| In this monograph we explore several interconnected topics in multivariate cryptography. We start with asymmetric multivariate cryptography, providing a classification of field maps exhibiting the famous differential symmetry which undermined SFLASH, demonstrating that HFE and other "big field" schemes cannot possess such an invariant, and verifying that the projection modifier always removes this symmetry. Different forms of symmetry remaining over subfields are then treated. We develop the dual theory of finding what linear symmetries a field map can possess, specifically providing strong constraints on which linear symmetries are possible for the projected analogue of the SFLASH core map. Further, we establish the mechanics of an attack utilizing the pseudoinverse of singular affine operators. A new algorithm, extending the efficacy of the distillation method of the SFLASH attack, is then presented with the application of effectuating an assault on cryptosystems which structurally must retain a form of subfield symmetry. We continue, summarizing the status of pC*- systems, suggesting viable parameters, and drawing conclusions about the duality of the projection and minus modifiers. Finally, we turn to symmetric multivariate cryptography, studying the Advanced Encryption Standard (AES), verifying that the Big Encryption System (BES) transformation of the AES equations is equivalent to the application of a generalized relinearization algorithm to the set of conjugates of the AES equations. Since relinearization is widely discredited as a tool for ideal resolution, this fact, philosophically, indicates that the BES transformation doesn't reveal any fundamentally unknown structure of AES. |
