Aes

Front 1

What is this? ^

Back 1

A circumflex

Front 2

Why don't we need to inverse on addencryptkey?

Back 2

It's its own inverse

Front 3

How do you convert a 128 bits to a 4 byte by 4 byte matrix?

Back 3

32 43 f6 88 a8 5a 30 8d 31 31 98 a2 e0 37 07 34

becomes

32 a8 31 e0

43 5a 31 37

f6 30 98 07

88 8d a2 34

This happens to the plain text and the key

Front 4

What does the function addroundkey(state, roundkey[0]) do?

Back 4

Take the state key and xor it with the round key

Front 5

What does the function shiftrows() do?

Back 5

Front 6

How do you decrypt AES?

Back 6

Do inverse transformations in reverse order

Every Encryption Transformation Must Be Invertible

Front 7

Why do we want to encrypt using AES?

Back 7

We want diffusion and confusion thats fast and easy to implement on a small device. Want a finite number too, like 256. Its a nice number for computers

Front 8

ℤ

Back 8

{… , −2, −1, 0, 1, 2, … }

All integers

Front 9

ℤ+

Back 9

{1, 2, … }

All positive integers

Front 10

ℚ

Back 10

{ 𝑝/𝑞 | 𝑝, 𝑞 ∈ ℤ, 𝑞 ≠ 0 n}

Rational Numbers

Front 11

∀

Back 11

“for each” or “for every”

Front 12

∈

Back 12

“member of” or “in”

Front 13

∃

Back 13

“there exists”

Front 14

∋

Back 14

“such that”

Front 15

∴

Back 15

“therefore”

Front 16

⇒

Back 16

“implies”

Front 17

What does this notation mean? {G, *}

Back 17

A group of elements in the set G, that's used on the binary operator * (multiplication)

Front 18

What is the first group axiom in terms of {G, *}?

Back 18

Closed under *

Rule: For each and every a and b that is a member of G, a * b is also a member of G

∀𝑎, 𝑏 ∈ 𝐺, 𝑎⦁𝑏 ∈ 𝐺

Front 19

What is the second group axiom in terms of {G,*}?

Back 19

Is Associative

Rule: For each and every a and b and c that is a member of set G, (a*b) * c = a (b*c)

∀𝑎, 𝑏, 𝑐 ∈ 𝐺, 𝑎 ⦁ 𝑏 ⦁ 𝑐 = 𝑎 ⦁ (𝑏 ⦁ 𝑐)

Front 20

What is the third group axiom in terms of {G,*}?

Back 20

Identity Element (with respect to *)

Rule: You can multiply any 2 things in your set and it will be in your set

∃ 𝑒 ∈ G ∋ 𝑎 ⦁ 𝑒 = 𝑒 ⦁ 𝑎 = 𝑎 ∀𝑎 ∈ G

Front 21

What is the fourth group axiom in terms of {G,*}?

Back 21

Left inverse Element (with respect to *)

Rule: For each and every a, that is a member of set G there exists a negative of a that is a member of set G and that there exists a negative of a * a that equal e

∀𝑎 ∈ 𝐺 ∃ 𝑎(hat) ∈ 𝐺 ∋ a(hat) ⦁ 𝑎 = e

Front 22

Prove the first axiom such that {E,+} is true

(Even integers that are added together will still be in the set E)

Back 22

Suppose you have a,b ∈ E

(a,b are members of E)

This implies that

𝑎, 𝑏 ∈ 𝐸 ⇒ ∃𝑚, 𝑛 ∈ ℤ ∋ 𝑎 = 2𝑚, 𝑏 = 2n

(A,b, are members of E. This implies that there exists m,n that are members of the intergers such that a = 2m, and b = 2n)

If this is true, then a + b = 2m + 2n = 2(m+n) are members of E. Thus this implies that a + b are members of E (are even)

Front 23

Prove the second axiom such that {E,+} is true

(even integers a,bc. (a+b) + c == a + (b+c) )

Back 23

Suppose a,b,c ∈ E

Then this implies that a,b,c ∈ ℤ

Then that implies that a + (b+c) = (a+b) + c

Front 24

Prove the third axiom such that {E,+} is true

Back 24

Let e = 0

Suppose a is in E

Then a is in E implies that A is in Z

This implies that a + e = a + 0 = a for every a in E

Front 25

Prove the fourth axiom such that {E,+} is true

Back 25

Front 26

What is an abelian group?

Back 26

Is a group (contains axioms A1-4) as well as axiom 5

Front 27

What is Axiom 5?

Back 27

Commutes

∀a, b ∈ 𝐺, a ⦁ b = b ⦁ a

Front 28

Show a basic example of a cyclic group

Back 28

𝐺 = {… , 𝑔 ^−2, 𝑔^−1, 𝑔^0, 𝑔^1, 𝑔^2, … }

Front 29

What does g stand of in a cyclic group?

Back 29

called the generator

Front 30

What is a Ring?

