CSE 20
DISCRETE MATH
Prof. Shachar Lovett
http://cseweb.ucsd.edu/classes/wi15/cse20-a/
Clicker
frequency:
CA
Todays topics
• Modular arithmetic
• Section 6.2 in Jenkyns, Stephenson
Recall: equivalence relations
• Saw two equivalent definitions
• xRy is an equivalence relation if…
• Symmetric: ∀𝑥, 𝑦, 𝑥𝑅𝑦 ⇔ 𝑦𝑅𝑥
• Reflexive: ∀𝑥, 𝑥𝑅𝑥
• Transitive: ∀𝑥, 𝑦, 𝑧, 𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧 → 𝑥𝑅𝑧
• Equivalently: there is a partition of the universe to
equivalence classes, and xRy iff x,y are in the same class
• Can be phrased as: ∃𝐶, 𝐶𝑙𝑎𝑠𝑠: 𝑈 → 𝐶 such that
𝑥𝑅𝑦 ⇔ 𝐶𝑙𝑎𝑠𝑠 𝑥 = 𝐶𝑙𝑎𝑠𝑠(𝑦)
Today: modular arithmetic
• A very important (and very useful) example of an
equivalence relation
• Recall our example of {even,odd} classes:
• 𝑥𝑅𝑦 ⇔ 2| 𝑥 − 𝑦 ⇔x,y are both even, or both odd
• Modular arithmetic generalizes this from 2 to any divisor
Modular arithmetic: definition
• Fix 𝑚 ≥ 2
• Define a relation: integers x,y are equivalent modulo m, if
𝑥 𝑚𝑜𝑑 𝑚 = 𝑦 𝑚𝑜𝑑 𝑚
• Equivalently, 𝑚|𝑥 − 𝑦
• Defines a partition of Z to m equivalence classes
𝑎 𝑚 = {𝑥 ∈ 𝑍: 𝑥 𝑚𝑜𝑑 𝑚 = 𝑎}
where 𝑎 ∈ {0,1, … , 𝑚 − 1}
Modular arithmetic: examples
• 𝑎 𝑚 = {𝑥 ∈ 𝑍: 𝑥 𝑚𝑜𝑑 𝑚 = 𝑎}
• What is [0]2?
A. All integers
B. 0
C. Even numbers
D. Odd numbers
E. Other
Modular arithmetic: examples
• 𝑎 𝑚 = {𝑥 ∈ 𝑍: 𝑥 𝑚𝑜𝑑 𝑚 = 𝑎}
• What is [1]2?
A. All integers
B. 1
C. Even numbers
D. Odd numbers
E. Other
Modular arithmetic: examples
• 𝑎 𝑚 = {𝑥 ∈ 𝑍: 𝑥 𝑚𝑜𝑑 𝑚 = 𝑎}
• What is [0]3?
A. All integers
B. All integers divisible by 3
C. All integers whose value mod 3 is equal to 1
D. All integers whose value mod 3 is equal to 2
E. Other
Modular arithmetic: examples
• 𝑎 𝑚 = {𝑥 ∈ 𝑍: 𝑥 𝑚𝑜𝑑 𝑚 = 𝑎}
• What is [1]3?
A. All integers
B. All integers divisible by 3
C. All integers whose value mod 3 is equal to 1
D. All integers whose value mod 3 is equal to 2
E. Other
Modular arithmetic: examples
• 𝑎 𝑚 = {𝑥 ∈ 𝑍: 𝑥 𝑚𝑜𝑑 𝑚 = 𝑎}
• What is [4]3?
A. All integers
B. All integers divisible by 3
C. All integers whose value mod 3 is equal to 1
D. All integers whose value mod 3 is equal to 2
E. Other
Modular arithmetic: notation
• Notation: to say that 𝑥 ∈ 𝑎 𝑚 , we typically write
𝑥 ≡ 𝑎 𝑚𝑜𝑑 𝑚
• More generally, for any two integers x,y we can write
𝑥 ≡ 𝑦 𝑚𝑜𝑑 𝑚
which means that x,y belong to the same equivalence class
modulo m, which means 𝑚|𝑥 − 𝑦.
Modular addition and multiplication
• Why is modular arithmetic so important? Because we can
operate directly on the equivalence classes, adding and
multiplying them (for a fixed modulus m)
• Definition: for any fixed 𝑚 ≥ 2,
𝑎
𝑚
+ 𝑏
𝑚
=
𝑎 + 𝑏 𝑚𝑜𝑑 𝑚
∗ 𝑏
𝑚
=
𝑎 ∗ 𝑏 𝑚𝑜𝑑 𝑚
𝑚
And
𝑎
𝑚
• Theorem: this is well defined
• What does this even mean???
𝑚
Modular addition
• Theorem: 𝑎 𝑚 + 𝑏 𝑚 =
𝑎 + 𝑏 𝑚𝑜𝑑 𝑚
𝑚
• Interpretation: if we take any 𝑥 ∈ 𝑎 𝑚 , and any 𝑦 ∈ 𝑏 𝑚 ,
then it is always true that 𝑥 + 𝑦 ∈
𝑎 + 𝑏 𝑚𝑜𝑑 𝑚
𝑚
• Equivalently:
𝑥 𝑚𝑜𝑑 𝑚 + 𝑦 𝑚𝑜𝑑 𝑚
𝑚𝑜𝑑 𝑚 = 𝑥 + 𝑦 𝑚𝑜𝑑 𝑚
Modular addition: proof
• Theorem: 𝑎 𝑚 + 𝑏 𝑚 =
𝑎 + 𝑏 𝑚𝑜𝑑 𝑚
𝑚
• Proof: we need to show
∀𝑥 ∈ 𝑎
𝑚, 𝑦
∈ 𝑏
𝑚,
𝑥 + 𝑦 𝑚𝑜𝑑 𝑚 = 𝑎 + 𝑏 𝑚𝑜𝑑 𝑚
Let 𝑥 = 𝑝𝑚 + 𝑎, 𝑦 = 𝑞𝑚 + 𝑏 and 𝑎 + 𝑏 = 𝑟𝑚 + 𝑐, with
𝑎, 𝑏, 𝑐 ∈ {0, … , 𝑚 − 1}. Then:
𝑥+𝑦 = 𝑝+𝑞 𝑚+ 𝑎+𝑏 = 𝑝+𝑞+𝑟 𝑚+𝑐
So 𝑥 + 𝑦 ∈ 𝑐
𝑚,
and also a + 𝑏 ∈ 𝑐
𝑚.
