (p(C / m), ceC) for every m e M;
Conditional probability distribution P(K / c) = pk / c)'k e K;
c e C
Современные информационные технологии и ИТ-образование
Том 14 № 3 (2018) ISSN 2411-1473 sitito.cs.msu.ru
A[exander V. Babash, va[ery A. Sizov, Theoretical questions of computer science, computational mathematics,
Elena K. Baranova, Andrey A. Mikrukov computer science and cognitive information technologies
Conditional probability distribution P(C / k) = (p(c / k), c^C) for every k £ K .
By way of the cipher model i fk )keK, P(M),
P(K)) example, we would refer to the random keystream cipher (Ver-nam cipher) model.
Suppose that I={1,2,...,n-1,0}, M=In, K= In, C= I". For fk(m)=c an encryption equation is determined by the equality c=yry2,...,yL, where y=i+Y, (mod n), J £
{1,2,..., L}
, and decryption equation is determined by the equalities y.-y.+n=i.(modn), j £ {1,2,...,L}. A probability distribution P(M) is established over the plaintext space M, and an equiprobable distribution P(M) is established over the key space K.
The work [13] provides more clarity on the definition 2.3 of perfectly secret cipher, which was presented in [1] earlier, as follows.
Definition 2. A cipher model over a message space M and the key space K is the perfect cipher, which is secure against a plaintext recovery ciphertext-only attack if for established probability distributions over spaces M and K for every message m £ M, and every cipher-text c £ C it is true, that:
p(m/c)=p(m).
The need for such refinement of the definition is dictated by the fact that each cipher can have several models. For example, the simple substitution cipher is considered as an imperfect one, although according to the models used to encrypt a plaintext of unit length, when "units" may be single letters, this cipher must be considered as perfectly secret [13].
Definition 3. An attack (mode, method) to determine a plaintext m of cipher (i. e. solutions of the equations f(m,k) = c in the unknown m from the space M), that requires non-zero expenditures because of time needed and complexity aspects and leads to a meaningful effect with a non-zero probability is called an effective attack for recovery of plaintext m of cipher model from established ciphertext c. Otherwise the attack is called ineffective.
Note 1. A plaintext recovery attack by guessing is not called effective.
Definition 4. Cipher model is called undecipherable if the effective attacks to determine the plaintext of this cipher model from established ciphertext do not exist. Otherwise, a cipher model is called decipherable.
Hypothesis 1. An effective ciphertext-only attack for plaintext recovery really does exist for imperfectly secret cipher model, which does not perfectly secure against a plaintext recovery
Effective attacks on ciphers are divided into two classes: keyless when attacks allow to determine a plaintext without determining a secret key, and attacks based on preliminary determining a secret key In the latter case, at first a secret key must be determined, and then a plaintext is read through decryption of ciphertext by recovered key
Definition 5. A cipher model over a message space M and the key space K is the perfect cipher, which is secure against a key recovery ciphertext-only attack if for established probability distributions over spaces M and K for every message m £ M, and every ciphertext c £ C it is true, that:
p(k/c) = p(k).
Similarly to the definitions 3 and 4, a concept of effective attack on the key space of the cipher model and a concept of undecipherable cipher model in relation to the key recovery ciphertext-only attack were introduced.
Hypothesis 2. An effective ciphertext-only attack for key recovery really does exist for imperfectly secret cipher model, which does not perfectly secure against a key recovery.
Theorem 1. An effective plaintext recovery attack does exist for the perfectly secret cipher model which is secure against a plaintext recovery ciphertext-only attack, and for the imperfectly secret cipher model, which is not perfectly secure against a key recovery cipher-text-only attack, if the conditions of hypothesis 2 are fulfilled.
Proof. According to the hypothesis 2, an effective key recovery attack does exist. Having a key it is possible to decrypt a ciphertext encrypted by this key and recover a plaintext.
A random keystream cipher is a decipherable one
Let's formulate and prove the following theorem.
Theorem 1. In the cases of non-equiprobable probability distribution over the plaintext space and equiprobable probability distribution over the key space, a model of random keystrean cipher is not perfectly secret against key recovery ciphertext-only attacking.