Back 30

{𝑅, +,×}

The Set of elements R operated on by the binary operators + and x

Front 31

What rules does a ring contain?

Back 31

Axioms 1-5 as well as M1-3

Front 32

What does the first m rule state?

Back 32

Closed Under x

∀a, b ∈ R, a × b ∈ R

Front 33

What does the second m rule state?

Back 33

x Is Associative

∀a,b,c ∈ R, a × b × c = a × b × c

∀a,b,c ∈ R, a(bc) = (ab)c

Front 34

What does the third m rule state?

Back 34

× Distributes Over +

∀ a,b,c ∈ R, 𝑎 (𝑏 + 𝑐) = ab + ac

+ Distributes Over ×

a, b, c ∊ R, (a + b)c = ac + bc

Front 35

What is a commutative ring?

Back 35

A ring that has rules A1-5, M1-3, and an additional rule m4

Front 36

What does the fourth m rule state?

Back 36

× Commutes

∀ a, b ∈ R, a × b = b × a

Front 37

Do 2 x 2 matrices commute?

Back 37

Front 38

Do 2 x 2 diagonal matrices commute?

Back 38

Front 39

What is a divisor?

Back 39

a × b = c, “a and b are divisors of c”

In ℤ, 2 × 3 = 6, “2 and 3 are divisors of 6”

Front 40

What is a zero divisor?

Back 40

a,b != 0, but a x b = 0

So In ℤ8, 2,4 ≠ 0 but 2 × 4 = 0

(2 × 4 mod 8 = 8 mod 8 = 0)

Front 41

What is an integral domain?

Back 41

A commutative ring (A1-5, M1-4) that also has M5 and M6

Front 42

What does the fifth m rule state?

Back 42

Multiplicative Identity (with Respect to x)

∃1 ∈ R ∋ a × 1 = 1 × a = a ∀ a ∈ R

Front 43

What does the sixth m rule state?

Back 43

No Zero Divisors

If a, b ∈ R and a × b = 0, then a = 0 or b = 0

Front 44

What is a Field?

Back 44

{F, +,×}

Set of Elements F

Two Binary Operators

Contains A1-5, M1-6 and M7

Examples:

{ℚ, +,×} – Rational Numbers

{ℝ, +,×} – Real Numbers

{ℑ, +,×} – Imaginary Numbers

Front 45

Is Z a field?

Back 45

No, only 1 and -1 have an inverse (M7)

Front 46

What does the seventh m rule state?

Back 46

∃ Left Multiplicative Inverse (With Respect to x)

∀a ≠ 0 ∈ F, ∃a^−1 ∈ F ∋ a^−1 × a = 1

Front 47

How do you make/find finite fields?

Back 47

Start with infinite set with well known set of elements and operators (+ and x) and then reduce infinite to a finite set (somehow)

Front 48

Give an example of how you'd make/find a finite field

Back 48

Modulus Operator

Z = {… , −2, −1,0,1,2, … }

Becomes

Z7 = {0,1,2,3,4,5,6}

Front 49

What does congruence mean?

Back 49

a mod n = (b mod n)

denoted by triple equals sign: a ≡ b mod n

Front 50

Give an example of congruence

Back 50

73 ≡ 4 mod 23

4 mod 23 = 4 since 4 = (0) × 23 + 4

73 mod 23 = 4 since 73 = 3 × 23 + 4

Front 51

How do you notate divisors?

Back 51

b|a

a = mb for some m

b divides a, so b is a divisor of a

Front 52

Prove this theorem:

Theorem: Zero Divisor ⇒ No Inverse

Back 52

Front 53

Prove 𝐹 = 5k | k ∈ ℤ} has Closure Under +

Back 53

Front 54

Prove {ℚ, +} has an identity element (A3)

Back 54

Front 55

Prove {ℚ, +} has a left inverse (A4)

Back 55

Front 56

Prove {Even functions, +} has a identity element

Back 56

Front 57

Prove {Even functions, +} has a left inverse element

Back 57

Front 58

Prove {Even functions, +} is closed by +

Back 58

Front 59

Whats an example of the integral domain?

Back 59

𝑅 = ℤ = {… , −3, −2, −1, 0, 1, 2, 3, … }

+ = Integer Addition

* = Integer Multiplication

Front 60

Whats an example of a field?

Back 60

{ℚ, +,×} – Rational Numbers

{ℝ, +,×} – Real Numbers

{ℑ, +,×} – Imaginary Numbers

Front 61

Whats an example of a commutative ring?

Back 61

𝑅 = {2k | k ∈ ℤ} = {… , −4, −2, 0, 2, 4, … }

+ = Integer Addition

* = Integer Multiplication

Front 62

What is a finite group?

Back 62

Has a finite number of elements called "the order" and is written as Order(G), that is otherwise infinite

{F, +,x}

Front 63

What is an example of a finite group?

Back 63

Integers mod n

𝐺 = ℤn = {0, 1, 2, … , n − 1}

* = + mod n

Order G = n

Front 64

What axioms does a finite group use?

