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Invariant-Based Cryptography: Toward a General Framework

Stanislav Semenov

TL;DR

The paper addresses building symmetric cryptographic primitives using invariant preservation instead of traditional one-way functions. It extends prior oscillatory four-point invariants to a general framework that includes discriminants and cross-ratios, yielding concrete schemes such as a discriminant-based method and a cross-ratio scheme, each with recoverability and integrity properties. It analyzes how invariant reuse enables compact session modes and discusses extensions to finite fields and algebras, as well as puzzles and a taxonomy of invariants for future exploration. The work argues that invariant-based design can provide compact, self-validating cryptographic primitives with practical potential for parameter exchange, commitments, and forward-secure streaming, offering a versatile alternative to opacity-based secrecy in symmetric protocols.

Abstract

We develop a generalized framework for invariant-based cryptography by extending the use of structural identities as core cryptographic mechanisms. Starting from a previously introduced scheme where a secret is encoded via a four-point algebraic invariant over masked functional values, we broaden the approach to include multiple classes of invariant constructions. In particular, we present new symmetric schemes based on shifted polynomial roots and functional equations constrained by symmetric algebraic conditions, such as discriminants and multilinear identities. These examples illustrate how algebraic invariants -- rather than one-way functions -- can enforce structural consistency and unforgeability. We analyze the cryptographic utility of such invariants in terms of recoverability, integrity binding, and resistance to forgery, and show that these constructions achieve security levels comparable to the original oscillatory model. This work establishes a foundation for invariant-based design as a versatile and compact alternative in symmetric cryptographic protocols.

Invariant-Based Cryptography: Toward a General Framework

TL;DR

The paper addresses building symmetric cryptographic primitives using invariant preservation instead of traditional one-way functions. It extends prior oscillatory four-point invariants to a general framework that includes discriminants and cross-ratios, yielding concrete schemes such as a discriminant-based method and a cross-ratio scheme, each with recoverability and integrity properties. It analyzes how invariant reuse enables compact session modes and discusses extensions to finite fields and algebras, as well as puzzles and a taxonomy of invariants for future exploration. The work argues that invariant-based design can provide compact, self-validating cryptographic primitives with practical potential for parameter exchange, commitments, and forward-secure streaming, offering a versatile alternative to opacity-based secrecy in symmetric protocols.

Abstract

We develop a generalized framework for invariant-based cryptography by extending the use of structural identities as core cryptographic mechanisms. Starting from a previously introduced scheme where a secret is encoded via a four-point algebraic invariant over masked functional values, we broaden the approach to include multiple classes of invariant constructions. In particular, we present new symmetric schemes based on shifted polynomial roots and functional equations constrained by symmetric algebraic conditions, such as discriminants and multilinear identities. These examples illustrate how algebraic invariants -- rather than one-way functions -- can enforce structural consistency and unforgeability. We analyze the cryptographic utility of such invariants in terms of recoverability, integrity binding, and resistance to forgery, and show that these constructions achieve security levels comparable to the original oscillatory model. This work establishes a foundation for invariant-based design as a versatile and compact alternative in symmetric cryptographic protocols.
Paper Structure (21 sections, 1 theorem, 13 equations)

This paper contains 21 sections, 1 theorem, 13 equations.

Table of Contents

  1. Symmetric Discriminant-Based Scheme
  2. Common Setup
  3. Alice’s Generation
  4. Bob's Verification and Recovery of h
  5. Remarks on Generalization
  6. Session Modes with Invariant Reuse
  7. Derived Invariant Mode (Minimal Transmission)
  8. Shared Root Mode (Invariant-Free Continuation)
  9. Initialization phase.
  10. Streaming phase.
  11. Note.
  12. Symmetric Cross-Ratio Scheme
  13. Common Setup
  14. Alice’s Generation
  15. Bob's Recovery of z4
  16. ...and 6 more sections

Key Result

Theorem 3.1

Let $M$ be a large prime modulus, and let $I \in \mathbb{Z}_M^\times$ be a fixed value. Suppose that for each session $i = 1, \dots, N$, a cross-ratio identity holds: and that only the masked triples are observable to an adversary, where each $f_i \in \mathrm{PGL}_2(\mathbb{Z}_M)$ is independently and uniformly chosen. Then, in the absence of knowledge of $z_4^{(i)}$, any polynomial-time adversa

Theorems & Definitions (2)

  • Theorem 3.1: Invariant Indistinguishability under Projective Masking
  • proof : Sketch of proof.