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Post-Quantum Cryptography & Shor's Algorithm Qubit Simulator (2026)

Execute Shor's quantum period-finding algorithm ($a^r \equiv 1 \pmod N$) step-by-step on toy RSA moduli and calculate logical vs physical surface-code qubits required to break RSA-2048 and ECC P-256.

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Post-Quantum Cryptography & Shor's Algorithm Qubit Simulator — Interactive Console
Runs locally in your browser • Instant output
Logical vs Physical Qubits
4,099 Logical Qubits
~20,000,000 Physical (d=27)
Post-Quantum Security
0 bits (Broken in polynomial time O((log N)^3))
2.7 × 10^10 Toffoli gates (~8 hours on Surface Code)
"Harvest Now, Decrypt Later" Risk
CRITICAL (Decryptable by CRQC)
NIST Fix: ML-KEM-768 (FIPS 203) + ML-DSA-65 (FIPS 204)
NIST FIPS 203 (ML-KEM-768) Drop-In Migration Snippet (Nginx & OpenSSH 9.9+)
# Nginx / OpenSSL 3.5+ Post-Quantum Hybrid Key Exchange
ssl_protocols TLSv1.3;
ssl_ecdh_curve X25519MLKEM768:X25519:secp384r1;

# OpenSSH 9.9+ sshd_config (Defends against Harvest-Now-Decrypt-Later)
KexAlgorithms mlkem768x25519-sha256,sntrup761x25519-sha512@openssh.com
Ready
Embed / Cite This Tool (Markdown & HTML)
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Blog / Documentation HTML Citation<a href="https://www.zerosuniverse.com/tools/quantum-shors-algorithm-rsa-simulator/">Post-Quantum Cryptography & Shor's Algorithm Qubit Simulator — ZerosUniverse</a>

2026 Quick-Reference Cheat Sheet & Benchmark Table: Post-Quantum Cryptography & Shor's Algorithm Qubit Simulator

Quick Answer & 2026 Technical Summary (quantum computing rsa qubit calculator)Updated 2026 Standard

RSA security relies on the hardness of factoring a large semiprime $N = p \times q$. Number theory shows that factoring $N$ can be reduced to finding the even period $r$ of the modular exponentiation function $f(x) = a^x \bmod N$ (such that $a^r \equiv 1 \pmod N$). Once a quantum computer finds $r$ using Quantum Phase Estimation and the Quantum Fourier Transform (QFT), a classical computer immediately extracts the factors via Euclidean $\gcd(a^{r/2} - 1, N)$ and $\gcd(a^{r/2} + 1, N)$. Use this interactive quantum computing rsa qubit calculator above to test shors algorithm simulator online, qubits needed to break rsa 2048, and nist post quantum cryptography ml-kem ml-dsa locally in your browser with zero server uploads.

Target Keyword Spec: quantum computing rsa qubit calculator | Modules: Interactive Shor's Period-Finding ($a^x \bmod N$) Engine • Logical vs Physical Surface-Code Qubit Calculator • Grover's Algorithm vs Symmetric AES-128/256 Analyzer
Primary Focus: quantum computing rsa qubit calculator
Core Capability: shors algorithm simulator online
Privacy Mode: 100% Client-Side (Zero Upload)
Technical Parameter / ModuleStandard / Keyword SpecArchitecture & Validation RuleOperational Use Case (2026)
Interactive Shor's Period-Finding ($a^x \bmod N$) Engineshors algorithm simulator onlinePick a semiprime $N = p \times q$ (15, 21, 33, 35, 55, 77, 91, 143, 221) an...'Harvest Now, Decrypt Later' (HNDL) Threat Modeling
Logical vs Physical Surface-Code Qubit Calculatorqubits needed to break rsa 2048Calculates the $2n + 3$ Beauregard logical qubits, Toffoli gate depth ($O(n...Quantum Computing & Cryptography University Education
Grover's Algorithm vs Symmetric AES-128/256 Analyzernist post quantum cryptography ml-kem ml-dsaDemonstrates why Shor's algorithm devastates asymmetric RSA/ECC (exponentia...Enterprise TLS & PKI Post-Quantum Bandwidth Planning
Tokenizer & Model Architecturetiktoken (o200k_base / cl100k_base) + GGUF1 Token ≈ 0.75 English Words (~4 Chars)Calibrated for 2026 Frontier & Open-Weight LLMs
Context Window & KV Cache Scaling8k / 32k / 128k / 1M+ Token ContextsFP16 vs Q8_0 vs Q4_K_M QuantizationAccounts for FlashAttention & prompt caching
Inference Cost & Throughput MetricUSD per 1M Input / Cached / Output TokensMemory Bandwidth (GB/s) ÷ Model Size (GB)Optimizes self-hosted GPU vs cloud API ROI
In-Depth ZerosUniverse Tutorial

Why Quantum Computing Threatens RSA Encryption & NIST Post-Quantum Guide

Read our complete step-by-step editorial guide, architecture breakdown, and defensive best practices on ZerosUniverse.

Read Full Guide

How to Use Post-Quantum Cryptography & Shor's Algorithm Qubit Simulator

01

Select an Algorithm to Audit in the CRQC Qubit Estimator

Pick RSA-2048, RSA-4096, ECC P-256, AES-128, AES-256, or ML-KEM-768 (Kyber) to inspect logical qubits, physical qubits, and quantum resistance status.