Back 64

A1-4

Front 65

a ∈ Zn

∃b ∈ ℤ𝑛 ∋ ab = 1 iff gcd a, n = 1

What does this theorem state?

Back 65

There is an integer a in the integers mod n,

This means that there exists some b in the integers mod n such that a times b = 1 if and only if the greatest common denominator of a and n = 1

Front 66

p ∈ ℤ+

ℤp a Field iff p prime

What does this theorem state?

Back 66

There is a p in the positive integers

Then the integers mod p is a field if and only if p is prime

Front 67

p ∈ ℤ+

ℤp a Field iff p prime

Prove this theorem

Back 67

ℤp is a commutative ring

ℤp is a field:

iff a^-1 exists for each a != 0 in Zp

iff gcd(a,p) = 1 for each a != 0 in Zp

iff p is prime

Front 68

What does Euclids Algorithm state?

Back 68

If a, b in Z+, and a > b > 0, then

gcd(a,b) = gcd(b,a mod b)

Front 69

Give an example of Euclids Algorithm

Back 69

gcd(55, 22)

= gcd(22, 55 mod 22) = gcd(22, 11)

= gcd(11, 22 mod 11) = gcd(11, 0)

= 11

Front 70

Whats the point of the mod operator?

Back 70

We want a finite field of order 256

Using the mod operator can reduce an infinite set of elements to a finite set of elements (called residue classes)

Front 71

What is a Galois Field?

Back 71

Its a finite field with a finite number of elements

Front 72

What is Galois Field Theorem?

Back 72

Let F be a finite field

Then order(F) = p^n, p, n in Z+, p prime

All finite fields of order p^n are isomorphic

Front 73

log2(16) = ?

Back 73

4 => 2^4 = 16

Front 74

log5(125) = ?

Back 74

3 => 5^3 = 125

Front 75

logb(b) = ?

Back 75

1

Front 76

logb(b^x) = ?

Back 76

x

Front 77

logb(x*y) = ?

Back 77

logb(x) + logb(y)

Front 78

logb(x/y)

Back 78

logb(x) - logb(y)

Front 79

logb(x^p)

Back 79

plogb(x)

Front 80

Prove logb(xy) = logb(x) + logb(y)

Back 80

Let s = logb(x) and t = logb(y)

Then b^s = x and b^t = y

So b^(s+t) = xy

Thus logb(xy) = logb(b^s+t)

= s + t

= logb(x) + logb(y)

Front 81

How is exp (exponentials) written?

Back 81

exp2(4) = 2^4 = 16

Front 82

What is a cyclic finite field

Back 82

There exists a g in F such that for each a in f there exists a k in the positive integers where a = g^k

g is a generator of F

Can define log(g) and exp(g)

Front 83

How is g a generator of F?

Back 83

ℱ = {𝑔^0, 𝑔^1, 𝑔^2, … , 𝑔^(Order (ℱ )−1)

Front 84

Give an example of a cyclic generator

Back 84

𝐺+ = ℤ7, + mod 7

where G+ is an abelian group

Front 85

Is Dr. D's new spartan field note worthy?

Back 85

No, the order(D) = 7 which means D ≡ GF(7) ≡ ℤ7

Front 86

3 × 5 mod 7

Write this out using log and exp

Back 86

3x5 =

exp3 log3 3 × 5

exp3 log3 3 + log3 5

exp3 1 + 5

exp3 6

1

Front 87

Is 2 a generator of GF(7)

Back 87

No. 124 show. We dont get all numbers in mod 7 once

Front 88

Is 3 a generator of GF(7)

Back 88

Yes. 123456 all show and repeat

Front 89

How does + and - work in Z2

Back 89

Front 90

What are the rules that govern doing modulo on a polynomial?

Back 90

f(x), g(x), q(x), r(x) ∈ A(x) , g(x) ≠ 0

f(x) = g(x)q(x) + r(x), 0 ≤ deg r < deg(g)

r = remainder when f divided by g = f mod g

Front 91

What if we chose a different m(x) for a Rijndael field?

Back 91

The cipher text bit might change, but the properties wont change. It would have the exact same strength

Front 92

Multiply x^6 + x^4 + x^2 + x + 1 and x^7 + x + 1 without polynomial division

Back 92

Front 93

Where does S-box come from?

Back 93

{rs} ∈ ℤ2[𝑥]𝑚(𝑥)

{rs} polynomial inRijndael field

{rs} → 𝐴{rs}^−1 + 𝑐

Front 94

How does 𝑥 × 𝑔(𝑥) in GF 2 work?

Back 94

Front 95

How does 𝑥 × 𝑔(𝑥) in ℤ2[𝑥]𝑚(𝑥) =𝑥3+𝑥+1

Back 95

Front 96

What are the necessary and sufficient conditions on a and n for an element a in Zn to have a multiplicative inverse?

Back 96

iff gcd(a,n) = 1 for every a in Zn

Front 97

What are the necessary and sufficient conditions on n for a in Zn to be a field?

Back 97

iff n is prime