Modular Arithmetic & Cryptography

CSC2110 Tutorial 8

Darek Yung

Outline

Quick Review Examples Q & A

Quick Review

Prime Modular Arithmetic Multiplicative Inverse Turing’s Code RSA

Prime

If p is a prime, GCD(a, p) = 1 unless a is multiple of p

If p is a prime, p | a1 * a2 * … * aN implies

p | ai for some i Every natural number n > 1 has a unique pri

me factorization

Modular Arithmetic

Multiplicative Inverse

Multiplicative Inverse

Multiplicative inverse can be calculated by Repeated squaring Integer linear combination

Turing’ Code (Version 1)

Secret: key k Encryption: m’ = m * k Decryption: m’ / k = (m * k) / k = m Attacked by

GCD(m1’, m2’) = GCD(m1k, m2k) = k

Turing’ Code (Version 2) Public: large prime p Secret: key k, k < p Encryption: m’ = rem(mk, p), m < p Decryption:

multiplicative inverse of k (mod p) k’ m’ mk (mod p) m’k’ mkk’ (mod p) m (mod p)

Plain text attack Given m’, m, p multiplicative inverse of m (mod p) m’’ m’m’’ kmm’’ (mod p) k (mod p)

RSA

Generate two large primes p, q Let n = pq, T = (p-1)(q-1) Select e such that GCD(e,T) = 1 Let d = multiplicative inverse of e (mod T)

RSA

Public: n, public key e Secret: private key d Encryption: m’ = me (mod n), m < n Decryption: (m’)d med m (mod n)

Q20: False Q21: False, modular arithmetic can be appli

ed to INTEGER a, b Q22: True, sum of digits is divisible by 9

Examples

Examples

Q23: True, difference between sum of even digits and sum of odd digits is divisible by 11

Q24: False, either of a, a’ can be negative.

e.g. n = 5, a = 3, a’ = -3 k = -2 Q25: True

Examples

Q26: False, c may be multiple of n Q27: False, k may be multiple of p

Examples

Q28: True Q29: False, k may be multiple of p Q30: False, k may be multiple of p and there

is no multiplicative inverse of k (mod p)

Examples

Q31: True (false if p1 = p2) Q32: True Q33: False, Version 1.0 use simple

multiplication

Examples

Q34: True (will show in long question) Q35: True Q36: False, unique m with respect to a

specific k. A large number of (m, k) pair.

Examples

Q37: True Q38: False, encrypted by public key Q39: True Q40: False, it is just to compute the power

of an integer modulo another number

Examples

It is now 5pm on Sunday, What’s the time and the day of the week after 17965 hours

17965 = 758(24) + 13 758 = 108(7) + 2 Tuesday 13 + (5+12) = 30 = 1(24) + 6 6am

Wed

Examples

Calculate 579572 (mod 21) by repeated squaring

72 = 64 + 8 5795 20 (mod 21) 57952 1 (mod 21)

57954 57958 579516 1 (mod 21)

579532 579564 1 (mod 21) 579572 579564 * 57958 20 (mod 21)

Examples

Calculate multiplicative inverse of 47981 (mod 95963)

95963 = 2(47981) + 1

1(95963) -2(47981) = 1

-2(47981) 1 (mod 95963)

required multiplicative inverse = -2

Examples

Prove ac bc (mod p) and GCD(c, p) = 1 imply a b (mod p) for prime p

Since GCD(c, p) = 1, there exists c’ = multiplicative inverse of c (mod p)

ac bc (mod p) acc’ bcc’ (mod p) a b (mod p)

Examples

Given a b (mod p) and b c (mod p) implies a c (mod p)

And a b (mod p) implies a+cb+c (mod p) Prove a b (mod p) and c d (mod p) imply

a + c b + d (mod p) for prime p

Examples

Prove a b (mod p) and c d (mod p) imply a + c b + d (mod p) for prime p

a b (mod p) a + c b + c (mod p) c d (mod p) c + b d + b (mod p)

a + c b + c c + b d + b (mod p)

a + c b + d (mod p)

Examples Explain why a number written in decimal is divisible by

9 if and only if the sum of its digits is a multiple of 9

10 1 (mod 9) 10k 1k 1 (mod 9)

All decimal number, d, can be written in form:dk(10k) + dk-1(10k-1) + … + d1(101) +d0

d dk + dk-1 + … + d0 (mod 9) divisible by 9 if and only if the sum of its digits is a multiple of 9

Examples Explain why a number written in decimal is divisible by 11 if and

only if the difference between the sum of its odd digits and sum of its even digits is a multiple of 11

10 -1 (mod 11) 10k (-1)k (mod 11)

All decimal number, d, can be written in form:dk(10k) + dk-1(10k-1) + … + d1(101) +d0

d (-1)kdk + (-1)k-1dk-1 + … + (-1)0d0 (mod 11) d dk - dk-1 + dk-2… - d1 + d0 (mod 11) divisible by 11 if and only if the difference between the sum of its odd digits and sum of its even digits is a multiple of 11

Examples

Under Turing’s Code (Version 1.0), given key k = 47 Encrypt message m1 = 569

Encrypt message m2 = 751

Get k by having m1 and m2

Example

Encrypt message m1 = 569

m1’ = k * m1 = 26743

Encrypt message m2 = 751

m2’ = k * m2 = 35297

How to crack?

Examples

GCD(m1’, m2’) = GCD(m1k, m2k) = k 35297 = 1(26743) + 8554 26743 = 3(8554) + 1081 8554 = 7(1081) + 987 1081 = 1(987) + 94 987 = 10(94) + 47 94 = 2(47) + 0

GCD(35297, 26743) = 47 = k

Examples

Under Turing’s Code (Version 2.0), given prime p = 89, key k =48 Encrypt message m = 78 Compute multiplicative inverse of m (mod p) Show how to perform plain-text attack

Examples Encrypt message m = 78 m’ = rem (mk, p) = rem (3744, 89) = 6

Compute m’’ = multiplicative inverse of m (mod p) 89 = 78 + 11 11 = 89 - 78 78 = 7(11) + 1 1 =78 – 7(11)

1 = 8(78) – 7(89)

m’’ = 8

Examples

Plain-text attack: given m’, m and p m’ m * k (mod p)

m’’ * m’ m’’ * m * k k (mod p) m’’ * m’ = 6 * 8 48 (mod p)

k = 48