QED.
Modular addition: examples
• What is 0 2 + 1 2 ?
A.
B.
C.
D.
E.
02
12
21
All integers
Other
Modular addition: examples
• What is 4 7 + 5 7 ?
A.
B.
C.
D.
E.
17
27
72
74
Other
Modular addition: examples
• What is 4 7 + 1 2 ?
A.
B.
C.
D.
E.
57
5 14
72
1 14
Other
Be careful!
• We can only add 𝑎 𝑚 + 𝑏 𝑛 if 𝑛 = 𝑚. Otherwise, this is
not defined!!!
• Also: in the special case of 𝑚 = 2, addition modulo 2
corresponds to XOR
• 0 2+ 0 2= 0 2
• 0 2+ 1 2= 1 2
• 1 2+ 0 2= 1 2
• 1 2+ 1 2= 0 2
Modular multiplication
• Theorem: 𝑎 𝑚 ∗ 𝑏 𝑚 =
𝑎 ∗ 𝑏 𝑚𝑜𝑑 𝑚
𝑚
Modular multiplication: proof
• Theorem: 𝑎 𝑚 ∗ 𝑏 𝑚 =
𝑎 ∗ 𝑏 𝑚𝑜𝑑 𝑚
𝑚
• Proof: we need to show
∀𝑥 ∈ 𝑎
𝑚, 𝑦
∈ 𝑏
𝑚,
𝑥 ∗ 𝑦 𝑚𝑜𝑑 𝑚 = 𝑎 ∗ 𝑏 𝑚𝑜𝑑 𝑚
Let 𝑥 = 𝑝𝑚 + 𝑎, 𝑦 = 𝑞𝑚 + 𝑏 and 𝑎 ∗ 𝑏 = 𝑟𝑚 + 𝑐, with 𝑎, 𝑏, 𝑐 ∈
{0, … , 𝑚 − 1}. Then:
𝑥 ∗ 𝑦 = 𝑝𝑚 + 𝑎 ∗ 𝑞𝑚 + 𝑏
= 𝑝𝑞𝑚2 + 𝑎𝑞𝑚 + 𝑏𝑝𝑚 + 𝑎𝑏
= 𝑝𝑞𝑚2 + 𝑎𝑞𝑚 + 𝑏𝑝𝑚 + 𝑟𝑚 + 𝑐
= 𝑚 𝑝𝑞𝑚 + 𝑎𝑞 + 𝑏𝑝 + 𝑟 + 𝑐
So 𝑥 ∗ 𝑦 ∈ 𝑐
𝑚,
and also a ∗ 𝑏 ∈ 𝑐
𝑚.
QED.
Applications of modular arithmetic
• For a fixed 𝑚 ≥ 2, we can add, multiply and take modulo
m, and the value (modulo m) will stay invariant
• This is very useful; many algorithmic applications
• Example 1: verifying calculations
• Prove that 12294793 ∗ 2329373 ≠ 92739273
• Hint: use values modulo 10
Applications of modular arithmetic
• Example 1: prove that 12294793 ∗ 2329373 ≠ 92739273
• Solution: compute values modulo 10
• 12294793 ≡ 3 𝑚𝑜𝑑 10
• 2329373 ≡ 3 𝑚𝑜𝑑 10
• 92739273 ≡ 3 𝑚𝑜𝑑 10
• But 3 ∗ 3 ≡ 9 𝑚𝑜𝑑 10 , so the equation cannot be correct
Applications of modular arithmetic
• Example 2: number theory
• Theorem: there are no 𝑥, 𝑦 ∈ 𝑍 such that 𝑥 2 + 𝑦 2 = 1234567
• Hint: use values modulo 4
Applications of modular arithmetic
• Theorem: there are no 𝑥, 𝑦 ∈ 𝑍 such that 𝑥 2 + 𝑦 2 =
1234567
• Hint: use values modulo 4
• Solution: for any 𝑥 ∈ 𝑍, 𝑥 2 ∈ 0 4 or 𝑥 2 ∈ 1 4 .
• There are 4 cases to check:
•
0
4
∗ 0
4
= 0 4; 1
4
∗ 1
4
= 1 4; 2
4
∗ 2
4
= 0 4; 3
4
∗ 3
4
= 1 4;
• So, 𝑥 2 + 𝑦 2 𝑚𝑜𝑑𝑢𝑙𝑜 4 can either be 0 4 , 1 4 or 2 4
• But 1234567 ≡ 3 (𝑚𝑜𝑑 4), so they cannot be equal!
Applications of modular arithmetic
• Cryptography
• Your online transactions are encrypted, based on modular
arithmetic
• RSA public key crpytosystem:
• Very large m, product of two large primes m=p*q
• Message: 𝑥 ∈ {0,1, … , 𝑚 − 1}
• Encryption: 𝑥 𝑒 𝑚𝑜𝑑 𝑚
• Security is based on the assumption that factorization is hard