DOCSLIB.ORG

S-Box Construction Based on Linear Fractional Transformation and Permutation Function

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
S-Box Construction Based on Linear Fractional Transformation and Permutation Function S S symmetry Article S-box Construction Based on Linear Fractional Transformation and Permutation Function Liyana Chew Nizam Chew 1,2,* and Eddie Shahril Ismail 1 1 Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600 UKM, Selangor, Malaysia; [email protected] 2 Cryptography Development Department, CyberSecurity Malaysia, Cyberjaya 63000, Selangor, Malaysia * Correspondence: [email protected] Received: 7 April 2020; Accepted: 7 May 2020; Published: 17 May 2020 Abstract: Substitution boxes (S-box) with strong and secure cryptographic properties are widely used for providing the key property of nonlinearity in block ciphers. This is critical to be resistant to a standard attack including linear and differential cryptanalysis. The ability to create a cryptographically strong S-box depends on its construction technique. This work aims to design and develop a cryptographically strong 8 8 S-box for block ciphers. In this work, the construction of the S-box × is based on the linear fractional transformation and permutation function. Three steps involved in producing the S-box. In step one, an irreducible polynomial of degree eight is chosen, and all roots of the primitive irreducible polynomial are calculated. In step two, algebraic properties of linear fractional transformation are applied in Galois Field GF (28). Finally, the produced matrix is permuted to add randomness to the S-box. The strength of the S-box is measured by calculating its potency to create confusion. To analyze the security properties of the S-box, some well-known and commonly used algebraic attacks are used. The proposed S-box is analyzed by nonlinearity test, algebraic degree, differential uniformity, and strict avalanche criterion which are the avalanche effect test, completeness test, and strong S-box test. S-box analysis is done before and after the application of the permutation function and the analysis result shows that the S-box with permutation function has reached the optimal properties as a secure S-box. Keywords: cryptography; substitution box; block cipher 1. Introduction With the rapid growth of digital communication and data exchange, there is hence an urgent need for the protection of data that is sensitive and confidential. The cryptographic encryption algorithms can be categorized as symmetric encryption algorithms and asymmetric encryption algorithms. The most well-known symmetric encryption algorithms are the Advanced Encryption Standard (AES) and Data Encryption Standard (DES) [1]. The DES was originally developed by International Business Machines (IBM) and later on adopted as a standard by the United States in 1977. The use of DES has now been withdrawn. The use of DES is permitted only as a component function of Triple DES (TDES) [2]. In 2001, the National Institute of Standards and Technology (NIST) announces the Rijndael cipher as a standard of AES. From 2001 until now AES has been successfully applied as a standard not only in the United States but also worldwide. The key size of any encryption algorithms is important for determining the strength of the algorithms. Thus, AES has the flexibility to have three keys that are 128 bits, 192 bits, and 256 bits. The foundation of modern cryptography by Claude Shannon indicates that two properties that a good cryptosystem should have are confusion and diffusion [2]. An important component in cryptographic algorithms that provide confusion by the non-linear component is the S-box. In most Symmetry 2020, 12, 826; doi:10.3390/sym12050826 www.mdpi.com/journal/symmetry Symmetry 2020, 12, 826 2 of 16 cryptographic algorithms such as AES and DES, S-box is the only nonlinear component providing a complex relationship between plaintext and ciphertext. In the one round of AES, there are four steps namely SubByte, ShiftRow, MixColumn, and AddRoundKey [3–5]. The S-box transformation is in the SubByte which is the only nonlinear part out of the four steps. Many of the past studies have shown that DES is broken due to its weak S-boxes. This implies that the security of cryptosystems is also measured by the security of its S-boxes. Thus, to develop secure cryptographic algorithms, researchers have focused on the formula of constructing a secure S-box and assessing the strength of the particular S-boxes against the analysis such as nonlinearity test, algebraic degree, differential uniformity, and strict avalanche criterion. The objective of this work is to develop a cryptographically strong 8 8 S-box and analyses the × S-box to prove its security properties. The rest of this paper is structured as follows. In Section2, we listed the literature review on various techniques and improved techniques in constructing the S-box. The review of S-box properties and analysis on nonlinearity test, algebraic degree, differential uniformity, and strict avalanche criterion are presented in Section3. Section4 is a brief introduction to linear fractional transformation and permutation function. In Section5, an S-box is constructed using linear fractional transformation and added permutation function to the S-box. Analysis of the cryptographic characteristics of the improved produce S-box was compared before and after the permutation function in Section6, followed by the conclusions in Section7. 