02

Choose a Semiprime N & Coprime Base a in the Shor's Simulator

Select a composite number $N = p \times q$ (e.g., $N = 15$, $21$, $35$, or $91$) and a coprime base $a$ in the interactive period-finding lab.

03

Inspect the Modular Exponentiation Wave & Period r

Trace the repeating sequence of $f(x) = a^x \bmod N$ across $x = 0, 1, 2, \dots$ to see how quantum superposition and QFT isolate the period $r$.

04

Review Classical GCD Factor Extraction & NIST PQC Replacements

Verify how $\gcd(a^{r/2} - 1, N)$ and $\gcd(a^{r/2} + 1, N)$ reveal the secret prime factors $p$ and $q$, then review the NIST FIPS 203/204 migration guide.

Key Capabilities & Technical Architecture

Interactive Shor's Period-Finding ($a^x \bmod N$) Engine

Pick a semiprime $N = p \times q$ (15, 21, 33, 35, 55, 77, 91, 143, 221) and base $a$ to visualize the modular exponentiation wave, Quantum Fourier Transform (QFT) period $r$, and $\gcd(a^{r/2} \pm 1, N)$ factor extraction.

Logical vs Physical Surface-Code Qubit Calculator

Calculates the $2n + 3$ Beauregard logical qubits, Toffoli gate depth ($O(n^3)$), and Gidney-Ekerå physical surface-code qubits needed to factor RSA-1024, RSA-2048, RSA-4096, and ECDSA P-256.

Grover's Algorithm vs Symmetric AES-128/256 Analyzer

Demonstrates why Shor's algorithm devastates asymmetric RSA/ECC (exponential speedup) while Grover's algorithm only halves symmetric key bit-security ($O(2^{n/2})$), leaving AES-256 and SHA-384 quantum-safe.

NIST FIPS 203 / 204 / 205 PQC Migration Matrix

Compare public-key, ciphertext, and signature byte sizes across classical RSA/ECDSA and NIST-standardized Post-Quantum Lattice/Hash algorithms (ML-KEM-768 Kyber, ML-DSA-65 Dilithium, SLH-DSA Sphincs+).

Practical Use Cases

'Harvest Now, Decrypt Later' (HNDL) Threat Modeling

Evaluate whether long-lived confidential data encrypted over classic TLS/RSA/ECDHE must transition immediately to hybrid X25519MLKEM768 key exchange.

Quantum Computing & Cryptography University Education

See exactly how integer factorization reduces to finding the period $r$ of $f(x) = a^x \bmod N$ and why odd periods or $a^{r/2} \equiv -1 \pmod N$ require selecting a new base $a$.

Enterprise TLS & PKI Post-Quantum Bandwidth Planning

Compare the network packet overhead of migrating from 32-byte X25519 keys to 1,184-byte ML-KEM-768 public keys and 3,309-byte ML-DSA-65 signatures.

Frequently Asked Questions (FAQs)

How does Shor's algorithm break RSA encryption?+

RSA security relies on the hardness of factoring a large semiprime $N = p \times q$. Number theory shows that factoring $N$ can be reduced to finding the even period $r$ of the modular exponentiation function $f(x) = a^x \bmod N$ (such that $a^r \equiv 1 \pmod N$). Once a quantum computer finds $r$ using Quantum Phase Estimation and the Quantum Fourier Transform (QFT), a classical computer immediately extracts the factors via Euclidean $\gcd(a^{r/2} - 1, N)$ and $\gcd(a^{r/2} + 1, N)$.

Why is there such a huge difference between 4,099 logical qubits and ~20 million physical qubits for RSA-2048?+

Physical superconducting or trapped-ion qubits are noisy and suffer from decoherence and gate errors when executing billions of sequential Toffoli gates. To run an 8-hour Shor's circuit without a single bit-flip ruining the calculation, thousands of physical qubits must be entangled into a fault-tolerant topological Surface Code patch to represent just 1 error-corrected logical qubit (as modeled by Gidney and Ekerå).

Is Elliptic Curve Cryptography (ECC P-256 / Ed25519) safer against quantum computers than RSA-2048?+

No—in fact, ECC requires fewer logical qubits to break than RSA! Because ECC keys are much shorter (256 bits vs 2048 bits) and the Elliptic Curve Discrete Logarithm Problem (ECDLP) is also solved by Shor's algorithm, breaking P-256 requires roughly ~2,330 logical qubits compared to ~4,099 logical qubits for RSA-2048.

Does quantum computing break AES-256 or SHA-256?+

No. Symmetric ciphers (AES) and cryptographic hash functions (SHA-2/SHA-3) do not rely on algebraic period-finding structures, so Shor's algorithm does not apply to them. Grover's quantum search algorithm provides only a quadratic speedup ($O(\sqrt{N})$), which effectively halves the key length: AES-256 retains 128 bits of post-quantum security ($2^{128}$ quantum operations), which remains computationally unbreakable.

What are the official NIST Post-Quantum Cryptography standards (FIPS 203, 204, 205)?+

In August 2024, NIST finalized its core Post-Quantum Cryptography standards based on Module-Lattice and stateless hash problems: FIPS 203 (`ML-KEM`, formerly CRYSTALS-Kyber) for general encryption and TLS key establishment; FIPS 204 (`ML-DSA`, formerly CRYSTALS-Dilithium) for digital signatures; and FIPS 205 (`SLH-DSA`, formerly SPHINCS+) for stateless hash-based signatures.