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The Most-Taught and Most-Misimplemented Algorithm
RSA is likely the first public-key cryptosystem most developers encounter, and it is routinely misused. The textbook version — compute m^e mod n to encrypt, m^d mod n to decrypt — is broken in practice due to algebraic structure that allows attacks without ever factoring the key. Real RSA requires carefully designed padding schemes. Understanding why reveals that secure cryptography is not just about the algorithm but about every detail of how it is applied.


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The Mathematical Foundation: Euler's Theorem
Euler's theorem states that for any integer a coprime to n, a raised to the power of Euler's totient phi(n) is congruent to 1 modulo n. RSA uses this to construct a pair of exponents e and d such that encrypting and then decrypting returns the original me

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RSA Key Generation Step by Step
Generate two distinct large primes p and q, each typically 1024 bits for a 2048-bit key. Compute n = p*q — this is the modulus, public. Compute phi(n) = (p-1)*(q-1) — this is kept secret. Choose the public exponent e = 65537, a Fermat prime chosen for eff

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Why Factoring Breaks RSA
If an attacker can factor n into p and q, they can compute phi(n) = (p-1)*(q-1) and then recover d = e^(-1) mod phi(n) — the private key. The security of RSA reduces directly to the hardness of factoring n. For a 2048-bit RSA modulus, the best known facto

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Why Textbook RSA Is Broken
Textbook RSA has a multiplicative homomorphism: if c1 = m1^e mod n and c2 = m2^e mod n, then c1*c2 mod n = (m1*m2)^e mod n. An attacker who sees a ciphertext can multiply it by a chosen value, ask for it to be decrypted, and recover information about the 

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RSA-2048 is being phased out of many standards not because it has been broken, but because the sub-exponential scaling of the Number Field Sieve means that as computers become faster and factoring research advances, the safety margin shrinks. ECDSA and Ed25519 provide equivalent or better security with keys 10 to 15 times smaller, which also makes certificate chains and TLS handshakes smaller and faster.


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Go Deeper: Elliptic Curve Cryptography
RSA's security rests on the hardness of factoring — but there is a newer approach to public-key cryptography that gives the same security with keys an order of magnitude smaller, using the algebraic geometry of elliptic curves. Understanding ECC from its geometric foundations reveals why 256 bits is enough, and why the choice of curve parameters matters enormously for security.


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https://slatesource.com/s/1011
Same security as RSA with a 10x smaller key — here is why the geometry works.

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https://crypto.stanford.edu/~dabo/cryptobook/
A Graduate Course in Applied Cryptography (Boneh and Shoup)