2. Related Work Shannon has suggested two methods for a cryptographic algorithm to be resistant to cryptanalysis attacks. These methods are called confusion and diffusion [6]. The method of confusion in the cryptographic algorithm is to complicate the relationship between the ciphertext and symmetric key, meanwhile, the idea of diffusion is to hide the relationship between the ciphertext and the plaintext. The simplest way to achieve both confusion and diffusion in the cryptographic algorithm is to use a substitution function and permutation function. The most difficult step in verifying the strength of a cryptographic algorithm against cryptanalysis is the selection of cryptographically secure S-box. Therefore, understanding the design and properties of an S-box for applications in the encryption algorithm is essential [7]. Researchers have been challenged by the improved efficiency of the S-Box to develop confusion ability in the block cipher. In literature, there are several methods and tools implemented for the construction of cryptographically powerful S-boxes. It is an extremely required property for S-box to demonstrate a good resistance towards linear and differential cryptanalysis [8,9]. The linear cryptanalysis is a known-plaintext attack based on finding an affine approximation to the action of a cipher which connects in one expression for some bits of the randomly chosen plaintext and fixed key [10]. By collecting known plaintext and ciphertext pairs the attack can try to guess the value of bits key, as more plaintext and ciphertext pairs are collected the guessing will become more reliable. Differential cryptanalysis is a process that analyzes the effect of different in plaintext pairs on the resulting pairs of ciphertext. Such differences can be used to assign probabilities and to identify the most likely key. Using the resulting ciphertext pairs this approach typically works on many pairs of plaintexts that have the same particular difference. Mohamed et al. have suggested several properties to be present in an S-box to be able to resist various cryptanalytic attacks [11]. An S-box that has a majority of these properties offers greater security. To be considered as cryptographically strong and secure, an S-box requires high nonlinearity, low differential uniformity, high algebraic degree, balancing, low linear approximation, high algebraic complexity, and low/no fixed and opposite fixed points. An S-box has high nonlinearity will offer greater resistance to linear cryptanalysis [12]. AES uses extremely nonlinear S-box for the encryption and decryption processes in its various rounds. S-box of the AES is operating independently on each byte of the input, this S-box is invertible and developed by assembling two transformations: multiplication inverse and affine transformation [4]. In [13], Jie et al. have proposed an improved Symmetry 2020, 12, 826 3 of 16 of AES S-box by changing the affine transformation and adding an affine transformation. The other research that improved AES S-box is changing the complexity of the algebraic expression increases from 9 to 255 and preserve the existing irreducible polynomial, affine transformation matrix, and affine constant with the ability to resist against differential cryptanalysis invariable. Another research on constructing the S-box that caught attention is the S-box structure namely Affine-Power-Affine [3]. The S-box structure named Affine-Power-Affine aimed to increase the algebraic expressions term of AES S-box which is simple. In [14], Mamadolimov et al. have proposed to develop an S-box from power and binomial functions over the finite field and the resulting S-box has Differential Uniformity (DU) 8 and Nonlinearity (NL) 102. This method has been extended and improved by expending the range of the power function into trinomial and including the addition and multiplication as the manipulation techniques [15]. The obtain S-box has improved the analysis results to DU 4 and NL 108. Zahid and Arshad proposed the cubic polynomial mapping to produce an 8 8 S-box. The tested strength of the S-box shows the × maximum value of NL is 108 [16]. Construction of the S-box using linear fractional transformation has been introduced by [17–19]. The proposed S-box has been structured by a simple and direct algorithm with a single step function. The strength analysis shows that the S-box fits the criteria for cryptographically strong and is protected against differential and linear cryptanalysis. In this work, we have applied the method in constructing the S-box that involves the technique of linear fractional transformation. Then applied permutation function to increase the non-linear properties of this S-box. Security analysis is done to the S-box before and after the application of the permutation function to observe the effectiveness of permutation function in increasing the security of the S-box.