Proof. We know that an initial condition that the equality p(k/ c)=p(k) for any values of the keys k in the ciphertext c is equivalent to condition that p(c/k)=p(c) for any values of k and c. The equality p(c/ k)=p(m) is obvious for the plaintext m, which was encrypted by the key k into ciphertext c. The statement of the theorem follows straight-ly from the condition of non-equiprobable probability distribution over the plaintext space. Consequently, a random keystream cipher must be considered as decipherable one.
Note 2. A key space is a subset of the key space of the random keystream ciphers. Let's call them weak keys. Known attacks on the Vigenere cipher are the attacks on the random keystream cipher with a key of smaller length. For example, an attack "favorable event method", i.e. searching applied weak key through the use of attack «the Father or Dean of American Cryptology William Frederick Friedman», can be applied [13]. Description of refining of the attack is presented in [13]. Such an attack provides the opportunity to find cipher texts encrypted by locally periodic stream [13]. Along with attack «William Frederick Friedman» another attack «Friedrich Wilhelm Kasiski» (see «Die Geheimschriften und die Dechiffrirkunst») can be applied.
Calculation of the complexity parameters for decryption of the random keystream cipher goes beyond the scope of the present study Therefore, we believe that some special study should focus on that question.
Theorem 2 (it had been formulated according to the personal communication with Aliev F. K.). A random keystream cipher I={1,... ,¡11-1,0} is undecipherable one in the cases of equiprobable probability distribution over the plaintext space M=IN and over key space K=IN.
The proof is obvious.
The results. On the basis of the reasoning submitted, we proved that the random keystream cipher is a decipherable one. Theoretically, it is necessary to develop appropriate methods for labor consumption estimation and reliability of deciphering the random keystream cipher.
Conclusion
Formalization of concepts of breakability and unbreakability presented in this article, as well as statements submitted on their basis, allow us to conclude that an opinion about unbreakability of all perfectly secret ciphers is wrong. Further developing and improving research in the field of decryption techniques can provide a good solution of their use to determine a key of random keystream cipher from ciphertext.
Vol. 14, no 3. 2018 ISSN 2411-1473 sitito.cs.msu.ru
Modern Information Technologies and IT-Education
___ Теоретические вопросы информатики, прикладной математики, А.В. Бабаш, В.А. Сизов,
компьютерных наук и когнитивно-информационных технологий Е.К. Баранова, А.А. Микрюков
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Современные информационные технологии и ИТ-образование
Том 14 № 3 (2018) ISSN 2411-1473 sitito.cs.msu.ru
A[exander V. Babash, va[ery A. Sizov, Theoretical questions of computer science, computational mathematics,
Elena K. Baranova, Andrey A. Mikrukov computer science and cognitive information technologies
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Поступила 12.05.2018; принята в печать 10.08.2018; опубликована онлайн 30.09.2018.
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Бабаш Александр Владимирович, доктор физико-математических наук, профессор, кафедра прикладной информатики и информационной безопасности, Институт цифровой экономики и информационных технологий, Российский экономический университет имени Г.В. Плеханова (115093, Россия, г. Москва, Стремянный пер., д. 36), ORCID: http://orcid.org/0000-0001-7578-6923, [email protected] Сизов Валерий Александрович, доктор технических наук, профессор, кафедра прикладной информатики и информационной безопасности, Институт цифровой экономики и информационных технологий, Российский экономический университет имени Г.В. Плеханова (115093, Россия, г. Москва, Стремянный пер., д. 36), ORCID: http://orcid.org/0000-0002-4844-4714, [email protected]
Баранова Елена Константиновна, доцент, кафедра информационной безопасности, Национальный исследовательский университет «Высшая школа экономики» (101000, Россия, г. Москва, ул. Мясницкая, д. 20), ORCID: http://orcid.org/0000-0003-4650-2623, [email protected] Микрюков Андрей Александрович, кандидат технических наук, доцент, кафедра прикладной информатики и информационной безопасности, Институт цифровой экономики и информационных технологий, Российский экономический университет имени Г.В. Плеханова (115093, Россия, г. Москва, Стремянный пер., д. 36), ORCID: http://orcid.org/0000-0002-8206-677X, [email protected]
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Modern Information Technologies and IT-Education
