Efficient probabilistic algorithm for estimating the algebraic properties of Boolean functions for large nn

Although several methods for estimating the resistance of a random Boolean function against (fast) algebraic attacks were proposed, these methods are usually infeasible in practice for relative large input variables $n$ (for instance $n\geq 30)$ due to increased computational complexity. An efficient estimation the resistance of Boolean function (with relative large input variables $n$) against (fast) algebraic attacks appears to be a rather difficult task. In this paper, the concept of partial linear relations decomposition is introduced, which decomposes any given nonlinear Boolean function into many linear (affine) subfunctions by using the disjoint sets of input variables. Based on this result, a general probabilistic decomposition algorithm for nonlinear Boolean functions is presented which gives a new framework for estimating the resistance of Boolean function against (fast) algebraic attacks. It is shown that our new probabilistic method gives very tight estimates (lower and upper bound) and it only requires about $O(n^22^n)$ operations for a random Boolean function with $n$ variables, thus having much less time complexity than previously known algorithms

Ongoing Research Areas in Symmetric Cryptography

This report is a deliverable for the ECRYPT European network of excellence in cryptology. It gives a brief summary of some of the research trends in symmetric cryptography at the time of writing. The following aspects of symmetric cryptography are investigated in this report: • the status of work with regards to different types of symmetric algorithms, including block ciphers, stream ciphers, hash functions and MAC algorithms (Section 1); • the recently proposed algebraic attacks on symmetric primitives (Section 2); • the design criteria for symmetric ciphers (Section 3); • the provable properties of symmetric primitives (Section 4); • the major industrial needs in the area of symmetric cryptography (Section 5)

D.STVL.9 - Ongoing Research Areas in Symmetric Cryptography

This report gives a brief summary of some of the research trends in symmetric cryptography at the time of writing (2008). The following aspects of symmetric cryptography are investigated in this report: • the status of work with regards to different types of symmetric algorithms, including block ciphers, stream ciphers, hash functions and MAC algorithms (Section 1); • the algebraic attacks on symmetric primitives (Section 2); • the design criteria for symmetric ciphers (Section 3); • the provable properties of symmetric primitives (Section 4); • the major industrial needs in the area of symmetric cryptography (Section 5)

Algebraic attacks on certain stream ciphers

To encrypt data streams of arbitrary lengths, keystream generators are used in modern cryptography which transform a secret initial value, called the key, into a long sequence of seemingly random bits. Many designs are based on linear feedback shift registers (LFSRs), which can be constructed in such a way that the output stream has optimal statistical and periodical properties and which can be efficiently implemented in hardware. Particularly prominent is a certain class of LFSR-based keystream generators, called (ι,m)-combiners or simply combiners. The maybe most famous example is the E0 keystream generator deployed in the Bluetooth standard for encryption. To evaluate the combiner’s security, cryptographers adopted an adversary model where the design and some parts of the input and output are known. An attack is a method to derive the key using the given knowledge. In the last decades, several kinds of attacks against LFSR-based keystream generators have been developed. In 2002 a new kind of attacks came up, named ”algebraic attacks”. The basic idea is to model the knowledge by a system of equation whose solution is the secret key. For several existing combiners, algebraic attacks represent the fastest theoretical attacks publicly known so far. This thesis discusses algebraic attacks against combiners. After providing the required mathematical fundament and a background on combiners, we describe algebraic attacks and explore the two main steps (generating the system of equations and computing the solution) in detail. The efficiency of algebraic attacks is closely connected to the degree of the equations. Thus, we examine the existence of low-degree equations in several situations and discuss multiple design principles to thwart their existence. Furthermore, we investigate ”fast algebraic attacks”, an extension of algebraic attacks.To encrypt data streams of arbitrary lengths, keystream generators are used in modern cryptography which transform a secret initial value, called the key, into a long sequence of seemingly random bits. Many designs are based on linear feedback shift registers (LFSRs), which can be constructed in such a way that the output stream has optimal statistical and periodical properties and which can be efficiently implemented in hardware. Particularly prominent is a certain class of LFSR-based keystream generators, called (ι,m)-combiners or simply combiners. The maybe most famous example is the E0 keystream generator deployed in the Bluetooth standard for encryption. To evaluate the combiner’s security, cryptographers adopted an adversary model where the design and some parts of the input and output are known. An attack is a method to derive the key using the given knowledge. In the last decades, several kinds of attacks against LFSR-based keystream generators have been developed. In 2002 a new kind of attacks came up, named ”algebraic attacks”. The basic idea is to model the knowledge by a system of equation whose solution is the secret key. For several existing combiners, algebraic attacks represent the fastest theoretical attacks publicly known so far. This thesis discusses algebraic attacks against combiners. After providing the required mathematical fundament and a background on combiners, we describe algebraic attacks and explore the two main steps (generating the system of equations and computing the solution) in detail. The efficiency of algebraic attacks is closely connected to the degree of the equations. Thus, we examine the existence of low-degree equations in several situations and discuss multiple design principles to thwart their existence. Furthermore, we investigate ”fast algebraic attacks”, an extension of algebraic attacks

Cryptanalysis of LFSR-based Pseudorandom Generators - a Survey

Pseudorandom generators based on linear feedback shift registers (LFSR) are a traditional building block for cryptographic stream ciphers. In this report, we review the general idea for such generators, as well as the most important techniques of cryptanalysis

