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Cryptography: RSA & DES

Cryptography: RSA & DES. Marcia Noel Ken Roe Jaime Buccheri. Cryptography is the science of writing in secret code. The first documented use of cryptography in writing dates back to circa 1900 B.C. by an Egyptian scribe. Primary users of cryptography were…. Government classified systems

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Cryptography: RSA & DES

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  1. Cryptography: RSA & DES Marcia Noel Ken Roe Jaime Buccheri

  2. Cryptography is the science of writing in secret code. The first documented use of cryptography in writing dates back to circa 1900 B.C. by an Egyptian scribe.

  3. Primary users of cryptography were…. • Government classified systems • Military • Financial Institutions Today it is used for… • email • credit card transactions • sensitive company info • personal info • passwords

  4. The basis of cryptography is to protect data from theft or alteration when communicating over an un-trusted medium.

  5. Some specific security requirements for application to application communication: . • Authentication: proving ones identity. • Privacy/confidentiality: ensuring that no one except intended receiver can read the message. • Integrity: assuring the receiver, the received message has not been altered in any way from the original. • Non-repudiation: proving that the sender really sent the message. Prevents the sender from denying having sent the message.

  6. Three types of cryptographic schemes used to accomplish these goals: • Secret key cryptography (symmetric) uses a single key for both encryption and decryption. • Public key (asymmetric) uses one key for encryption and another for decryption. • Hash functions uses mathematical transformation to irreversibly “encrypt information”.

  7. Two Key Words In Cryptography Encryption: transformation of data into a form that is close to impossible to read without a key. Decryption: the transformation of encrypted data back to an intelligible form.

  8. Public Key Encryption Alice wants to send a message (plaintext) to Bob… • She encrypts the message (ciphertext) using the public key ( which can be advertised to the public), and sends it to Bob. • Using his private key, Bob converts the ciphertext back into plaintext.

  9. Message Authentication This could also be used to prove that Alice sent the message to Bob… • Alice encrypts the message with her private key, and sends the message with the digital signature attached, to Bob. • Bob would decrypt the signature using the public key. If the result is the same, he knows Alice sent the message.

  10. Private (Secret) Key Encryption • A single key is used to both encrypt and decrypt the message. • Sender and receiver must agree and exchange a key without someone else eavesdropping.

  11. Cryptographic Hash Functions • Cryptographic hash functions are typically used for creating digital signatures. • A hash function compresses the bits of a message to a fixed-size hash value.

  12. Sender computes the hash value of the message, then encrypts with their private key. • Recipient decrypts the hash value with the senders public key. • Then computes the hash value of the message and compares the calculated hash value with the received hash value. A match proves the message and signature are authentic.

  13. Components of Cryptographic Algorithms Cryptographic algorithms, also called ciphers are based on the underlying mathematics and the possible number and length of the keys (which are expressed as a number of bits). The above statement determines the strength of a system.

  14. Information theory:allows us to examine communication and understand what message is really being sent, and how much is unnecessary. • Complexity theory: the study of the computational complexity of different cryptographic techniques.

  15. Number Theory: • prime numbers… gcd(m, n) • modular arithmetic… a = b(mod n) if a = b + k*n for some integer k • Factoring: finding prime factors of a number. ex. 80 2 * 2 * 2 * 2 * 5

  16. Discrete Logarithms: inverse of modular exponentiation. find x where ax = b(mod n) ex. If 3x = 15 mod 17, then x = 6

  17. Cryptanalysis • The art of deciphering encrypted communications without knowing the proper keys. • Basically an unauthorized third party wages an attack against a key or a message.

  18. Cryptanalysis Techniques • Ciphertext-only attack: The attacker must work from the ciphertext only. • Known-plaintext attack: The attacker knows or can guess the plaintext for some parts of the ciphertext. The task is to decrypt the rest of the ciphertext blocks using this info. • Brute-force attack: where every possible key combination is tried.

