A new robust PKC encryption method based on invertible matrix multiplication for HIE in medical IoT systems

Abstract Electronic health information exchange (HIE) allows doctors, nurses, pharmacists, other health care providers and patients to appropriately access and securely share a patient’s vital medical information electronically—improving the speed, quality, safety and cost of patient care. At present, public key cryptography (PKC) is the most secure and practical cryptographic techniques to achieve this function in healthcare information exchange for medical IoT systems. ElGamal cryptosystem is one of the most well-known public key cryptosystems (they are also called asymmetric key cryptosystems), which is based on the discrete logarithm problem. At present, with the development of quantum computer technology, the ElGamal cryptosystem may be attacked by quantum algorithms. At the same time, since ElGamal requires several secure random integers to resist cryptographic analysis and ensure communication security and securing data sharing. Especially, when the encrypted information is large, multiple random numbers need to be used for grouping encryption, which makes the efficiency of ElGamal need to be improved. It is necessary to construct an alternative cryptographic algorithm that is more secure and efficient than ElGamal. In this paper, we construct a new public key cryptosystem (PKC) based on the discrete logarithm problem in $$GL\left(n,p\right)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mi>L</mml:mi> <mml:mfenced> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> </mml:mfenced> </mml:mrow> </mml:math> , which is constructed by the invertible matrix multiplication and can become an alternative version of the ElGamal public key cryptosystem. We call it Matrix ElGamal cryptosystem (M-EPKC). It is proved that the proposed PKC is computationally secure, which can provide the same security as the ElGamal cryptosystem in a much smaller finite field and use fewer random integers when encrypting large amounts of messages. For matrices of size $$n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> , ElGamal PKC requires $$n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> times more random numbers to encrypt plaintext with the same amount of data. The proposed M-EPKC is not only proved to be resistant to Shor’s algorithm attack (Shor in SIAM Rev 41: 303–332, 1999) on the integer field, but it can also improve its own computational efficiency by accelerating the decryption algorithm. Therefore, compare with the ElGamal cryptosystem, our proposed M-EPKC can provide a more secure and efficient method in healthcare information exchange for medical IoT systems.