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A new post-quantum multivariate polynomial public key encapsulation algorithm

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Abstract

We propose a new quantum-safe cryptosystem called multivariate polynomial public key (MPPK). Its security stems from the hardness of finding integer solutions to multivariate equations over a prime field GF(p). Indeed, for a large prime p, solving modular Diophantine equations is an NP-complete problem. MPPK introduces a novel way of key pair generation that involves multiplying a base n-degree multiplicand multivariate polynomial with respect to a message variable and two univariate multiplier polynomials, solvable by radicals over GF(p). The coefficients of the two resulting polynomial products are used to construct the public key, except for the coefficients of the constant and highest degree terms with respect to the message variable. The base multivariate polynomial’s constant and highest degree terms are used to form two noise functions, as parts of the public key, through multiplications with random variables. The private key consists of the two multiplier polynomials and the two random noise constants. MPPK encryption performs multivariate polynomial evaluations with a randomly chosen secret as the message variable and multiple noise values for other variables. The ciphertext tuple is created by calculating the values of two product multivariate polynomials and two noise functions. MPPK decryption eliminates the base multivariate polynomial by dividing by the two-product multivariate polynomial values and then extracting the secret from the resulting univariate polynomial with a radical. For adversarial extraction of the private key from the public key alone, the best complexity is exponential with respect to the bit length of the prime finite field. The same holds for the adversarial extraction of the plaintext from the ciphertext.

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Notes

  1. By introducing two noise functions over a hidden ring structure, we overcome a security weakness of the previous MPPK where the noise functions are defined in the same prime field GF(p) as the public key polynomials \(\Phi (.)\) and \(\Psi (.)\) [3, 4].

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Acknowledgements

The third author acknowledges the financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC). Authors acknowledge Ryan Toth for the performance data generated. The benchmarking data were partially taken from more detailed benchmarking performance papers to be published separately [50, 51]. Ryan Toth was a co-op student at Quantropi in the Summer of 2022.

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  1. Quantropi Inc., 1545 Carling Ave, Suite 620, Ottawa, ON, K1Z 8P9, Canada

    Randy Kuang & Maria Perepechaenko

  2. School of Computer Science, Carleton University, 1125 Colonel By Drive, Ottawa, ON, K1S 5B6, Canada

    Michel Barbeau

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  1. Randy Kuang
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  2. Maria Perepechaenko
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Correspondence to Randy Kuang.

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Kuang, R., Perepechaenko, M. & Barbeau, M. A new post-quantum multivariate polynomial public key encapsulation algorithm. Quantum Inf Process 21, 360 (2022). https://doi.org/10.1007/s11128-022-03712-5

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