Quantitative Evaluation of Chaotic CBC Mode of Operation

📝 Abstract

The cipher block chaining (CBC) block cipher mode of operation presents a very popular way of encrypting which is used in various applications. In previous research work, we have mathematically proven that, under some conditions, this mode of operation can admit a chaotic behavior according to Devaney. Proving that CBC mode is chaotic is only the beginning of the study of its security. The next step, which is the purpose of this paper, is to develop the quantitative study of the chaotic CBC mode of operation by evaluating the level of sensibility and expansivity for this mode.

💡 Analysis

The cipher block chaining (CBC) block cipher mode of operation presents a very popular way of encrypting which is used in various applications. In previous research work, we have mathematically proven that, under some conditions, this mode of operation can admit a chaotic behavior according to Devaney. Proving that CBC mode is chaotic is only the beginning of the study of its security. The next step, which is the purpose of this paper, is to develop the quantitative study of the chaotic CBC mode of operation by evaluating the level of sensibility and expansivity for this mode.

📄 Content

Quantitative Evaluation of Chaotic CBC Mode of Operation

Abdessalem Abidi1, Qianxue Wang3, Belgacem bouallègue1, Mohsen Machhout1 and Christophe Gyeux2 1Electronics and Microelectronics Laboratory University of Monastir, Faculty of Sciences of Monastir, Tunisia
2 FEMTO-ST Institute, UMR 6174 CNRS DISC
Computer Science Department
University of Franche Comté, 16, Route de Gray, 25000, Besançon France
3College of Automation Guangdong University of Technology,
Guangzhou 510006 China e-mail: abdessalemabidi9@gmail.com

Abstract—The cipher block chaining (CBC) block cipher mode of operation presents a very popular way of encrypting which is used in various applications. In previous research work, we have mathematically proven that, under some conditions, this mode of operation can admit a chaotic behavior according to Devaney. Proving that CBC mode is chaotic is only the beginning of the study of its security. The next step, which is the purpose of this paper, is to develop the quantitative study of the chaotic CBC mode of operation by evaluating the level of sensibility and expansivity for this mode. Keywords—Cipher Block Chaining; mode of operation; Block cipher; Devaney’s chaos; sensivity; expansivity. I. INTRODUCTION
Block ciphers have a very simple principle. They do not treat the original text bit by bit but they manipulate blocks of text for example, a block of 64 bits for the DES (Data Encryption Standard) or a block of 128 bits for the AES (Advanced Encryption Standard) algorithm. In fact, the original text is broken into blocks of N bits. For each block, the encryption algorithm is applied to obtain an encrypted block which has the same size. Then we gather all blocks, which are encrypted separately, to obtain the complete encrypted message. For decryption, we precede in the same way but this time starting from the cipher text to obtain the original message using the decryption algorithm instead of the encryption function. So, it is not sufficient to put anyhow a block cipher algorithm in a program. We can instead use these algorithms in various ways according to their specific needs. These ways are called the block cipher modes of operation. There are several modes of operation and each mode has owns characteristics and its specific security properties. In this article, we will consider only one of these modes, which is the cipher block chaining (CBC) mode. The chaos theory we consider in this paper is the Devaney’s topological one [1]. In addition to being recognized as one of the best mathematical definition of chaos, this theory offers a framework with qualitative and quantitative tools to evaluate the notion of unpredictability [2]. As an application of our fundamental results, we are interested in the area of information safety and security. In this paper, which is an extension of our previous article [3], the theoretical study of the chaotic behavior for the CBC mode of operation is deepened by evaluating its level of sensibility and expansivity [4]. Our fundamental study is motivated by the desire to produce chaotic programs in the area of information security. The remainder of this research work is organized as follows. In Section 2, we will recall some basic definitions concerning chaos and cipher-block chaining mode of operation. Section 3 is devoted to the results of our previous research works. In Section 4 quantitative topological properties for chaotic CBC mode of operation is studied in detail. This research work ends by a conclusion section in which our contribution is recalled and some intended future work are proposed. II. BASIC RECALLS This section is devoted to basic definitions and terminologies in the field of topological chaos and in the one of block cipher mode of operation. A. Devaney’s chaotic dynamical systems In the remainder of this article, S𝑛 denotes the n𝑡ℎ term of a sequence S while χℕ is the set of all sequences whose elements belong to χ. 𝑉𝑖 stands for the i𝑡ℎ component of a vector V. ƒ𝑘 = ƒ ° … °ƒ is for the 𝑘𝑡ℎ composition of a function ƒ. ℕ is the set of natural (non-negative) numbers, while ℕ* stands for the positive integers 1, 2, 3, . . . Finally, the following notation is used: ⟦1; 𝑁⟧= {1,2,… , 𝑁}.
Consider a topological space (𝜒, 𝜏) and a continuous function ƒ: 𝜒 → 𝜒 on (𝜒, 𝜏).  Definition 1. The function ƒ is topologically transitive if, for any pair of open sets 𝑈, 𝑉 ⊂𝜒 U, there exists an integer 𝑘> 0 such that ƒ𝑘 (U) ∩V ≠∅.  Definition 2. An element 𝑥 𝑖𝑠 𝑎 𝑝𝑒𝑟𝑖𝑑𝑖𝑐 𝑝𝑜𝑖𝑛𝑡 for ƒ of period 𝑛∈ℕ, n > 1, 𝑖𝑓 ƒn(𝑥) = 𝑥 and ƒk(𝑥) ≠𝑥 . ƒ is regular on (𝜒, 𝜏) if the set of periodic points for ƒ is dense in 𝜒 : for any point 𝑥 in 𝜒 , any neighborhood of 𝑥 contains at least one periodic point.  Definition 3. (Devaney’s formulation of chaos [1]) The function ƒ is chaotic on (𝜒, 𝜏) if ƒ is regular and topologically transitive. The chaos property is s

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