the mathematics of rsa encryption

THE MATHEMATICS OF RSA ENCRYPTION

Casco Bay .NET User GroupNov. 2016

Goals Make encryption less mysterious

Outline What is RSA Encryption? Math Refresher How does it work? Math Deep Dive Where do keys come from? Why does it work? Why is it secure?

What is RSA Encryption? Public Key Cryptography Scheme

Math Refresher - Modulus % “The Remainder” operator 546 % 31 = 19 546 = 17*31 + 19

Math Refresher - Primes A prime can only be divided by 1 and itself Every number can be factored into a list of

primes 360 = 2 * 2 * 2 * 3 * 3 * 5 11 = 11

Two numbers are coprime if they have no common prime factors 6 = 2 * 3, 35 = 5 * 7, so 6 and 35 are coprime 26 = 2 * 13, 4 = 2 * 2, so 26 and 4 are not

coprime

How Does it Work? Public key: (e, n) Private key: (d, n) Message: M Encrypted Message: EM = M**e % n Decrypted Message: DM = EM**d % n

Example Public key: (e, n) = (3593, 150349) Private key: (d, n) = (957, 150349) Message: M = 90001 Encrypted Message: EM = M**e % n

EM = 90001**3593 % 150349 131425

Decrypted Message: DM = EM**d % n DM = 131425**957 % 150349 90001

Example Public key: (e, n) = (3593, 150349) Private key: (d, n) = (957, 150349) Message: M = 22621 Encrypted Message: EM = M**e % n

EM = 22621**3593 % 150349 62033

Decrypted Message: DM = EM**d % n DM = 62033**957 % 150349 22621

Why did that work?

Why did that work? Math

Why did that work? Math

Cleverly chosen keys

Why did that work? Math

Cleverly chosen keys Euler’s Theorem

φ, the totient function φ(n) is called the totient of n

Number of integers less than n, coprime with n n = 15 = 5 * 3 φ(15) = Number of integers coprime with 15

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 1, 2, 4, 7, 8, 11, 13, 14

φ(15) = 8

Calculating φ(n) φ(15) = 8

Notice: φ(5 * 3) = (5-1) * (3-1) n = p * q

n is the product of 2 different primes, p and q

There are p multiples of q There are q multiples of p 0 is counted twice φ(n) = p*q – p – q + 1 = (p-1) * (q-1)

Euler’s Theorem x**φ(n) % n = 1, where x is coprime with

n Euler’s Theorem predicts x**8 % 15 = 1

For x coprime with 150 1 2 3 4 5 6 7 8 9 10

11

12

13

14

0 1 1 6 1 10

6 1 1 6 10

1 6 1 10 1 2 3 4 5 6 7 8 9 1

011

12

13

14

0 1 1 6 1 10

6 1 1 6 10

1 6 1 11 2 4 7 8 11 13 141 1 1 1 1 1 1 1

Proof of Euler’s Theorem

X 1 2 4 7 8 11

13

14

1 1 2 4 7 8 11

13

14

2 2 4 8 14

1 7 11

13

4 4 8 1 13

2 14

7 11

7 7 14

13

4 11

2 1 8

8 8 1 2 11

4 13

14

7

11

11

7 14

2 13

1 8 4

13

13

11

7 1 14

8 4 2

14

14

13

11

8 7 4 2 1

Proof of Euler’s Theorem

X 1 2 4 7 8 11

13

14

1 1 2 4 7 8 11

13

14

2 2 4 8 14

1 7 11

13

4 4 8 1 13

2 14

7 11

7 7 14

13

4 11

2 1 8

8 8 1 2 11

4 13

14

7

11

11

7 14

2 13

1 8 4

13

13

11

7 1 14

8 4 2

14

14

13

11

8 7 4 2 1

Proof of Euler’s Theorem Consider the product of each number in the

first row 1*2*4*7*8*11*13*14 % 15

What if we multiply this value by 7**8? 7**8 * (1*2*4*7*8*11*13*14) % 15

(7*1)*(7*2)*(7*4)*(7*7)*(7*8)*(7*11)*(7*13)*(7*14) % 15

7*14*13*4*11*2*1*8 % 15 1*2*4*7*8*11*13*14 % 15

It didn’t change the value, so 7**8 % 15 = 1

Key Generation How did we get our keys from the

example?

Public key: (e, n) = (3593, 150349) Private key: (d, n) = (957, 150349)

Key Generation p, q = 251, 599 n = p * q

150349 e = 3593 φ(n) = (p-1) * (q-1)

149500 d*e % φ(n) = 1 solve for d

d*3593 % 149500 = 1 d = 957 is the only solution

What makes those keys work? EM = M**e % n DM = EM**d % n

DM = (M**e % n)**d % n DM = M**(e * d) % n

Why does it work? DM = M**(e*d) % n

e*d % φ(n) = 1 e*d = 1 + k*φ(n) DM = M**(1 + k*φ(n)) % n = (M**1) * (M**φ(n))**k % n = M * (1**k) % n = M

Why is it secure? Can we get the private key from the

public key? e*d % φ(n) = 1 3593*d % φ(150349) = 1

Why is it secure? Can we get the private key from the

public key? e*d % φ(n) = 1 3593*d % φ(150349) = 1

No, because factoring appears to be difficult

Why is it secure? Can we get the private key from the public

key? e*d % φ(n) = 1 3593*d % φ(150349) = 1

No, because factoring appears to be difficult RSA-200

27997833911221327870829467638722601621070446786955428537560009929326128400107609345671052955360856061822351910951365788637105954482006576775098580557613579098734950144178863178946295187237869221823983

663 bits, 2 years real time, 75 years CPU time

Why is it secure? Can we solve for M given the encrypted

message? EM = M**e % n 131435 = M**3593 % 150349

Why is it secure? Can we solve for M given the encrypted

message? EM = M**e % n 131435 = M**3593 % 150349

No, because taking the eth root (The RSA Problem) appears to be difficult

Why is it secure? Can we solve for d given a decrypted

(authenticated) message? EM = M**d % n 131435 = 90001**d % 150349

Why is it secure? Can we solve for d given a decrypted

(authenticated) message? EM = M**d % n 131435 = 90001**d % 150349

No, because the discrete logarithm appears to be difficult

Why is it secure? Can we take a guess at M, given the

encrypted message? EM = M**e % n 131435 = M**3593 % 150349

Why is it secure? Can we take a guess at M, given the

encrypted message? EM = M**e % n 131435 = M**3593 % 150349 90001**3593 % 150329 = 131435 !!

Why is it secure? Can we take a guess at M, given the

encrypted message? EM = M**e % n 131435 = M**3593 % 150349 90001**3593 % 150329 = 131435 !!

Yes, that is a “chosen plaintext attack”, and that is why you must pad your messages

More References Video demonstrating Public Key

Cryptography https://www.youtube.com/watch?v=

GSIDS_lvRv4 Wikipedia Page

https://en.wikipedia.org/wiki/RSA_(cryptosystem)

Appendix: Properties of % A + B % n = (A % n) + (B % n) % n

517 + 878 % 10 = 7 + 8 % 10 A * B % n = (A % n) * (B % n) % n

318 * 73 % 10 = 8 * 3 % 10 A ** B % n = (A % n) ** B % n ≠ (A % n) ** (B %

n) 93 ** 57 % 10 = 3 ** 57 % 10 ≠ 3 ** 7 % 10

A ** B % n = A ** (B % phi(n)) % n (For A and n coprime) 93 ** 57 % 10 = 93 ** (57 % 4) % 10

A % n = A’ => A = A’ + k*n 382 % 10 = 2 => 382 = 2 + 38*10