There are several attempts to build asymmetric pubic key encryption schemes based on multivariate polynomials of degree two over a finite field. However, most of them are insecure. The common defect in many of them comes from the fact that certain quadratic forms associated with their central maps have low rank, which makes them vulnerable to the MinRank attack. We propose a new simple and efficient multivariate pubic key encryption scheme based on matrix multiplication, which does not have such a low rank property. The new scheme will be called Simple Matrix Scheme or ABC in short. We also propose some parameters for practical and secure implementation.

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In this paper, we propose an algebraic broadcast attack against NTRU, which recovers a single message encrypted multiple times using different NTRU public keys. Namely, when a message is broadcasted, under some reasonable assumptions, our attack can be completed in polynomial time and space. To the best of our knowledge, this is the first successful broadcast attack against NTRU.

Let X = (x_1,..,x_n) and Y = (y_1,...,y_m) be a pair of corresponding plaintext and ciphertext for a cryptosystem. We define an embedded surface of this cryptosystem as any polynomial equation: E(X,Y)=E(x_1,..,x_n,y_1,...,y_m)=0, which is satisfied by all such pairs. In this paper, we present a new attack on the multivariate public key cryptosystems from Diophantine equations developed by Gao and Heindl by using the embedded surfaces associated to this family of multivariate cryptosystems.

In this paper, we present and prove the first closed formula bounding the degree of regularity of an HFE system over an arbitrary finite field. Though these bounds are not necessarily optimal, they can be used to deduce 1. if D, the degree of the corresponding HFE polynomial, and q, the size of the corresponding finite field, are fixed, inverting HFE system is polynomial for all fields; 2. if D is of the scale O(n^α) where n is the number of variables in an HFE system, and q is fixed, inverting HFE systems is quasi-polynomial for all fields. We generalize and prove rigorously similar results by Granboulan, Joux and Stern in the case when q = 2 that were communicated at Crypto 2006.