What question did this study set out to answer?

The aim is to establish the importance of the Riemann Hypothesis through modular and geometric principles.

March 13, 2026Open Access

Internal State Reconstruction via Side-Channels: Universal Principle and Applications to Cryptographic Algorithms

Key Points

The aim is to establish the importance of the Riemann Hypothesis through modular and geometric principles.
Introduced the Modular Semi-Group Tλ,γ(x) for analysis.
Used proof by contradiction to assess the implications of an off-line zero.
Showed relationships involving generalized Zeta functions and perturbations.
Proved the critical line ℜ(s) = α/2 is a universal fixed point.
Demonstrated that the existence of an off-line zero injects a power-class perturbation.
Established that this perturbation creates a singularity, violating conditions for the error term R(s).

Abstract

This work establishes the structural necessity of the Riemann Hypothesis (RH) through the introduction of the Modular Semi-Group Tλ,γ(x). We prove that the critical line ℜ(s) = α/2 is the universal geometric fixed point for any generalized Zeta function ζa(s). Riemann’s functional equation, ξ(s) = ξ(1 − s), demands that the generalized error term R(s) be identically zero (R(s) ≡ 0), which forces the total cancellation of the scaling disagreement Δ(x). Using a proof by contradiction, we show that the existence of an off-line zero ρHL = σ + it (where σ > 1/2) injects a power-class perturbation O(x−σ) into Δ(x). The Mellin Transform of this perturbation forces the appearance of a singularity (a pole) at s = σ within the critical strip. The existence of this pole violates the holomorphy required for R(s) ≡ 0, demonstrating that ρHL is analytically impossible.

Internal State Reconstruction via Side-Channels: Universal Principle and Applications to Cryptographic Algorithms

Key Points

Abstract

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