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Non-Degenerate One-Time Pad and the integrity of perfectly secret messages

Alex Shafarenko

TL;DR

This work tackles the problem of achieving unconditional integrity for perfectly secret communications by augmenting the One-Time Pad with non-degenerate diffusion. It introduces a novel NDOTP that encodes plaintext and key as permutations in a Lehmer/factoradic framework, producing a diffusive, nonlocal impact from ciphertext perturbations. The approach combines a Big-endian PHT with CRT, a differentiation of Lehmer codes, and a Pseudo Foata Injection to inject robust redundancy, all while preserving perfect secrecy and maintaining quadratic-time complexity. The result is a practical toolkit—NDOTP, PHT/CRT, Derivative diffusion, and PFI—that yields unconditional integrity guarantees without requiring extra integrity keys, with performance suited to realistic message sizes.

Abstract

We present a new construction of a One Time Pad (OTP) with inherent diffusive properties and a redundancy injection mechanism that benefits from them. The construction is based on interpreting the plaintext and key as members of a permutation group in the Lehmer code representation after conversion to factoradic. The so constructed OTP translates any perturbation of the ciphertext to an unpredictable, metrically large random perturbation of the plaintext. This allows us to provide unconditional integrity assurance without extra key material. The redundancy is injected using Foata's "pun": the reading of the one-line representation as the cyclic one; we call this Pseudo Foata Injection. We obtain algorithms of quadratic complexity that implement both mechanisms.

Non-Degenerate One-Time Pad and the integrity of perfectly secret messages

TL;DR

This work tackles the problem of achieving unconditional integrity for perfectly secret communications by augmenting the One-Time Pad with non-degenerate diffusion. It introduces a novel NDOTP that encodes plaintext and key as permutations in a Lehmer/factoradic framework, producing a diffusive, nonlocal impact from ciphertext perturbations. The approach combines a Big-endian PHT with CRT, a differentiation of Lehmer codes, and a Pseudo Foata Injection to inject robust redundancy, all while preserving perfect secrecy and maintaining quadratic-time complexity. The result is a practical toolkit—NDOTP, PHT/CRT, Derivative diffusion, and PFI—that yields unconditional integrity guarantees without requiring extra integrity keys, with performance suited to realistic message sizes.

Abstract

We present a new construction of a One Time Pad (OTP) with inherent diffusive properties and a redundancy injection mechanism that benefits from them. The construction is based on interpreting the plaintext and key as members of a permutation group in the Lehmer code representation after conversion to factoradic. The so constructed OTP translates any perturbation of the ciphertext to an unpredictable, metrically large random perturbation of the plaintext. This allows us to provide unconditional integrity assurance without extra key material. The redundancy is injected using Foata's "pun": the reading of the one-line representation as the cyclic one; we call this Pseudo Foata Injection. We obtain algorithms of quadratic complexity that implement both mechanisms.
Paper Structure (20 sections, 8 theorems, 53 equations, 4 figures, 1 table)

This paper contains 20 sections, 8 theorems, 53 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Threat model
  3. Nondegenerate one-time pad
  4. Lehmer code and factoradic numbers
  5. Conversion to factoradic and back.
  6. Differential properties.
  7. Non-Degenerate One-Time Pad (NDOTP)
  8. Discussion.
  9. Preconditioning
  10. Big-endian Pseudo-Hadamard Transform
  11. Configuration
  12. Discussion
  13. Differentiating the Lehmer code
  14. Discussion.
  15. Injecting Redundancy
  16. ...and 5 more sections

Key Result

Proposition 1

Let $r$ to be a positive number; $0\le p,k,c < r$ an integer plaintext, key and ciphertext, respectively; and $\pi$ an arbitrary cyclic permutation from $S_r$. Then the tripartite relationWe use square brackets to index a cyclic permutation in one-line notation.$c = \pi^k[p]$ is a trijection under a

Figures (4)

  • Figure 1: Differential properties of the Lehmer code
  • Figure 2: One step of the recurrence relation in Eqs \ref{['eq:dif1']}-\ref{['eq:dif2']}
  • Figure 3: Histogram of the Cayley distance from the perturbed to original $a$ obtained by Monte Carlo method. The computed mean $\bar{T}(a,A)\approx90.84$, in keeping with the expected $\nu-\ln\nu\approx90.45$
  • Figure 4: Blue: penetration rate $R$ vs penetration depth $L$. The vertical axis is in logarithmic scale (base 60). Red: the same for random perturbation (Corollary \ref{['cor:dens']})

Theorems & Definitions (17)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Definition 1
  • Proposition 4
  • proof
  • Definition 2
  • ...and 7 more