Load more

Recommended publications
  • Linear Cryptanalysis: Key Schedules and Tweakable Block Ciphers
    Linear Cryptanalysis: Key Schedules and Tweakable Block Ciphers Thorsten Kranz, Gregor Leander and Friedrich Wiemer Horst Görtz Institute for IT Security, Ruhr-Universität Bochum, Germany {thorsten.kranz,gregor.leander,friedrich.wiemer}@rub.de Abstract. This paper serves as a systematization of knowledge of linear cryptanalysis and provides novel insights in the areas of key schedule design and tweakable block ciphers. We examine in a step by step manner the linear hull theorem in a general and consistent setting. Based on this, we study the influence of the choice of the key scheduling on linear cryptanalysis, a – notoriously difficult – but important subject. Moreover, we investigate how tweakable block ciphers can be analyzed with respect to linear cryptanalysis, a topic that surprisingly has not been scrutinized until now. Keywords: Linear Cryptanalysis · Key Schedule · Hypothesis of Independent Round Keys · Tweakable Block Cipher 1 Introduction Block ciphers are among the most important cryptographic primitives. Besides being used for encrypting the major fraction of our sensible data, they are important building blocks in many cryptographic constructions and protocols. Clearly, the security of any concrete block cipher can never be strictly proven, usually not even be reduced to a mathematical problem, i. e. be provable in the sense of provable cryptography. However, the concrete security of well-known ciphers, in particular the AES and its predecessor DES, is very well studied and probably much better scrutinized than many of the mathematical problems on which provable secure schemes are based on. This been said, there is a clear lack of understanding when it comes to the key schedule part of block ciphers.
    [Show full text]
  • Advanced Encryption Standard (Aes) Modes of Operation
    ADVANCED ENCRYPTION STANDARD (AES) MODES OF OPERATION 1 Arya Rohan Under the guidance of Dr. Edward Schneider University of Maryland, College Park MISSION: TO SIMULATE BLOCK CIPHER MODES OF OPERATION FOR AES IN MATLAB Simulation of the AES (Rijndael Algorithm) in MATLAB for 128 bit key-length. Simulation of the five block cipher modes of operation for AES as per FIPS publication. Comparison of the five modes based on Avalanche Effect. Future Work 2 OUTLINE A brief history of AES Galois Field Theory De-Ciphering the Algorithm-ENCRYPTION De-Ciphering the Algorithm-DECRYPTION Block Cipher Modes of Operation Avalanche Effect Simulation in MATLAB Conclusion & Future Work References 3 A BRIEF HISTORY OF AES 4 In January 1997, researchers world-over were invited by NIST to submit proposals for a new standard to be called Advanced Encryption Standard (AES). From 15 serious proposals, the Rijndael algorithm proposed by Vincent Rijmen and Joan Daemen, two Belgian cryptographers won the contest. The Rijndael algorithm supported plaintext sizes of 128, 192 and 256 bits, as well as, key-lengths of 128, 192 and 256 bits. The Rijndael algorithm is based on the Galois field theory and hence it gives the algorithm provable 5 security properties. GALOIS FIELD 6 GALOIS FIELD - GROUP Group/Albelian Group: A group G or {G, .} is a set of elements with a binary operation denoted by . , that associates to each ordered pair (a, b) of elements in G an element (a . b) such that the following properties are obeyed: Closure: If a & b belong to G, then a . b also belongs to G.
