Table of Links Abstract and 1. Introduction Abstract and 1. Introduction Contribution


Related Work


Brief of Homomorphic Polynomial Public Key Cryptography and 4.1 DPPK
4.2 MPPK Key Encapsulation Mechanism
4.3 HPPK KEM
4.4 MPPK DS


HPPK Digital Signature Scheme
5.1 HPPK DS Algorithm
5.2 Signing
5.3 Verify
5.4 A Variant of the Barrett Reduction Algorithm
5.5 A Toy Example


HPPK DS Security Analysis


Conclusion, References, Acknowledgements, and Author contributions statement Contribution Contribution Contribution Related Work Related Work Related Work Brief of Homomorphic Polynomial Public Key Cryptography and 4.1 DPPK
4.2 MPPK Key Encapsulation Mechanism
4.3 HPPK KEM
4.4 MPPK DS Brief of Homomorphic Polynomial Public Key Cryptography and 4.1 DPPK Brief of Homomorphic Polynomial Public Key Cryptography and 4.1 DPPK 4.2 MPPK Key Encapsulation Mechanism 4.2 MPPK Key Encapsulation Mechanism 4.3 HPPK KEM 4.3 HPPK KEM 4.4 MPPK DS 4.4 MPPK DS HPPK Digital Signature Scheme
5.1 HPPK DS Algorithm
5.2 Signing
5.3 Verify
5.4 A Variant of the Barrett Reduction Algorithm
5.5 A Toy Example HPPK Digital Signature Scheme HPPK Digital Signature Scheme 5.1 HPPK DS Algorithm 5.1 HPPK DS Algorithm 5.2 Signing 5.2 Signing 5.3 Verify 5.3 Verify 5.4 A Variant of the Barrett Reduction Algorithm 5.4 A Variant of the Barrett Reduction Algorithm 5.5 A Toy Example 5.5 A Toy Example HPPK DS Security Analysis HPPK DS Security Analysis HPPK DS Security Analysis Conclusion, References, Acknowledgements, and Author contributions statement Conclusion, References, Acknowledgements, and Author contributions statement Conclusion, References, Acknowledgements, and Author contributions statement 5 HPPK Digital Signature Scheme which reveals the unitary and reversible relation Indeed, this encryption operator serves as a distinctive permutation operator or a quantum permutation gate that can be implemented in a native quantum computing system[35, 36]. Furthermore, it adheres to the principles of non-commutativity, i.e. The non-commutativity is very important for us to reuse it multiple times to encrypt the entire coefficients of a product polynomial without reducing its security[37], unlike the case of One-Time-Pad encryption with XOR because XOR operator is commutable. 5.1 HPPK DS Algorithm 5.2 Signing The signing algorithm is very straightforward in HPPK DS: Following this, the signer appends the signature sig = {F,H} to the message, potentially including the public key if the recipient does not possess it. Upon receiving the message, the recipient can verify the signature using the public key of the HPPK DS. In scenarios where the hash code of the message exceeds the bit length of the finite field Fp, segmentation becomes necessary. Each segment is aligned with Fp, and a signature is independently generated for each segment. Subsequently, the segmented parts are concatenated to form the message’s signature. p p To prevent signature verification failure, it is advisable to conduct verification, as outlined in subsection 5.3. In the event of verification failure, it is recommended to randomly select a new α ∈ Fp and reevaluate F and H. p F H 5.3 Verify The signature verification consists of two stages: evaluating signature embedded coefficients and comparing polynomial values at the variable value x. The procedure is as follows 5.4 A Variant of the Barrett Reduction Algorithm 5.5 A Toy Example 5.5.1 Key Pair Generation 5.5.1 Key Pair Generation 5.5.1 Key Pair Generation Private Key: Private Key: Public Key: For simplification, we choose β = 1 to evaluate all public key elements. Public Key 5.5.2 Signing 5.5.2 Signing 5.5.2 Signing Assume that the hashed value of a message M is x = 9. Here are steps to generate the signature: 5.5.3 Verify 5.5.3 Verify 5.5.3 Verify So we obtain the verification polynomials Authors:
(1) Randy Kuang, Quantropi Inc., Ottawa, K1Z 8P9, Canada (randy.kuang@quantropi.com);
(2) Maria Perepechaenko, Quantropi Inc., Ottawa, K1Z 8P9, Canada;
(3) Mahmoud Sayed, Systems and Computer Engineering, Carleton University, Ottawa, K1S 5B6, Canada;
(4) Dafu Lou, Quantropi Inc., Ottawa, K1Z 8P9, Canada. Authors: Authors: (1) Randy Kuang, Quantropi Inc., Ottawa, K1Z 8P9, Canada (randy.kuang@quantropi.com); (2) Maria Perepechaenko, Quantropi Inc., Ottawa, K1Z 8P9, Canada; (3) Mahmoud Sayed, Systems and Computer Engineering, Carleton University, Ottawa, K1S 5B6, Canada; (4) Dafu Lou, Quantropi Inc., Ottawa, K1Z 8P9, Canada. This paper is available on arxiv under ATTRIBUTION-NONCOMMERCIAL-SHAREALIKE 4.0 INTERNATIONAL license. This paper is available on arxiv under ATTRIBUTION-NONCOMMERCIAL-SHAREALIKE 4.0 INTERNATIONAL license. available on arxiv available on arxiv

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