Factorization
Factorization is the decomposition of a mathematical object, such as an integer or joint distribution, into simpler multiplicative factors.
Hacking Cryptography
Integer factorization is the problem of decomposing an integer into a product of prime factors.
- How it is used
- Supports RSA and other classical public-key systems by making multiplication easy while keeping factor recovery difficult.
- How deep it goes
- Defines decomposition, illustrates trial division and classical methods, and stops short of detailed factoring algorithms or post-quantum replacements.
- Examples
- Uses 764,512, products of large primes, historical factor tables, RSA equations, batch GCD, and quantum factoring demonstrations.
- What you can do after
- Evaluate RSA's factorization assumption and identify weak randomness, deterministic encryption, and short-private-exponent risks.
Real-World Cryptography
The mathematical problem of decomposing an integer into its prime factors.
- How it is used
- Explains factorization as the hardness assumption protecting RSA's private exponent.
- How deep it goes
- Traces factorization through RSA's modulus, group order, and private-exponent derivation.
- Examples
- Uses RSA's N = p × q modulus and its public-exponent encryption and private-exponent decryption flow.
- What you can do after
- Evaluate/understand why keeping RSA's prime factors secret protects private-key derivation.
Software Security for Developers
Integer factoring is the decomposition of an integer into a product of prime numbers.
- How it is used
- Underpins RSA security by making recovery of prime factors computationally difficult.
- How deep it goes
- Defines the security problem and its limits but defers mathematical details and algorithmic proofs.
- Examples
- Uses the RSA-250 challenge and large RSA key-size recommendations as concrete reference points.
- What you can do after
- Evaluate RSA's security assumptions and recognize why future factoring advances could invalidate current protection.
Read how Software Security for Developers explains factorization →
Related concepts
- Shor's algorithmaddressed by · Shor's algorithm is a quantum algorithm that efficiently factors integers and solves discrete logarithms using quantum period finding.
- RSAused by · RSA is a public-key cryptosystem based on the computational difficulty of factoring large composite integers and supporting encryption and digital signatures.
- Markov propertyconfigured by · The Markov property is the condition that, given the current state, the distribution of future states is independent of the process’s earlier history.
- public key cryptographyused by · Public-key cryptography is asymmetric cryptography that uses mathematically related public and private keys for encryption, decryption, signing, and verification.
- conditional probabilityuses · Conditional probability is the probability of an event or variable value given that specified information, an event, or another variable’s value is known.
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