Square (cipher)

Square (cipher)
Name	Square
Designers	Joan Daemen, Lars Knudsen, Vincent Rijmen
Publish date	1997
Derived to	AES, SHARK
Key size	128 bits
Block size	128 bits
Structure	Substitution–permutation network
Cryptanalysis	Susceptible to a chosen-plaintext attack known as the Square attack.

Square (cipher). Square is a block cipher developed as a precursor to the AES. Designed by a team including Joan Daemen and Vincent Rijmen, it operates on 128-bit blocks using a 128-bit key within an 8-round SPN structure. Although not widely adopted as a standard itself, its innovative design principles and the cryptanalytic technique developed against it, the Square attack, directly influenced the development of the Rijndael cipher, which later became the AES.

Overview

The block cipher was first presented at the Fast Software Encryption workshop in 1997. Its architecture is a classic Substitution–permutation network, utilizing operations on bytes arranged in a 4x4 matrix, a structure that would become central to the AES. The cipher's primary contribution to cryptography was not its deployment but its role in advancing design and analysis methodologies. The NIST's selection process for a new encryption standard highlighted the need for ciphers resistant to novel attacks, a challenge Square helped to define.

History

Square was conceived by Joan Daemen, Lars Knudsen, and Vincent Rijmen during a period of intense cryptographic innovation following the public revelation of differential cryptanalysis. Its development was closely tied to the earlier cipher SHARK, also designed by Vincent Rijmen. The cipher was published in 1997, coinciding with the NIST's announcement seeking a successor to the DES. While Square itself was not submitted to the AES competition, its core ideas were refined into Rijndael, which was submitted by Joan Daemen and Vincent Rijmen and ultimately won the contest, becoming the FIPS standard in 2001.

Design

The design operates on a 128-bit state, viewed as a 4x4 array of bytes. Each round consists of four distinct transformations applied in sequence: ByteSub, ShiftRow, MixColumn, and a key addition layer. The ByteSub step is a non-linear substitution using an S-box derived from the multiplicative inverse over the finite field GF(2⁸). The ShiftRow step cyclically shifts rows of the state array, while MixColumn applies a linear transformation that mixes the columns using polynomial multiplication over GF(2⁸). This structure ensures both confusion and diffusion, principles articulated by Claude Shannon.

Operation

Encryption begins with an initial key addition (whitening). The eight main rounds then apply the four operations: ByteSub, ShiftRow, MixColumn, and AddRoundKey. The final round omits the MixColumn step, a common design feature in SPNs. Decryption inverts each operation using the inverse functions: InvByteSub, InvShiftRow, and InvMixColumn. The Key schedule expands the original 128-bit cipher key into a series of round keys used in each AddRoundKey step, employing a combination of S-box lookups and round constants.

Security

The primary cryptanalytic result for Square was the discovery of the Square attack, a dedicated chosen-plaintext attack effective against reduced-round versions. This attack is a form of integral cryptanalysis that exploits the byte-oriented structure and the properties of the MixColumn transformation. While the full 8-round cipher was designed to be resistant, the attack demonstrated vulnerabilities in simpler variants. This analysis was crucial, informing the strengthening of Rijndael against such techniques. The cipher's design also aimed to resist known attacks like differential cryptanalysis and linear cryptanalysis.

Variations

The most significant direct variation is the Rijndael cipher, which generalized Square's fixed block and key size to support multiple lengths. The AES is a specific subset of Rijndael. The earlier cipher SHARK shares a similar mathematical foundation. The design principles of Square, particularly its use of operations in GF(2⁸), influenced numerous subsequent ciphers in the AES era. The Square attack itself became a foundational technique in cryptanalysis, applied to other ciphers like Hierocrypt and Crypton.

Category:Block ciphers Category:Advanced Encryption Standard