The Hill Cipher: A Weakness Studied Through Group Action Theory

Florent Dewez (1) , Valentin Montmirail (2)

1 UVHC - Université de Valenciennes et du Hainaut-Cambrésis
2 CRIL - Centre de Recherche en Informatique de Lens

The Hill cipher is considered as one of the most famous symmetric-key encryption algorithm: based on matrix multiplication, it has some interesting structural features which, for instance, can be exploited for teaching both cryptology and linear algebra. On the other hand, these features have rendered it vulnerable to some kinds of attack, such as the known-plaintext attack, and hence inapplicable in cases of real application. Despite this weakness, it does not stop the community proposing different upgrades for application purposes. In the present paper , we show that the Hill cipher preserves an algebraic structure of a given text and we use group action theory to study in a convenient setting some consequences of this fact, which turns out to be a potentially exploitable weakness. Indeed, our study might lead to a ciphertext-only attack requiring only that the alphabet has a prime number of characters. The main feature of this potential attack is the fact that it is not based on a search over all possible keys but rather over an explicit set of texts associated with the considered ciphertext. Group action theory guarantees that there will be, at worst, as much texts to test as keys, implying especially a better complexity.

Mots clés: Symmetric Encryption; Hill Cipher; Mathematical Analysis; Cryptanalysis; Group Action Theory

Domaines: Cryptographie et sécurité [cs.CR]

Florent Dewez, Valentin Montmirail. The Hill Cipher: A Weakness Studied Through Group Action Theory. 2017. ⟨hal-01631232⟩