Breakthrough a step toward revealing hidden structure of prime numbers

Impact on Cryptography if Factoring Breakthrough Occurs

• Thread begins with a thought experiment: an efficient factoring algorithm on consumer hardware could instantly break RSA and similar public-key systems.
• Many expect this would be catastrophic for current security, especially for previously captured traffic encrypted with RSA (“store now, decrypt later”).
• Some argue industry disaster-recovery plans for such a sudden math breakthrough are minimal or unclear; others think systems would adapt under pressure, as with major outages.

RSA, ECC, and Alternative Cryptosystems

• Several comments note a shift from RSA to elliptic-curve cryptography (ECC), mainly for performance and smaller keys, not necessarily because primes are expected to be “cracked soon.”
• It’s emphasized that ECC also falls to large enough quantum computers (via Shor’s algorithm), and arguably with fewer qubits than RSA.
• Quantum-resistant (post-quantum) schemes and lattice-based cryptography are mentioned, but seen as early-stage and not yet widely deployed.

Historical and Ongoing Attacks on RSA

• RSA is described as having repeatedly “failed” at specific key sizes: 128-, 512-, and eventually 1024-bit keys became insecure as factoring algorithms improved (e.g., various number field sieves).
• Others counter that this is not “breaking RSA” but the normal process of increasing key sizes as attacks improve; conservatively large keys have not been obsoleted overnight.

Security Posture and Data Handling

• Comments stress that encryption alone is not enough; storing sensitive data on others’ computers (cloud) is inherently risky.
• Airgapping is discussed; numerous side-channel exfiltration methods from airgapped systems are cited (LEDs, fan noise, EM emissions), underscoring that physical isolation is not perfect.

Primes, Structure, and Harmonic Analysis

• Discussion touches on Ulam/Sacks spirals as visual patterns of primes; these are long known and used mainly as striking illustrations.
• Analytic number theory and the Riemann zeta function link primes to harmonic analysis; the new result is framed as a significant tightening of bounds, not an immediate path to breaking cryptography.
• Some see primes’ structure as conceptually simple yet computationally irreducible; others emphasize the deep unsolved complexity of error terms and distribution.

Complexity and Open Problems

• Integer factorization is described as in NP and co-NP, with its exact complexity (e.g., NP-complete or NP-intermediate) still unknown.
• Claims that “factoring breakthrough implies P=NP” are challenged as incorrect or at least unproven.