arXiv subjects
Publications and source records attributed to Ilia Toli.
It has been suggested that the algebraic structure of AES (and other similar block ciphers) could lead to a weakness exploitable in new attacks. In this paper, we use the algebraic structure of AES-like ciphers to construct a cipher embedding where the ciphers may lose their non-linearity. We show some examples and we discuss the limitations of our approach.
It is shown by Karp reduction that deciding the singularity of $(2^n - 1) \times (2^n - 1)$ sparse circulant matrices (SC problem) is NP-complete. We can write them only implicitly, by indicating values of the $2 + n(n + 1)/2$ eventually nonzero entries of the first row and can make all matrix operations with them. The positions are $0, 1, 2^{i} + 2^{j}$. The complexity parameter is $n$. Mulmuley's work on the rank of matrices \cite{Mulmuley87} makes SC stand alone in a list of 3,000 and growing NP-complete problems.
We propose public-key cryptosystems with public key a system of polynomial equations, algebraic or differential, and private key a single polynomial or a small-size ideal. We set up probabilistic encryption, signature, and signcryption protocols.
We propose variations of the class of hidden monomial cryptosystems in order to make it resistant to all known attacks. We use identities built upon a single bivariate polynomial equation with coefficients in a finite field. Indeed, it can be replaced by a ``small'' ideal, as well. Throughout, we set up probabilistic encryption protocols, too. The same ideas extend to digital signature algorithms, as well. Our schemes work as well on differential fields of positive characteristic, and elsewhere.