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Homomorphic Polynomial Public Key Cryptography for Quantum-secure Digital Signature

Randy Kuang, Maria Perepechaenko, Mahmoud Sayed, Dafu Lou

TL;DR

This work addresses the need for quantum-secure digital signatures by extending the Homomorphic Polynomial Public Key framework to a DS scheme with dual hidden rings. It introduces a Barrett reduction-based verification that converts modular multiplications into divisions and embeds signature data nonlinearly into public polynomials, mitigating forgery risks observed in prior MPPK/DS schemes. The authors provide a toy example to illustrate signing and verification, analyze security to show exponential-timeprivate-key-recovery and forgery against appropriately sized hidden rings, and discuss parameter choices and key/signature sizes aligned with NIST security levels. The result is a quantum-resistant DS construction that leverages homomorphic properties and nonlinear embeddings to improve security while maintaining compact public keys and signatures, with future work focused on benchmarking and comparative performance against PQC standards.

Abstract

In their 2022 study, Kuang et al. introduced Multivariable Polynomial Public Key (MPPK) cryptography, leveraging the inversion relationship between multiplication and division for quantum-safe public key systems. They extended MPPK into Homomorphic Polynomial Public Key (HPPK), employing homomorphic encryption for large hidden ring operations. Originally designed for key encapsulation (KEM), HPPK's security relies on homomorphic encryption of public polynomials. This paper expands HPPK KEM to a digital signature scheme, facing challenges due to the distinct nature of verification compared to decryption. To adapt HPPK KEM to digital signatures, the authors introduce an extension of the Barrett reduction algorithm, transforming modular multiplications into divisions in the verification equation over a prime field. The extended algorithm non-linearly embeds the signature into public polynomial coefficients, addressing vulnerabilities in earlier MPPK DS schemes. Security analysis demonstrates exponential complexity for private key recovery and forged signature attacks, considering ring bit length twice that of the prime field size.

Homomorphic Polynomial Public Key Cryptography for Quantum-secure Digital Signature

TL;DR

This work addresses the need for quantum-secure digital signatures by extending the Homomorphic Polynomial Public Key framework to a DS scheme with dual hidden rings. It introduces a Barrett reduction-based verification that converts modular multiplications into divisions and embeds signature data nonlinearly into public polynomials, mitigating forgery risks observed in prior MPPK/DS schemes. The authors provide a toy example to illustrate signing and verification, analyze security to show exponential-timeprivate-key-recovery and forgery against appropriately sized hidden rings, and discuss parameter choices and key/signature sizes aligned with NIST security levels. The result is a quantum-resistant DS construction that leverages homomorphic properties and nonlinear embeddings to improve security while maintaining compact public keys and signatures, with future work focused on benchmarking and comparative performance against PQC standards.

Abstract

In their 2022 study, Kuang et al. introduced Multivariable Polynomial Public Key (MPPK) cryptography, leveraging the inversion relationship between multiplication and division for quantum-safe public key systems. They extended MPPK into Homomorphic Polynomial Public Key (HPPK), employing homomorphic encryption for large hidden ring operations. Originally designed for key encapsulation (KEM), HPPK's security relies on homomorphic encryption of public polynomials. This paper expands HPPK KEM to a digital signature scheme, facing challenges due to the distinct nature of verification compared to decryption. To adapt HPPK KEM to digital signatures, the authors introduce an extension of the Barrett reduction algorithm, transforming modular multiplications into divisions in the verification equation over a prime field. The extended algorithm non-linearly embeds the signature into public polynomial coefficients, addressing vulnerabilities in earlier MPPK DS schemes. Security analysis demonstrates exponential complexity for private key recovery and forged signature attacks, considering ring bit length twice that of the prime field size.
Paper Structure (19 sections, 2 theorems, 39 equations, 1 figure, 1 table)

This paper contains 19 sections, 2 theorems, 39 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Contribution
  3. Related Work
  4. Brief of Homomorphic Polynomial Public Key Cryptography
  5. DPPK
  6. MPPK Key Encapsulation Mechanism
  7. HPPK KEM
  8. MPPK DS
  9. HPPK Digital Signature Scheme
  10. HPPK DS Algorithm
  11. Signing
  12. Verify
  13. A Variant of the Barrett Reduction Algorithm
  14. A Toy Example
  15. Key Pair Generation
  16. ...and 4 more sections

Key Result

Proposition 1

Assuming the same bit length $L$ for two hidden rings, there exists a private key recovery attack on the HPPK DS scheme with classical computational complexity of $\mathcal{O}(2^L)$, leveraging the HPPK DS public key and intercepted signatures.

Figures (1)

  • Figure 1: A semi-log graph illustration of the Barrett reduction results falling in $[n, 2n)$ per $10^8$ computations of $z=a*b \bmod{n}$, with randomly chosen $a < n$ and $b < n$, is plotted as a function of $\delta=k-\lceil log_2 n \rceil$ in bits. The blue line corresponds to $\lceil log_2 n \rceil= 208$ bits, the yellow line to $\lceil log_2 n \rceil= 292$ bits, and the grey line to $\lceil log_2 n \rceil= 400$ bits.

Theorems & Definitions (4)

  • Proposition 1
  • proof
  • Proposition 2
  • proof