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**2024/1480**

**Title:**

On Schubert cells of Projective Geometry and quadratic public keys of Multivariate Cryptography

**Authors:**
Vasyl Ustimenko

**Abstract:**

Jordan-Gauss graphs are bipartite graphs given by special quadratic equations over the commutative ring K with unity with partition sets

K^n and K^m , n ≥m such that the neighbour of each vertex is defined by the system of linear equation given in its row-echelon form.

We use families of this graphs for the construction of new quadratic and cubic surjective multivariate maps F of K^n onto K^m (or K^n onto K^n) with the trapdoor accelerators T , i. e. pieces of information which allows to compute the reimage of the given value of F in poly-nomial time. The technique allows us to use the information on the quadratic map F from K^s to K^r, s ≥ r with the trapdoor accelerator T for the construction of other map G from K^{s+rs} onto K^{r+rs} with trapdoor accelerator. In the case of finite field it can be used for construc-tion of new cryptosystems from known pairs (F, T).

So we can introduce enveloping trapdoor accelerator for Matsumoto-Imai cryptosystem over finite fields of characteristic 2, for the Oil and Vinegar public keys over F_q (TUOV in particular), for quadratic multivariate public keys defined over Jordan-Gauss graphs D(n, K) where K is arbitrary finite commutative ring with the nontrivial multiplicative group.

**ePrint:**
https://eprint.iacr.org/2024/1480

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