  Diffie-Hellman key-agreement and RSA cryptosystem are widely used to provide
security in internet protocols. But both of the two algorithms are totally
breakable using Shor's algorithms. This paper proposes two connected
matrix-based key-agreements: (a) Diffie-Hellman Key-Agreement with Errors and
(b) RSA-Resemble Key-agreement, which, respectively, bear resemblance to
Diffie-Hellman key-agreement and RSA cryptosystem and thereby they gain some of
the well-known security characteristics of these two algorithms, but without
being subject to Shor's algorithms attacks. That is, the new schemes avoid the
direct reliance on the hardness of Discrete Logarithm and Integer Factoring
problems which are solvable by Shor's algorithms. The paper introduces a new
family of quantum-safe hardness assumptions which consist of taking noisy
powers of binary matrices. The new assumptions are derived from Decisional
Diffie-Hellman (DDH) assumption in the general linear group GL(n,2) by
introducing random noise into a quadruple similar to that which define the DDH
assumption in GL(n,2(. Thereby we make certain that the resulting quadruple is
secure against Shor's algorithm attack and any other DLP-based attack. Thence,
the resulting assumptions, are used as basis for the two key-agreement schemes.
We prove that these key-agreements are secure -- in key indistinguishability
notion -- under the new assumptions.
