"Some asymmetric algorithms rely on one-way functions WITH trapdoor, others on one-way functions WITHOUT trapdoor."

* With a trapdoor, a secret shortcut makes the inverse easy for its owner; without one, the inverse is hard for everyone. *

This fundamental distinction divides all public-key cryptography:

With trapdoor means: there exists a secret (the factorization $N = p \cdot q$) that makes the inverse computation easy. Without it, the inverse is infeasible.

Without trapdoor means: the inverse is hard for everyone — there is no shortcut. Security relies on the mathematical structure, not a secret shortcut.

Go deeper:

• Trapdoor function (Wikipedia) — the precise definition of "easy one way, easy back only with a secret".
• Public-key cryptography (Wikipedia) — the whole family this distinction organises.

RSA was invented by Rivest, Shamir, and Adleman in 1978. Its security relies on the difficulty of factoring large numbers $N = p \cdot q$ and computing e-th roots mod N.

The RSA hard problem:

• Forward: $c \equiv m^e \mod N$ — easy (square-and-multiply)
• Inverse: $m \equiv \sqrt[e]{c} \mod N$ — hard without knowing $p$ and $q$

The factoring connection:

• If you can factor $N = p \cdot q$ → you can compute $\varphi(N) = (p-1)(q-1)$ → you can compute $d \equiv e^{-1} \mod \varphi(N)$ → you can decrypt
• Factoring ⇒ e-th root (proven)
• e-th root ⇒ factoring (believed but not proven)
• There might be a way to compute e-th roots without factoring — but nobody has found one

Current recommendations (BSI 2023): RSA modulus $N$ must be at least 3000 bits (practically 3072 bits). Primes $p, q$ are each ~1536 bits (~462 decimal digits), making $N$ about $10^{1000}$.

Go deeper:

• Rivest, Shamir & Adleman — the original 1978 paper (PDF) — "A Method for Obtaining Digital Signatures and Public-Key Cryptosystems", where it all began.
• RSA cryptosystem (Wikipedia) — full scheme, history and attacks.
• The RSA problem (Wikipedia) — why RSA security is the e-th-root problem, not exactly factoring.