CS 161 - Lecture 1

RSA : Key generation

1.Choose two distinct prime numbers p and q. 2.Compute n = pq. 3.Compute φ(n) = φ(p)φ(q) = (p − 1)(q − 1) = n - (p + q -1). 4.Choose an integer e such that 1 < e < φ(n) and gcd(e, φ(n)) = 1; i.e., e and φ(n) are coprime. (e is released as the public key exponent.) 5. Solve for d given d⋅e ≡ 1 (mod φ(n)) (d is kept as the private key exponent.)

Digital Certificates

A technology used to associate a user's identity to a public key. Certificate has minimum : name, verification key, expiration date, signed by a certificate authority if Signing Key Disclosed -people can sign for you what can you do? revocation

composite number

A whole number greater than 1 that has more than 2 factors.

prime numbers

A whole number greater than 1 with only two factors, 1 and itself (2 is the only even prime number)

RSA: Decryption

Alice can recover m from c by using her private key exponent d via computing

RSA: Encryption

Alice transmits her public key (n, e) to Bob and keeps the private key d secret. Bob then wishes to send message M to Alice. He first turns M into an integer m, such that 0 ≤ m < n by using an agreed-upon reversible protocol known as a padding scheme. He then computes the ciphertext c corresponding to

RSA

An asymmetric algorithm used to encrypt data and digitally sign transmissions. It is named after its creators, Rivest, Shamir, and Adleman, and RSA is also the name of the company they founded together. RSA relies on the mathematical properties of prime numbers when creating public and private keys.

fundamental theorem of arithmetic

Every polynomial equation with degree greater than zero has at least one root in the set of complex numbers. (can be factored into primes)

Bull Run

NSA use to defeat encryption / SSL

relatively prime

Two or more nonzero whole numbers whose greatest common factor is 1

extended euclidean algorithm

computes, besides the greatest common divisor of integers a and b, the integers x and y such that

GCD

greatest common divisor of two non negative, not both zero integers; largest integer that divides both m and n evenly with remainder of zero

Fermat-Euler Theorem

states that if n and a are coprime positive integers, then

number field sieve

the most efficient classical algorithm known for factoring integers larger than 100 digits. exponential running time : record factorization - 232 digit / 768 bit number (in 2009)

Digital signatures

they associate identity (signer) with text if you change the text, the signature is no longer valid not possible to change contracts great but... does the verifier check true value for d,n ? real sign in key associate with correct person? -Digital Certificates what about with large documents (m>n)? -cryptographic hashes what if we both want to encrypt and sign -use two sets of RSA keys

Euler's totient function

φ(n), is an arithmetic function that counts the totatives of n, that is, the positive integers less than or equal to n that are relatively prime to n. : φ(n)=# of positive integer's less than n and relatively prime to n