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We propose a new basic trapdoor lIC (l-Invertible Cycles) of the mixed field type for Multivariate Quadratic public key cryptosystems. This is the first new basic trapdoor since the invention of Unbalanced Oil and Vinegar in 1997. lIC can be considered an extended form of the well-known Matsumoto-Imai Scheme A (also MIA or C*), and share some features of stagewise triangular systems. However lIC has very distinctive properties of its own. In practice, lIC is much faster than MIA, and can even match the speed of single-field MQ schemes.

ℓ-Invertible Cycles for $\mathcal{M}$ ultivariate $\mathcal{Q}$ uadratic ( ${\mathcal{MQ}}$ ) Public Key Cryptography