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Eucliadean Algorithm

Divide and shift left until remainder is zero

Chinese Remainder Theorem

1. Multiply the inner numbers or don't if it's 2.

2. Set them equal to 1mod(the others)

3. Figure out x = 1mody (how many times you have to multiplg x and then mod y by the get a remainder of 1)

4. Multiply that by the mod and the original number

5. Profit

Fermat's Little Theorem

Given a prime p, and p doesn't divide a then a^(p - 1) = 1(modp)

Diffe-Helman

Given: (G, q, g)

A generates an x in Z1 g^x-> B generates a y in Zq <- (g^y)^x = g^yx - shared key > (g^x)^y = g^xy

AUTH: SIGN SKA(G^X AND G^Y)

MITM DHE

(1) Alice picks x ← Zq and sends gx to Bob

(2) Oscar gets gx from Alice, picks x′ ← Zq and sends gx′ to Bob (3) Bob picks y ← Zq and sends gy to Alice

(4) Oscar gets gy from Bob, picks y′ ← Zq and sends gy′ to Alice

Bob thinks the key is kB = (gx′ )y and Alice thinks the key is kA = (gy′ )x.

PUBLIC-KEY CRYPTOGRAPHY

Public Key Security Game

RSA

Hybrid Encryption

Digital Signatures

Sig-Forge

RSA-Sign

Attacks on RSA

Schnorr

Uniqueness CRT

Cert Chains

Cert Chains Continued

Hash Chains

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