Cryptography (DES+RSA) by Amit Konar Dept. of Math and CS, UMSL

Cryptography (DES+RSA) by Amit Konar Dept. of Math and CS, UMSL

Transpositional Ciphers-A Review Decryption 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 Encryption 1 2 3 4 5 6 7 8 A G O O D F R I E N D I S A T R E A S U R E O O A G D E N F I R D A I S A S T E R R E U Plaintexts Cyphertexts

Digital P-box 1 0 0 0 1 1 0 1 Enciphering 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 0 0 1 0 0 1 1 A P-box (Permutation box) performs transposition at the bit level. It can be implemented by hardware or software, but hardwired realization is faster.

S-Box 3 lines to 8 lines Decoder 8 lines to 8lines P-Box 8 lines to 3 lines Encoder An S-box (substitution box) performs substitution at the bit level. Example: Suppose 2 (010) is the input. Decoder s output is 00000010. P-box s output is 01000000. Encoder output is 6 (110)

Product Block P-Box S S S S S P-Box S S S P-Box The P-boxes and S-boxes can be combined to get more complex cipher blocks, called product blocks.

Data Encryption Standard (DES) 64-bit plaintexts A complex set of cipher blocks 64-bit ciphertexts 48-bits 56-bit key Key Processor

Iteration Block Left 32-bits Right 32-bits Function of right 32 bits and key 48-bits key XOR Left 32-bits Right 32-bits

The f-function R i-1 (32 bits) E K i (48-bits) R i -1 (48-bits) XOR S1 S2 S8 P 32-bits

Public Key Cryptography In public key cryptography, we have two keys: one private key and the other public key. A public Key cryptosystem must meet the following conditions: 1. Given the keys, the enciphering and deciphering processes should be simple. 2. Deriving the private key from the public key should be computationally infeasible. 3. Determining the private key from a chosen plaintext attack should be computationally infeasible.

Defining Totients in Public Key Cryptography Totient: Let p and q be two prime numbers and n = p q. Then the totient (n) is the number less than n with no factors common with n. Example: Let p=5 and q=2. So, n=10. Then the numbers that are less than 10 are relatively prime to 10 (i.e. no factors common with 10) are 1, 3, 7 and 9. So, (10)=4. Similarly, when n=21, the nos. that are relatively prime to 21 are 1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19 and 20. Therefore, (21) =12.

Defining Keys in Public Key Cryptography Given n, choose an integer e <n, that is relatively prime to (n). Find a second integer d, such that e. d mod (n) =1 then the public key is (e, n), and the private key is d.

Encryption and Decryption in Public Key Cryptography The RSA Method after its inventors (Rivest, Shamir and Adleman) Let m be a message. Then ciphertext c= m e mod n, and decrypted plaintext m= c d mod n where public key is (e, n) and private key is d or (d, n), such that e. d mod (n) =1.

Example: The RSA Method ALICE (e, n) = (5, 119) (d, n)= (77, 119) Bob 6 5 mod 119 41 77 mod 119 Plaintext m =6 Ciphertext c=41 Deciphered m=6 Public key (e, n) = (5, 119) Private key (d, n)= (77, 119)

Is RSA really Effective? If an intruder knows the decryption algorithm and n= 119, the only thing missing is d= 77. Why couldn t the intruder use trial and error to find d? The answer is yes, in this trivial example, an intruder could easily guess the value of d. But a major concept of the RSA algorithm is to use very large numbers for d and e. In practice, the numbers are so large (on the scale of tens of digits) that the trial-and-error approach of breaking the code takes a long time (several years) even with the fastest computer today.

Selection of Public and Private Keys 1. Choose two large prime numbers p and q. 2. Compute n= p x q. 3. Choose e (less than n) such that e and (p -1) (q -1) are relatively prime (having no common factors other than 1). 4. Choose d such that e x d mod [(p-1) (q- 1)] is equal to 1.

Confidentiality and authentication together in RSA 1. ONLY CONFIDENTIALITY C m Enciphering with recipient s public key Deciphering with recipients private key. m 2. BOTH CONFIDENTIALITY & AUTHENTICATION Enciphering with sender s Private key Enciphering with recipient s public key Deciphering with recipient s private key Authenticat ion with sender s public key

How does RSA provide data and origin authentication? Example: Let p=7, q= 11; so, n = 77. Let e= 17; so d= 53. Suppose, Allice wishes to send Bob HELLO WORLD. The plaintext is 07 04 11 11 14 26 22 14 17 11 03. Using Allice s private key(d, n), the ciphertext is 07 53 mod 77 = 35 04 53 mod 77 = 09. or, 35 09 44 44 93 12 24 94 04 05.

Providing Confidentiality and Authentication by RSA Providing both confidentiality and authentication requires enciphering with the sender s private key and the recipent s public key. Example: Suppose, Allice wishes to send Bob HELLO WORLD in confidence and authenticated. Assume Allice s private key is 53 and Bob s public key is 37. The plaintext is 07 04 11 11 and the encipherment is (07 53 mod 77) 37 mod 77 =07 (04 53 mod 77) 37 mod 77 =37 or, 07 37 44 44 14 59 22 14 61 44 47.

Deciphering confidential and authenticated message The recipient uses recipient s private key to decipher the message and the sender s public key to authenticate it. Bob receives the ciphertext: 07 37 44 44 14 59 22 14 61 44 47. The decipherment is (07 13 mod 77) 17 mod 77 =07 (37 13 mod 77) 17 mod 77= 04 Or, 07 04 11 11 14 26 22 14 17 11 03.

What is Digital Signature? A digital signature ensures authentication (sender s identity), integrity (no changes in data) and non-repudiation (i.e. the receiver is able to prove that the received message came from a specific sender). It can be realized by RSA through enciphering by sender s private key and deciphering by sender s public key.

Signing the Digest Signing the entire message is not time-efficient. The sender prepares a digest by Hashing and then encrypts the digest. The message plus the signed digest is communicated. From Allice Message Message To Bob Signed Digest Hashing (MD 5/ SHA- 1) Digest Encrypt Allice s private key Signed Digest

Receiver Site Message Signed Digest Hashing Decrypt Allice s public key Digest Digest Error Zero?