Cryptography & Security

Keeping secrets safe: modular arithmetic, RSA, entropy, hashing and error-correcting codes

Cryptography & Security brings together the formulas that keep data private, verify identities, and detect mistakes, from modular arithmetic and RSA to entropy, hashing, and error-correcting codes. It’s useful for students revising for exams, as well as anyone working with secure systems, passwords, or communications, because these formulas explain how modern protection works and how strong it really is.

Modular Exponentiation

NEWIntermediate

c = m^e \bmod n

Core operation of RSA encryption — raise a message to a power, modulo n.

RSA Decryption

m = c^d \bmod n

Recovers the message using the private exponent d.

Euler's Totient

NEWIntermediate

\varphi(n) = (p-1)(q-1)

Counts integers below n coprime to it, for n = pq — used to make RSA keys.

Shannon Entropy

H = -\sum_i p_i \log_2 p_i

Average information content — and the theoretical strength of a random source.

Password Entropy

NEWIntermediate

H = L \log_2 R

Bits of security from password length and character-set size.

Key Space Size

NEWIntermediate

K = 2^b

Number of possible keys for a b-bit key.

Birthday Bound

n \approx 1.177\sqrt{2^b}

Collisions become likely after about √ of the hash space.

Hamming Distance

d(x,y) = \sum_i [x_i \ne y_i]

Number of positions at which two strings differ.

Hamming Code Parity Bits

2^r \ge m + r + 1

Minimum parity bits r needed to protect m data bits.

XOR Cipher

c = m \oplus k

Bitwise XOR of message and key; XOR again with k to decrypt.

Diffie–Hellman Shared Secret

s = B^a \bmod p

Two parties derive the same secret over a public channel.

Brute-Force Time

NEWIntermediate

t = \frac{K}{r}

Time to try every key at a given guess rate.

Avalanche Effect Target

E[\Delta] = \frac{n}{2}

A good hash flips about half its output bits when one input bit changes.

Caesar Cipher Shift

c = (m + k) \bmod 26

Shift each letter forward by k positions.

Collision Probability

P \approx \frac{n^2}{2N}

Approximate chance of a collision among n items in space N.