Introduction to Modular Arithmetic and Public Key Cryptography

What is modular arithmetic? Modular arithmetic is arithmetic with the remainders upon division by a fixed number n. It is based upon the idea that the remainder of the sum/difference/product of two numbers is the remainder of the sum/difference/product of the remainders. For example, if n=5, (31+7)%5 = 38%5 = 3, and (31%5+7%5)%5 = 1+2=3

So, what is arithmetic mod n? Our “numbers” are 0, 1, 2, . . . (n-1). We add, subtract as usual, but subtract or add n as necesary to get an answer between 0 and n-1. For multiplication, the process is similar; multiply the two numbers together, and then take the remainder dividing by n.

Some examples, mod n = 6. 4+3=7 -6=1 3 – 5 = -2 + 6 = 4 4 * 5 = 20 % 6 = 2 WHAT ABOUT DIVISION? ? ? Let us say there is an x such that x * 2 = 1. Let us also say there is a y such that y * 3 = 1

Some examples, mod n = 6. 4+3=7 -6=1 3 – 5 = -2 + 6 = 4 4 * 5 = 20 % 6 = 2 WHAT ABOUT DIVISION? ? ? Let us say there is an x such that x * 2 = 1. Let us also say there is a y such that y * 3 = 1 Then x * y * 2 * 3 = 6 xy = 1.

Some examples, mod n = 6. 4+3=7 -6=1 3 – 5 = -2 + 6 = 4 4 * 5 = 20 % 6 = 2 WHAT ABOUT DIVISION? ? ? Let us say there is an x such that x * 2 = 1. Let us also say there is a y such that y * 3 = 1 Then x * y * 2 * 3 = 6 xy = 1. But 6*anything = 0!!!

Some examples, mod n = 6. 4+3=7 -6=1 3 – 5 = -2 + 6 = 4 4 * 5 = 20 % 6 = 2 WHAT ABOUT DIVISION? ? ? Let us say there is an x such that x * 2 = 1. Let us also say there is a y such that y * 3 = 1 Then x * y * 2 * 3 = 6 xy = 1. But 6*anything = 0!!! So 1 = 0 ? !? !?

Can we divide if n is a prime? Yes, but. . . From now on, our modulus will be a prime p. We will show to divide in arithmetic mod p. Devious method! We will need a result, called the “extended euclidean algorithm” to pull this off. But first, we need the euclidean algorithm to understand what is going on. The euclidean algorithm computes the greatest common divisor of two positive integers.

Elementary Euclidean Algorithm

Extended Euclidean Algorithm

What can we do with the egcd? Given two numbers a, b, the extended euclidean algorithm finds their gcd g and two numbers s and t such that as + bt = g. In particular, if a and b have no common factors (aside from 1) (i. e. they are “relatively prime”), we can find two numbers s, t such that as + bt = 1 For modular division, if p is prime, given a, we can find s and t such that as +tp = 1. s is then the “multiplicative inverse” of a (suitably reduced, if necesary).

Some more, strange, results.

Another result Chinese remainder theorem: Given m 1, m 2, m 3, . . mk and a 1, a 2 a 3, ak, where The mi, mj are positive, pairwise relatively prime The ai are positive integers less than mi respectively. Then, there exists a b such that mi divides b-ai for each i. If we require that b be less than the product of the mi, then this b is unique. (Proof in next slide)

Proof of Chinese Remainder Theorem Suffices to take k=2 by induction. Thus, need to prove that, for 0 < a < m and 0 < b < n if m and n are relatively prime, there exists a unique u between 0 and mn such that u % m = a; u % n = b Since m and n are relatively prime, there exist p , q such that pm + qn = 1. Then bpm + aqn % mn = u satisfies all the conditions.

The RSA Theorem

Proof of the RSA Theorem

How RSA works Take two primes, p, q, let n=pq Chose an e, relatively prime to (p-1)(q-1). Find a d such that de – k(p-1)(q-1) = 1 with the extended euclidean algorithm: then de = 1+k(p-1)(q-1) “Publish”, n, e as public key. Encryption: raise a to the e-th power Decryption: raise result to the d-th power.

“Efficient” powering to compute a^n

Another Crypto-system: Diffie. Hellman key exchange Let p be a large prime, s a number between 2 and p-2; p and s are “publicly known”. Each person has a private key a. Whenever two people want to exchange messages, they send each other s^a mod p They raise the number they receive to their private key power mod p, and have an exchange key for a symmetric crypto-system.

Another Crypto System: El-Gamal As before, let p be a large (publicly known) prime number, s some number between 2 and p-2. Each person chooses a private key e and “publishes” E = s raised to the e-th power mod p. To send message x, we first generate a “session key” k, and send t = s^k and y = E^k x mod p We decrypt by computing t^(-e) y = x mod p