    [Show full text]
  • Block Ciphers
    Block Ciphers Chester Rebeiro IIT Madras CR STINSON : chapters 3 Block Cipher KE KD untrusted communication link Alice E D Bob #%AR3Xf34^$ “Attack at Dawn!!” message encryption (ciphertext) decryption “Attack at Dawn!!” Encryption key is the same as the decryption key (KE = K D) CR 2 Block Cipher : Encryption Key Length Secret Key Plaintext Ciphertext Block Cipher (Encryption) Block Length • A block cipher encryption algorithm encrypts n bits of plaintext at a time • May need to pad the plaintext if necessary • y = ek(x) CR 3 Block Cipher : Decryption Key Length Secret Key Ciphertext Plaintext Block Cipher (Decryption) Block Length • A block cipher decryption algorithm recovers the plaintext from the ciphertext. • x = dk(y) CR 4 Inside the Block Cipher PlaintextBlock (an iterative cipher) Key Whitening Round 1 key1 Round 2 key2 Round 3 key3 Round n keyn Ciphertext Block • Each round has the same endomorphic cryptosystem, which takes a key and produces an intermediate ouput • Size of the key is huge… much larger than the block size. CR 5 Inside the Block Cipher (the key schedule) PlaintextBlock Secret Key Key Whitening Round 1 Round Key 1 Round 2 Round Key 2 Round 3 Round Key 3 Key Expansion Expansion Key Key Round n Round Key n Ciphertext Block • A single secret key of fixed size used to generate ‘round keys’ for each round CR 6 Inside the Round Function Round Input • Add Round key : Add Round Key Mixing operation between the round input and the round key. typically, an ex-or operation Confusion Layer • Confusion layer : Makes the relationship between round Diffusion Layer input and output complex.
    [Show full text]
  • Differential-Linear Crypt Analysis
    Differential-Linear Crypt analysis Susan K. Langfordl and Martin E. Hellman Department of Electrical Engineering Stanford University Stanford, CA 94035-4055 Abstract. This paper introduces a new chosen text attack on iterated cryptosystems, such as the Data Encryption Standard (DES). The attack is very efficient for 8-round DES,2 recovering 10 bits of key with 80% probability of success using only 512 chosen plaintexts. The probability of success increases to 95% using 768 chosen plaintexts. More key can be recovered with reduced probability of success. The attack takes less than 10 seconds on a SUN-4 workstation. While comparable in speed to existing attacks, this 8-round attack represents an order of magnitude improvement in the amount of required text. 1 Summary Iterated cryptosystems are encryption algorithms created by repeating a simple encryption function n times. Each iteration, or round, is a function of the previ- ous round’s oulpul and the key. Probably the best known algorithm of this type is the Data Encryption Standard (DES) [6].Because DES is widely used, it has been the focus of much of the research on the strength of iterated cryptosystems and is the system used as the sole example in this paper. Three major attacks on DES are exhaustive search [2, 71, Biham-Shamir’s differential cryptanalysis [l], and Matsui’s linear cryptanalysis [3, 4, 51. While exhaustive search is still the most practical attack for full 16 round DES, re- search interest is focused on the latter analytic attacks, in the hope or fear that improvements will render them practical as well.
    [Show full text]
  • Linear (Hull) and Algebraic Cryptanalysis of the Block Cipher PRESENT
    Linear (Hull) and Algebraic Cryptanalysis of the Block Cipher PRESENT Jorge Nakahara Jr1, Pouyan Sepehrdad1, Bingsheng Zhang2, Meiqin Wang3 1EPFL, Lausanne, Switzerland 2Cybernetica AS, Estonia and University of Tartu, Estonia 3Key Laboratory of Cryptologic Technology and Information Security, Ministry of Education, Shandong University, China Estonian Theory Days, 2-4.10.2009 Outline Why cryptanalysis?! Outline Outline Contributions The PRESENT Block Cipher Revisited Algebraic Cryptanalysis of PRESENT Linear Cryptanalysis of PRESENT Linear Hulls of PRESENT Conclusions Acknowledgements Outline Outline Contributions we performed linear analysis of reduced-round PRESENT exploiting fixed-points (and other symmetries) of pLayer exploiting low Hamming Weight bitmasks using iterative linear relations first linear hull analysis of PRESENT: 1st and 2nd best trails known-plaintext and ciphertext-only attack settings revisited algebraic analysis of 5-round PRESENT in less than 3 min best attacks on up to 26-round PRESENT (out of 31 rounds) Outline Outline The PRESENT Block Cipher block cipher designed by Bogdanov et al. at CHES’07 aimed at RFID tags, sensor networks (hardware environments) SPN structure 64-bit block size, 80- or 128-bit key size, 31 rounds one full round: xor with round subkey, S-box layer, bit permutation (pLayer) key schedule: 61-bit left rotation, S-box application, and xor with counter Outline Outline The PRESENT Block Cipher Computational graph of one full round of PRESENT Outline Outline Previous attack complexities on reduced-round
