Keccak sponge function family main document

One difficulty in the cryptanalysis of the Advanced Encryption Standard AES is the tension between operations in the two fields GF(28) and GF(2). This paper outlines a new approach that avoids this conflict. We define a new block cipher, the BES, that uses only simple algebraic operations in GF(28). Yet the AES can be regarded as being identical to the BES with a restricted message space and key space, thus enabling the AES to be realised solely using simple algebraic operations in one field GF(28). This permits the exploration of the AES within a broad and rich setting. One consequence is that AES encryption can be described by an extremely sparse overdetermined multivariate quadratic system over GF(28), whose solution would recover an AES key.

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Essential Algebraic Structure within the AES

Block ciphers encrypt blocks of plaintext, messages, into blocks of ciphertext under the action of a secret key, and the process of encryption is reversed by decryption which uses the same user-supplied key. Block ciphers are fundamental to modern cryptography, in fact they are the most widely used cryptographic primitive useful in their own right, and in the construction of other cryptographic mechanisms. In this book the authors provide a technically detailed, yet readable, account of the state of the art of block cipher analysis, design, and deployment. The authors first describe the most prominent block ciphers and give insights into their design. They then consider the role of the cryptanalyst, the adversary, and provide an overview of some of the most important cryptanalytic methods. The book will be of value to graduate and senior undergraduate students of cryptography and to professionals engaged in cryptographic design. An important feature of the presentation is the authors' exhaustive bibliography of the field, each chapter closing with comprehensive supporting notes.

The Block Cipher Companion

Since being officially selected as the new Advanced Encryption Standard (AES), Rijndael has continued to receive great attention and has had its security continuously evaluated by the cryptographic community.

Rijndael is a cipher with a simple, elegant and highly algebraic structure. Its selection as the AES has led to a growing interest in the study of algebraic properties of block ciphers, and in particular algebraic techniques that can be used in their cryptanalysis.

In these notes we will examine some algebraic aspects of the AES and consider a number of algebraic techniques that could be used in the analysis of the cipher. In particular, we will focus on the large, though surprisingly simple, systems of multivariate quadratic equations derived from the encryption operation, and consider some approaches that could be used when attempting to solve these systems.

These notes refer to an invited talk given at the Fourth Conference on the Advanced Encryption Standard (AES4) in May 2004, and are largely based on[4].

/pdf/algebraic-aspects-of-the-advanced-encryption-standard-1wx7fetiqq.pdf

Algebraic Aspects of the Advanced Encryption Standard

In this paper we ask the question what happens if we replace all the constants in Rijndael, including the replacement of the irreducible polynomial, the coefficients of the MixColumn operation, the affine transformation in the S box, etc. We show that such replacements can create new dual ciphers, which are equivalent to the original in all aspects. We present several such dual ciphers of Rijndael, such as the square of Rijndael, and dual ciphers with the irreducible polynomial replaced by primitive polynomials. We also describe another family of dual ciphers consisting of the logarithms of Rijndael.We then discuss self-dual ciphers, and extend our results to other ciphers.

In How Many Ways Can You Write Rijndael

For the block cipher RIJNDAEL with a block length of 128 bits group theoretic properties of the round functions are derived. Especially it is shown that these round functions generate the alternating group.

/pdf/the-round-functions-of-rijndael-generate-the-alternating-4sf6z6ul4y.pdf

The Round Functions of RIJNDAEL Generate the Alternating Group

Group theoretic properties of Rijndael-like ciphers

While it is known previously that the cycle lengths of individual components of the AES round function are very small, we demonstrate here that the cycle length of the S-box combined with the ShiftRow and MixColumn transformation is at least 10 205 . This result is obtained by providing new invariances of the complete AES round function without the key addition. Furthermore, we consider self-duality properties of the AES round function and derive a property analogous to the complementation property of the DES round function. These results confirm the assessments given in other publications that the AES components have several unexpected structural properties.

Complementation-like and cyclic properties of AES round functions

Abstract We show that the round functions of the KASUMI block cipher for odd and even round type generate the alternating group on the message space. Moreover, under the assumption of independent round keys, we prove that also the KASUMI two-round functions and the KASUMI encryption functions generate the alternating group.

The round functions of KASUMI generate the alternating group

While it is known previously that the cycle lengths of individual components of the AES round function are very small, we demonstrate here that the cycle length of the S-box combined with the ShiftRow and MixColumn transformation is at least 10205. This result is obtained by providing new invariances of the complete AES round function without the key addition. Furthermore, we consider self-duality properties of the AES round function and derive a property analogous to the complementation property of the DES round function.These results confirm the assessments given in other publications that the AES components have several unexpected structural properties.

Ralph Wernsdorf

Papers

The Round Functions of RIJNDAEL Generate the Alternating Group

Group theoretic properties of Rijndael-like ciphers

Complementation-like and cyclic properties of AES round functions

The round functions of KASUMI generate the alternating group

Complementation-Like and cyclic properties of AES round functions