A Chaos-Based Authenticated Cipher with Associated Data

In recent years, there has been a rising interest in authenticated encryptionwith associated data (AEAD)which combines encryption and authentication into a unified scheme. AEAD schemes provide authentication for a message that is divided into two parts: associated data which is not encrypted and the plaintext which is encrypted. However, there is a lack of chaos-based AEAD schemes in recent literature. This paper introduces a new128-bit chaos-based AEAD scheme based on the single-key Even-Mansour and Type-II generalized Feistel structure. The proposed scheme provides both privacy and authentication in a single-pass using only one 128-bit secret key. The chaotic tent map is used to generate whitening keys for the Even-Mansour construction, round keys, and random s-boxes for the Feistel round function. In addition, the proposed AEAD scheme can be implemented with true randomnumber generators to map a message tomultiple possible ciphertexts in a nondeterministic manner. Security and statistical evaluation indicate that the proposed scheme is highly secure for both the ciphertext and the authentication tag. Furthermore, it has multiple advantages over AES-GCM which is the current standard for authenticated encryption

On the algebraic immunity of direct sum constructions

In this paper, we study sufficient conditions to improve the lower bound on the algebraic immunity of a direct sum of Boolean functions. We exhibit three properties on the component functions such that satisfying one of them is sufficient to ensure that the algebraic immunity of their direct sum exceeds the maximum of their algebraic immunities. These properties can be checked while computing the algebraic immunity and they allow to determine better the security provided by functions central in different cryptographic constructions such as stream ciphers, pseudorandom generators, and weak pseudorandom functions. We provide examples for each property and determine the exact algebraic immunity of candidate constructions

Optimization and Guess-then-Solve Attacks in Cryptanalysis

In this thesis we study two major topics in cryptanalysis and optimization: software algebraic cryptanalysis and elliptic curve optimizations in cryptanalysis. The idea of algebraic cryptanalysis is to model a cipher by a Multivariate Quadratic (MQ) equation system. Solving MQ is an NP-hard problem. However, NP-hard problems have a point of phase transition where the problems become easy to solve. This thesis explores different optimizations to make solving algebraic cryptanalysis problems easier. We first worked on guessing a well-chosen number of key bits, a specific optimization problem leading to guess-then-solve attacks on GOST cipher. In addition to attacks, we propose two new security metrics of contradiction immunity and SAT immunity applicable to any cipher. These optimizations play a pivotal role in recent highly competitive results on full GOST. This and another cipher Simon, which we cryptanalyzed were submitted to ISO to become a global encryption standard which is the reason why we study the security of these ciphers in a lot of detail. Another optimization direction is to use well-selected data in conjunction with Plaintext/Ciphertext pairs following a truncated differential property. These allow to supplement an algebraic attack with extra equations and reduce solving time. This was a key innovation in our algebraic cryptanalysis work on NSA block cipher Simon and we could break up to 10 rounds of Simon64/128. The second major direction in our work is to inspect, analyse and predict the behaviour of ElimLin attack the complexity of which is very poorly understood, at a level of detail never seen before. Our aim is to extrapolate and discover the limits of such attacks, and go beyond with several types of concrete improvement. Finally, we have studied some optimization problems in elliptic curves which also deal with polynomial arithmetic over finite fields. We have studied existing implementations of the secp256k1 elliptic curve which is used in many popular cryptocurrency systems such as Bitcoin and we introduce an optimized attack on Bitcoin brain wallets and improved the state of art attack by 2.5 times

DoubleMod and SingleMod: Simple Randomized Secret-Key Encryption with Bounded Homomorphicity

An encryption relation f Z Z with decryption function f 1 is “group-homomorphic” if, for any suitable plaintexts x1 and x2, x1+x2 = f 1( f (x1)+f (x2)). It is “ring-homomorphic” if furthermore x1x2 = f 1( f (x1) f (x2)); it is “field-homomorphic” if furthermore 1=x1 = f 1( f (1=x1)). Such relations would support oblivious processing of encrypted data. We propose a simple randomized encryption relation f over the integers, called DoubleMod, which is “bounded ring-homomorphic” or what some call ”somewhat homomorphic.” Here, “bounded” means that the number of additions and multiplications that can be performed, while not allowing the encrypted values to go out of range, is limited (any pre-specified bound on the operation-count can be accommodated). Let R be any large integer. For any plaintext x 2 ZR, DoubleMod encrypts x as f (x) = x + au + bv, where a and b are randomly chosen integers in some appropriate interval, while (u; v) is the secret key. Here u > R2 is a large prime and the smallest prime factor of v exceeds u. With knowledge of the key, but not of a and b, the receiver decrypts the ciphertext by computing f 1(y) = (y mod v) mod u. DoubleMod generalizes an independent idea of van Dijk et al. 2010. We present and refine a new CCA1 chosen-ciphertext attack that finds the secret key of both systems (ours and van Dijk et al.’s) in linear time in the bit length of the security parameter. Under a known-plaintext attack, breaking DoubleMod is at most as hard as solving the Approximate GCD (AGCD) problem. The complexity of AGCD is not known. We also introduce the SingleMod field-homomorphic cryptosystems. The simplest SingleMod system based on the integers can be broken trivially. We had hoped, that if SingleMod is implemented inside non-Euclidean quadratic or higher-order fields with large discriminants, where GCD computations appear di cult, it may be feasible to achieve a desired level of security. We show, however, that a variation of our chosen-ciphertext attack works against SingleMod even in non-Euclidean fields