    [Show full text]
  • Visualization of the Avalanche Effect in CT2
    University of Mannheim Faculty for Business Informatics & Business Mathematics Theoretical Computer Science and IT Security Group Bachelor's Thesis Visualization of the Avalanche Effect in CT2 as part of the degree program Bachelor of Science Wirtschaftsinformatik submitted by Camilo Echeverri [email protected] on October 31, 2016 (2nd revised public version, Apr 18, 2017) Supervisors: Prof. Dr. Frederik Armknecht Prof. Bernhard Esslinger Visualization of the Avalanche Effect in CT2 Abstract Cryptographic algorithms must fulfill certain properties concerning their security. This thesis aims at providing insights into the importance of the avalanche effect property by introducing a new plugin for the cryptography and cryptanalysis platform CrypTool 2. The thesis addresses some of the desired properties, discusses the implementation of the plugin for modern and classic ciphers, guides the reader on how to use it, applies the proposed tool in order to test the avalanche effect of different cryptographic ciphers and hash functions, and interprets the results obtained. 2 Contents Abstract .......................................... 2 Contents .......................................... 3 List of Abbreviations .................................. 5 List of Figures ...................................... 6 List of Tables ....................................... 7 1 Introduction ..................................... 8 1.1 CrypTool 2 . 8 1.2 Outline of the Thesis . 9 2 Properties of Secure Block Ciphers ....................... 10 2.1 Avalanche Effect . 10 2.2 Completeness . 10 3 Related Work ..................................... 11 4 Plugin Design and Implementation ....................... 12 4.1 General Description of the Plugin . 12 4.2 Prepared Methods . 14 4.2.1 AES and DES . 14 4.3 Unprepared Methods . 20 4.3.1 Classic Ciphers, Modern Ciphers, and Hash Functions . 20 4.4 Architecture of the Code . 22 4.5 Limitations and Future Work .
    [Show full text]
  • A Block Cipher Algorithm to Enhance the Avalanche Effect Using Dynamic Key- Dependent S-Box and Genetic Operations 1Balajee Maram and 2J.M
    International Journal of Pure and Applied Mathematics Volume 119 No. 10 2018, 399-418 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu Special Issue ijpam.eu A Block Cipher Algorithm to Enhance the Avalanche Effect Using Dynamic Key- Dependent S-Box and Genetic Operations 1Balajee Maram and 2J.M. Gnanasekar 1Department of CSE, GMRIT, Rajam, India. Research and Development Centre, Bharathiar University, Coimbatore. [email protected] 2Department of Computer Science & Engineering, Sri Venkateswara College of Engineering, Sriperumbudur Tamil Nadu. [email protected] Abstract In digital data security, an encryption technique plays a vital role to convert digital data into intelligible form. In this paper, a light-weight S- box is generated that depends on Pseudo-Random-Number-Generators. According to shared-secret-key, all the Pseudo-Random-Numbers are scrambled and input to the S-box. The complexity of S-box generation is very simple. Here the plain-text is encrypted using Genetic Operations and S-box which is generated based on shared-secret-key. The proposed algorithm is experimentally investigates the complexity, quality and performance using the S-box parameters which includes Hamming Distance, Balanced Output and the characteristic of cryptography is Avalanche Effect. Finally the comparison results motivates that the dynamic key-dependent S-box has good quality and performance than existing algorithms. 399 International Journal of Pure and Applied Mathematics Special Issue Index Terms:S-BOX, data security, random number, cryptography, genetic operations. 400 International Journal of Pure and Applied Mathematics Special Issue 1. Introduction In public network, several types of attacks1 can be avoided by applying Data Encryption/Decryption2.
