Public Key Cryptography MO03

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Public Key Cryptography

Nor Shahida Seberi

Siti Fatimah SaadT4MT1

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Cryptography

• The art of protecting information bytransforming it (encrypting it) into an

unreadable format, called cipher text.

• Only those who possess a secret key can

decipher (or decrypt ) the message into plain

text.

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Types of 

encryptions

Symmetricencryption Asymmetricencryption

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Symmetric encryption

• The same key is used to encrypt and decrypt

the message.

• Example:

k = 4

Turn plain text SECRET into cipher text

S+4=W, E+4=I, C+4=G, R+4=V, E+4=I, T+4=X

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Asymmetric encryption

• Also known as public key encryption

• Invented in 1976 by Whitfield Diffie and MartinHellman.

• it uses two keys

public key, k  

private key, k’  

• Private key not required for both parties

• The system is extremely secure 

• One very popular public-key encryption programis Pretty Good Privacy (PGP) 

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Diffie-Hellman Example

• Block cipher• Block size of 7 bits. Possible 27 combinations 

• Private key (a’1, a’2, … , a’n) of 7 integers: (1, 2, 5, 11, 32, 87, 141)

• Chose two special integers, w and m, such that w and m are relatively prime,meaning gcd(w ,m) = 1: w = 901, m = 1234 

• Public key (a1, a2, … , an) of 7 integers using the equation: ai = w * a’i mod m:(901, 568, 803, 39, 450, 645, 1173)

• Partition SECRET into 7 bit blocks each block consisting of xn bits ( x 1, x 2, …, x n)

S1010011

E1000101

C1000011

R 1010010

E1000101

T1010100

• B x = ∑  x iai i=1

n

• S = 1 X (901) + 0 X (568) + 1 X (803) + 0 X (39) + 0 X (450) + 1 X (645) + 1 X (1173)

• S = 3522

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Diffie-Hellman Example

• Encrypted blocks Bx received. Special version of subset-sum problem 

• Which subset of (a’1, a’2, … , a’n) sums to B’ x where B’ x = B x  * w -1 mod m 

• w -1 is the modular inverse of w for m, w  * w -1 mod m = 1

• B’ x = 3522 X (901)-1 mod 1234• B’ x = 3522 X 1171 mod 1234• B’ x = 234

1. sum ← 0 2. for i = n step -1 until 1 do

if ai + sum <= B’ x 

then sum ← sum + ai;

subset(i)← 1

else subset(i)← 03. if sum = B’ x then exit with subset

else exit with “failure” 

• Private key (1, 2, 5, 11, 32, 87, 141), B’ x = 234, find subset (1, 0, 1, 0, 0, 1, 1) = S

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RSA

• Based on the difficulty of factoring

large numbers

• Developed by Ron Rivest, AdiShamir, and Leonard Adleman

at MIT in 1977.

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Example

• Bob chooses two very large (distinct) prime numbers p and q ;

• n= pq , m= lcm { p−1, q −1} (lcm is the least common multiple );• Bob chooses r , where r >1 and r is coprime with m (i.e. r and m have no

factors in common);

• Bob then finds the unique s such that rs≡1(modm)

• Bob now tells everyone what n and r are, but does NOT say what p, q  

or s are.

• Alice wants to send the message M (a single number) where M and n 

are coprime and 0<M<n.

• Alice finds Mc, where Mc≡Mr (modn), and sends the message Mc to Bob.

• Bob receives the message Mc from Alice and decodes it.• Now Bob knows p,q ,m,n,r ,s, and he uses these to decode the message

Mc from Alice so as to find M. To do this Bob uses the theorem that

(Mc)s≡M(modn)

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(1) Alice wishes to send the message M  to Bob

(2) Bob chooses  p=17, q=23; so n=391, m=176,

r =3 and s=59.

(3) Bob then tells Alice that n=391 and r =3.

(4) Note: It does not matter how many people

have this information, they still won't be able tofind s.

(5) Alice computes Mc and finds that Mc =180.

(5) Bob receives the coded message 180 from

 Alice

(6) Bob now calculates M ≡18059(mod391), and

finds Alice's secret message M .

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