RSA Algorithm and Modular Arithmetic - Prof. Krzysztof Gaj, Study notes of Cryptography and System Security

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Mathematical Background:

Modular Arithmetic

ECE 646 - Lecture 5

Motivation:

Public-key ciphers

RSA as a trap-door one-way function

M C = f(M) = Me^ mod N C

M = f-1(C) = Cd^ mod N

PUBLIC KEY

PRIVATE KEY

N = P ⋅ Q P, Q - large prime numbers

e ⋅ d ≡ 1 mod ((P-1)(Q-1))

message ciphertext

RSA keys

PUBLIC KEY (^) PRIVATE KEY

{ e, N } { d, P, Q }

N = P ⋅ Q

e ⋅ d ≡ 1 mod ((P-1)(Q-1))

P, Q - large prime numbers

gcd(e, P-1) = 1 and gcd(e, Q-1) = 1

d:

P, Q:

N:

e:

Basic definitions

General Notation

Z – integers

∃ - there exists

• for all

∃! - there exists unique

∈ - belongs to (^) ∉ - does not belong to

Divisibility

a | b iff ∃ c ∈ Z such that b = c ⋅ a

a | b a divides b a is a divisor of b

a | b a does not divide b a is not a divisor of b

True or False?

Greatest common divisor

Greatest common divisor of a and b , denoted by gcd( a , b ) ,

is the largest positive integer that divides both a and b.

d = gcd ( a , b ) iff 1) d | a and d | b

2. if c | a and c | b then c ≤ d

gcd (8, 44) =

gcd (-15, 65) =

gcd (45, 30) =

gcd (31, 15) =

gcd (0, 40) =

gcd (121, 169) =

Relatively prime integers

Two integers a and b are relatively prime or co-prime

if gcd( a , b ) = 1

Properties of the greatest common divisor

gcd ( a , b ) = gcd ( a - kb , b ) for any k ∈∈∈∈ Z

Integers coungruent modulo n

Two integers a and b are congruent modulo n ( equivalent modulo n )

written a ≡≡≡≡ b iff

a mod n = b mod n or a = b + kn , k ∈∈∈∈ Z or

n | a - b

Laws of modular arithmetic

Rules of addition, subtraction and multiplication

modulo n

a + b mod n = (( a mod n ) + ( b mod n )) mod n

a - b mod n = (( a mod n ) - ( b mod n )) mod n

a ⋅ b mod n = (( a mod n ) ⋅ ( b mod n )) mod n

9 · 13 mod 5 =

25 · 25 mod 26 =

Algorithms

Euclid’s Algorithm

for computing gcd(a,b)

i

• 0 1

…

t - t

ri

r -2 = max (a, b) r -1 = min (a, b) r 0 r 1

…

rt-1 = gcd(a, b) rt =

qi

q - q 0 q 1

…

qt-

qi = ri - ri ri +1 = ri -1 - qi ⋅⋅⋅⋅ ri

ri +1 = ri -1 mod ri

Euclid’s Algorithm

Example: gcd(36, 126)

i

• 0 1

ri

r -2 = max (a, b) = r -1 = min (a, b) = r 0 = 18 = gcd(36, 126) r 1 = 0

qi

q -1 = 3 q 0 = 2 q 1 qi = ri - ri ri +1 = ri -1 - qi ⋅⋅⋅⋅ ri

ri +1 = ri -1 mod ri

Multiplicative inverse modulo n

The multiplicative inverse of a modulo n is an integer [!!!]

x such that

a ⋅⋅⋅⋅ x ≡≡≡≡ 1 (mod n )

The multiplicative inverse of a modulo n is denoted by a -1^ mod n (in some books a or a*).

According to this notation: a ⋅⋅⋅⋅ a -1^ ≡≡≡≡ 1 (mod n )

Extended Euclid’s Algorithm

for computing z = a -1^ mod n

i

• 0 1

…

t - t

ri

r -2 = n r -1 = a r 0 r 1

…

rt-1 = 1 rt =

xi

x -2= x -1= x 0 x 1

…

xt -1 = a -1^ mod n xt = ± n

qi

q -1 =

n/a
q 0 q 1

…

qt-

qi = ri - ri ri +1 = ri -1 - qi ⋅⋅⋅⋅ ri xi +1 = xi -1 - qi ⋅⋅⋅⋅ xi

Note: If rt -1 ≠≠≠≠ 1 the inverse does not exist

Extended Euclid’s Algorithm

Example z = 20-1^ mod 117

i

• 0 1 2 3 4

ri

r -2 = 117 r -1 = 20 r 0 = 17 r 1 = 3 r 2 = 2 r 3 = 1 r 4 = 0

xi

x -2= 0 x -1= 1 x 0 =- x 1 = 6 x 2 = - x 3 = 41 = 20-1^ mod 117 x 4 = -

qi

q -1 = 5 q 0 = 1 q 1 = 5 q 2 = 1 q 3 = 2

qi = ri - ri ri +1 = ri -1 - qi ⋅⋅⋅⋅ ri xi +1 = xi -1 - qi ⋅⋅⋅⋅ xi

Check: (^20) ⋅ 41 mod 117 = 1

Motivation

Breaking ciphers

Historical ciphers

Affine Cipher

ci = f(mi) = k 1 ⋅ mi + k 2 mod 26

mi = f-1(ci) = k 1 -1^ ⋅ (ci - k 2 ) mod 26

Key = (k 1 , k 2 ) k 1 , k 2 ∈ [0, 25], gcd (k 1 , 26)=

Encryption transformation:

Decryption transformation:

Key:

Historical ciphers

Affine Cipher – Example (2)

N

S

A

encryption

13

3 ⋅ 13 + 11 mod 26 = 24

coding decoding

Y

3 ⋅ 18 + 11 mod 26 = 13 N

3 ⋅ 0 + 11 mod 26 = 11 L

Historical ciphers

Affine Cipher – Example (3)

Y

N

L

decryption

24

9 ⋅ (24 – 11) mod 26 = 13

coding decoding

N

9 ⋅ (13 – 11) mod 26 = 18 S

9 ⋅ (11 – 11) mod 26 = 0 A

Ciphertext:

FMXVE DKAPH FERBN DKRXR SREFM ORUDS DKDVS HVUFE DKAPR KDLYE VLRHH RH

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

R - 8 D - 7 E, H, K - 5

Breaking the affine cipher (1)

Step 1: Establish a relative frequency of letters in the ciphertext

Most frequent single letters

Average frequency in a long English text: E — 13% T, N, R, I, O, A, S — 6%-9% D, H, L — 3.5%-4.5% C, F, P, U, M, Y, G, W, V — 1.5%-3% B, X, K, Q, J, Z — < 1%

Average frequency in a random string of letters: 1 26