    [Show full text]
  • FPGA Implementation and Analysis of the Block Cipher Mode Architectures for the PRESENT Light Weight Encryption Algorithm
    ISSN (Print) : 0974-6846 Indian Journal of Science and Technology, Vol 9(38), DOI: 10.17485/ijst/2016/v9i38/90314, October 2016 ISSN (Online) : 0974-5645 FPGA Implementation and Analysis of the Block Cipher Mode Architectures for the PRESENT Light Weight Encryption Algorithm A. Prathiba* and V. S. Kanchana Bhaaskaran School of Electronics Engineering, VIT University Chennai, Chennai - 600127, India; prathi_communication@yahoo. co.in, [email protected] Abstract Objective: This paper presents the Field Programmable Gate Array (FPGA) implementations of the different block cipher mode architectures of the ISO standardized light weight block cipher PRESENT, designed for resource constrained devices. Methods/ Statistical Analysis: The performance evaluations compare the implementations of the different block cipher modes, namely Electronic Code Book (ECB) mode, Cipher Block Chaining (CBC) mode, Cipher Feedback Mode (CFB), Output Feed Back Mode (OFB) and CounTeR (CTR) mode for the PRESENT cipher. The throughput of encryption of three successive 64 bit blocks of data ranges from 565.312Mbps to 574.784Mbps for the modes other than the cipher feedback arrives as 68.912 Mbps, 155.392Mbps and300.8 Mbps for a 64 bit block of data for the input streams of size 8 bits, 16 bits andmode 32 in bits the respectively.Spartan-3 FPGA. Findings: The throughput The throughput for providing of the confidentialityblock cipher mode through hardware encryption architectures in the cipher of the feedback light weight mode cipher PRESENT demonstrates the high speed performance of the cipher in encryption/decryption of data as blocks and streams. Application/ Improvement: different modes of operation for the light weight block cipher PRESENT.
    [Show full text]
  • Linear Cryptanalysis of DES with Asymmetries
    Linear Cryptanalysis of DES with Asymmetries Andrey Bogdanov and Philip S. Vejre Technical University of Denmark {anbog,psve}@dtu.dk Abstract. Linear cryptanalysis of DES, proposed by Matsui in 1993, has had a seminal impact on symmetric-key cryptography, having seen massive research efforts over the past two decades. It has spawned many variants, including multidimensional and zero-correlation linear crypt- analysis. These variants can claim best attacks on several ciphers, in- cluding present, Serpent, and CLEFIA. For DES, none of these vari- ants have improved upon Matsui’s original linear cryptanalysis, which has been the best known-plaintext key-recovery attack on the cipher ever since. In a revisit, Junod concluded that when using 243 known plain- texts, this attack has a complexity of 241 DES evaluations. His analysis relies on the standard assumptions of right-key equivalence and wrong- key randomisation. In this paper, we first investigate the validity of these fundamental as- sumptions when applied to DES. For the right key, we observe that strong linear approximations of DES have more than just one dominant trail and, thus, that the right keys are in fact inequivalent with respect to linear correlation. We therefore develop a new right-key model us- ing Gaussian mixtures for approximations with several dominant trails. For the wrong key, we observe that the correlation of a strong approx- imation after the partial decryption with a wrong key still shows much non-randomness. To remedy this, we propose a novel wrong-key model that expresses the wrong-key linear correlation using a version of DES with more rounds.
    [Show full text]
  • Combined Algebraic and Truncated Differential Cryptanalysis on Reduced-Round Simon
    Combined Algebraic and Truncated Differential Cryptanalysis on Reduced-round Simon Nicolas Courtois1, Theodosis Mourouzis1, Guangyan Song1, Pouyan Sepehrdad2 and Petr Susil3 1University College London, London, U.K. 2Qualcomm Inc., San Diego, CA, U.S.A. 3Ecole´ Polytechnique Fed´ ereale´ de Lausanne, Lausanne, Switzerland Keywords: Lightweight Cryptography, Block Cipher, Feistel, SIMON, Differential Cryptanalysis, Algebraic Cryptanal- ysis, Truncated Differentials, SAT Solver, Elimlin, Non-linearity, Multiplicative Complexity, Guess-then- determine. Abstract: Recently, two families of ultra-lightweight block ciphers were proposed, SIMON and SPECK, which come in a variety of block and key sizes (Beaulieu et al., 2013). They are designed to offer excellent performance for hardware and software implementations (Beaulieu et al., 2013; Aysu et al., 2014). In this paper, we study the resistance of SIMON-64/128 with respect to algebraic attacks. Its round function has very low Multiplicative Complexity (MC) (Boyar et al., 2000; Boyar and Peralta, 2010) and very low non-linearity (Boyar et al., 2013; Courtois et al., 2011) since the only non-linear component is the bitwise multiplication operation. Such ciphers are expected to be very good candidates to be broken by algebraic attacks and combinations with truncated differentials (additional work by the same authors). We algebraically encode the cipher and then using guess-then-determine techniques, we try to solve the underlying system using either a SAT solver (Bard et al., 2007) or by ElimLin algorithm (Courtois et al., 2012b). We consider several settings where P-C pairs that satisfy certain properties are available, such as low Hamming distance or follow a strong truncated differential property (Knudsen, 1995).
