1. Discrete Math for CS CMPSC 360 LECTURE 12 Last time: Stable matching
Modular arithmetic
Today:
Exponentiation
Multiplicative Inverses
Extended Euclid’s algorithm
CMPSC
360
11/29/2018

2. Modular exponentiation
Naive algorithm to compute 𝑥 𝑦 mod 𝑚.
mod-exp-naive(𝒙,𝒚,𝒎)
Input: integer 𝒙,
natural number 𝒚,
positive integer 𝒎.
if 𝒚=𝟎 then return 1
else
𝒛= mod-exp-naive(𝒙,𝒚−𝟏,𝒎)
return (𝒛⋅𝒙 mod 𝒎)
11/29/2018

3. I-clicker problem (frequency: BC)
If 𝑦 has 𝑏 bits, how many multiplications mod 𝑚 are performed when we compute 𝑥 𝑦 mod 𝑚 by the naive algorithm?
Find the largest correct bound:
At least √𝑏.
At least 𝑏.
At least 2𝑏.
At least 𝑏 2 .
At least 2 𝑏−1 .
11/29/2018

4. Modular exponentiation
Uses repeated squaring to compute 𝑥 𝑦 mod 𝑚.
mod-exp(𝒙,𝒚,𝒎)
Input: integer 𝒙,
natural number 𝒚,
positive integer 𝒎.
if 𝒚=𝟎 then return 1
else
𝒛= mod-exp(𝒙,𝒚 div 𝟐,𝒎)
if 𝒚 mod 2 =𝟎 then return (𝒛⋅𝒛 mod 𝒎)
else return (𝒛⋅𝒛⋅𝒙 mod 𝒎)
11/29/2018

5. I-clicker problem (frequency: BC)
If 𝑦 has 𝑏 bits, how many multiplications mod 𝑚 are performed when we compute 𝑥 𝑦 mod 𝑚 by repeated squaring?
Find the smallest correct bound:
At most √𝑏.
At most 𝑏.
At most 2𝑏.
At most 𝑏 2 .
At most 2 𝑏−1 .
11/29/2018

6. Definitions
If 𝑥⋅𝑦≡1 (mod 𝑚) then 𝑦 is a multiplicative inverse of 𝑥 modulo 𝑚.
The greatest common divisor of natural numbers 𝑥 and 𝑦, denoted gcd(𝑥,𝑦), is the largest natural number that divides both.
If gcd(𝑥,𝑦)=1 then 𝑥 and 𝑦 are relatively prime (also called coprime).
11/29/2018