1. Modular Arithmetic
ICS 6D
Sandy Irani

2. DIV and MOD functions d an integer d ≥ 1 n an integer
There are unique integers
q for “quotient”
r for “remainder”
Such that
r ∈ {0, 1, 2,…,d-1}
n = d·q + r

3. DIV and MOD functions for n < 0
q = floor(n/d)
r = n - q·d
Example: n = -25, d = 6

4. DIV and MOD functions for n < 0
q = floor(n/d)
r = n - q·d
Example n = -75, d = 12

5. DIV and MOD functions for n < 0
q = 0
while (r < 0)
r = r + d
q = q - 1
Example n = -25, d = 6

6. DIV and MOD functions for n < 0
q = 0
while (r < 0)
r = r + d
q = q -1
Example n = -75, d = 12

7. Modular Arithmetic “Mod n” is a function from ℤ to {0, 1, …, n-1}
Addition mod n:
(x + y) mod n
Multiplication mod n: xy mod n

8. Modular Arithmetic
In computing arithmetic expressions mod n, can compute partial results mod n and the result is the same:
((x mod n) + (y mod n)) mod n = (x + y) mod n
( ) mod 5 =
((x mod n) · (y mod n)) mod n = (x · y) mod n
(158 · 219) mod 5 =

9. Modular Arithmetic
( ) mod 11
(56· ) mod 7

10. Modular Arithmetic
210 mod 7 = (25 mod 7) (25 mod 7) mod 7

11. Modular Arithmetic Any multiple of n acts like 0 mod n:

12. Rings
A ring is a closed mathematical system with addition and multiplication operations that
Obeys certain laws (associative, distributive, etc.)
Has identities:
0 + x = x
1·x = x
The elements of a ring can be different kinds of objects:
Polynomials, sequences, numbers, etc.

13. The ring ℤn The ring ℤn is the set {0, 1, 2, …, n-1} along with
Addition mod n
Multiplication mod n
Example: ℤ5 + x
01234
01234

14. Equivalence mod n
x mod n = y mod n ↔ (x-y) = integer multiple of n ↔ x ≡ y mod n ↔ “x is equivalent to y mod n”
Example: n = 5 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9