    [Show full text]
  • How Far Can We Go Beyond Linear Cryptanalysis?
    How Far Can We Go Beyond Linear Cryptanalysis? Thomas Baign`eres, Pascal Junod, and Serge Vaudenay EPFL http://lasecwww.epfl.ch Abstract. Several generalizations of linear cryptanalysis have been pro- posed in the past, as well as very similar attacks in a statistical point of view. In this paper, we de¯ne a rigorous general statistical framework which allows to interpret most of these attacks in a simple and uni¯ed way. Then, we explicitely construct optimal distinguishers, we evaluate their performance, and we prove that a block cipher immune to classical linear cryptanalysis possesses some resistance to a wide class of general- ized versions, but not all. Finally, we derive tools which are necessary to set up more elaborate extensions of linear cryptanalysis, and to general- ize the notions of bias, characteristic, and piling-up lemma. Keywords: Block ciphers, linear cryptanalysis, statistical cryptanalysis. 1 A Decade of Linear Cryptanalysis Linear cryptanalysis is a known-plaintext attack proposed in 1993 by Matsui [21, 22] to break DES [26], exploiting speci¯c correlations between the input and the output of a block cipher. Namely, the attack traces the statistical correlation between one bit of information about the plaintext and one bit of information about the ciphertext, both obtained linearly with respect to GF(2)L (where L is the block size of the cipher), by means of probabilistic linear expressions, a concept previously introduced by Tardy-Corfdir and Gilbert [30]. Soon after, several attempts to generalize linear cryptanalysis are published: Kaliski and Robshaw [13] demonstrate how it is possible to combine several in- dependent linear correlations depending on the same key bits.
    [Show full text]
  • Authentication Key Recovery on Galois/Counter Mode (GCM)
    Authentication Key Recovery on Galois/Counter Mode (GCM) John Mattsson and Magnus Westerlund Ericsson Research, Stockholm, Sweden {firstname.lastname}@ericsson.com Abstract. GCM is used in a vast amount of security protocols and is quickly becoming the de facto mode of operation for block ciphers due to its exceptional performance. In this paper we analyze the NIST stan- dardized version (SP 800-38D) of GCM, and in particular the use of short tag lengths. We show that feedback of successful or unsuccessful forgery attempts is almost always possible, contradicting the NIST assumptions for short tags. We also provide a complexity estimation of Ferguson’s authentication key recovery method on short tags, and suggest several novel improvements to Fergusons’s attacks that significantly reduce the security level for short tags. We show that for many truncated tag sizes; the security levels are far below, not only the current NIST requirement of 112-bit security, but also the old NIST requirement of 80-bit security. We therefore strongly recommend NIST to revise SP 800-38D. Keywords. Secret-key Cryptography, Message Authentication Codes, Block Ciphers, Cryptanalysis, Galois/Counter Mode, GCM, Authentica- tion Key Recovery, AES-GCM, Suite B, IPsec, ESP, SRTP, Re-forgery. 1 Introduction Galois/Counter Mode (GCM) [1] is quickly becoming the de facto mode of op- eration for block ciphers. GCM is included in the NSA Suite B set of cryp- tographic algorithms [2], and AES-GCM is the benchmark algorithm for the AEAD competition CAESAR [3]. Together with Galois Message Authentication Code (GMAC), GCM is used in a vast amount of security protocols: – Many protocols such as IPsec [4], TLS [5], SSH [6], JOSE [7], 802.1AE (MAC- sec) [8], 802.11ad (WiGig) [9], 802.11ac (Wi-Fi) [10], P1619.1 (data storage) [11], Fibre Channel [12], and SRTP [13, 14]1 only allow 128-bit tags.
    [Show full text]