Would Proving the Riemann Hypothesis Break RSA?

Would a proof of the Riemann Hypothesis break RSA encryption? The rumour is old. For the known consequences and the algorithms examined here, the answer is no, but the connection is more interesting than the rumour suggests. A hypothesis about zeros of L-functions could touch RSA in two places: factoring a public key, and generating its secret primes. The factoring algorithms we ran on small test keys never consult the hypothesis; what decides their success is the size of the prime factors and the work they are allowed. Key generation is different. If the Extended Riemann Hypothesis holds, Miller's test proves a number prime once every prime base up to an explicit bound has been tried. For the 1024-bit primes inside an RSA-2048 key, the sharpest bound we know needs 41,749 bases, about a thousand times the 40 random rounds of a generous probabilistic test. Proving the extended hypothesis would make this existing, expensive test unconditional; the ordinary Riemann Hypothesis alone does not supply that guarantee. The mathematics is known; we put numbers on it, with small experiments anyone can rerun in minutes. Version v0.3.1 is the article version prepared for The Mathematical Intelligencer (expository essay; same experiments and numbers as v0.2, text rewritten for a general mathematical audience). Files. Kulma_2026_RH_RSA_Intelligencer_v0.3.1.pdf: the article (preprint v0.3.1). Kulma_2026_RH_RSA_LabRSA_v0.3.1.zip: frozen snapshot of the Lab RSA repository at tag v0.3.1 (22 experiment scripts exp13-exp34, all artefacts, figure script paper/make_fig1.py, LaTeX source of the article, release gate python lab_rsa_contract.py release, 7 checks). Code under the MIT licence, text under CC BY 4.0. All moduli are generated inside the scripts; the harness accepts no external keys.