  19. Chosen-plaintext attack: The attacker is able to have any text he likes encrypted with the unknown key. • Man-in-the-middle attack:This attack is relevant for cryptographic communication and key exchange protocols. An adversary intercepts the communication and alters it.

  20. Time Complexity • Classical complexity theory deals with worst-case complexity, a concept useless in cryptography where breaking a system must be hard for almost all instances, not just for some of them. • Much of modern cryptography is based on NP-complete problems. If these algorithms could be solved in polynomial time (P) cryptographic systems would be worthless.

  21. Secure Hash Algorithm (SHA) A hash algorithm which is the US federal standard for use with DSA (Digital Signature Algorithm) for digital signatures and storing passwords.

  22. SHA Overview • Pad the messages so the length is a multiple of 512 bits. This is done with a series of 1’s and 0’s. • Initialize the 5 word (160 bits) buffer(A,B,C,D,E) to a set of standard hex values. word A: 67 45 23 01 word B: EF CD AB 89

  23. Each 32 bit register (A,B,C,D,E) is an input variable into a set of four auxiliary functions which perform bitwise operations. F(X,Y,Z) = XY v not(X) Z G(X,Y,Z) = XZ v Y not(Z) H(X,Y,Z) = X xor Y Xor Z I(X,Y, Z) = Y xor (X v not(Z))

  24. Process the message in 16 word(512 bit) chunks, using four rounds of 20 bit operations on each buffer and the chunks….

  25. for(i to 15) do{ /*i = 0, j = 0*/ for(j to 15) copy into X[j] = (32) bits from 512 bit message block /* let [abcd k s i] denote an operation*/ a = b + ((a + F(b,c,d) + X[k] + T [i])<<<s) /*do the following 16 operations*/ [ABCD 0 7 1] [DABC 1 12 2] [CDAB 2 17 3] [BCDA 3 22 4] [ABCD 4 7 5] [DABC 5 12 6] [CDAB 6 17 7] [BCDA 7 22 8] [ABCD 8 7 9] [DABC 9 12 10] [CDAB 10 17 11] [BCDA 11 22 12] [ABCD 12 7 13] [DABC 13 12 14] [CDAB 14 17 15] [BCDA 15 22 16] Increment the buffers }end .. /*T is a pre-determined table constructed from the sin function.*/

  26. The output is a 160 bit message digest which is entered as a variable into the DSA algorithm. • It would take about 2n steps to find a message with the same message digest.

  27. RSA Background • RSA was invented by three MIT researchers: Ronald Rivest, Adi • Shamir, Leonard Adleman • Searching for a more complete Public Key Cryptography approach • than Diffie-Hellman. • First published in 1977 in Scientific American. • NSA asked them to stop distributing their report detailing RSA, but • had no legal reason for them to stop distribution. • Patented until September 2000.

  28. RSA Overview • RSA is a public key, or asymetric cryptography system. • Two sets of keys, one public and one private. The public key pair is published, while the private is not. • Strength of RSA comes from the difficulty of factoring large prime numbers. • A message encrypted with the public key pair, can only be decrypted using the private key pair. A message encrypted using the private key pair can only be decrypted using the public key pair.

  29. RSA Concepts We have two functions, P(M) using the public key pair (e, n) and S(M) using the private key pair (d, n). • The two functions are inverse functions. EXAMPLE – Function:X 2 Inverse Function: sqrt (X) • Either of the functions can be used to encrypt a message, but the opposite must be used to decrypt. EncryptP(M)  C Decrypt S(C)  M --------------------------------------------------------------------------- EncryptS(M)  C Decrypt P(C)  M M is the message to be encrypted C is the ciphertext (encrypted message)

  30. RSA Concepts – Key Generation Properly generating keys is essential to the strength of RSA We are trying to generate two pairs of keys, the public key pair and the private key pair. They will be denoted as follows: Public:(e, n)Private:(d, n) e is the public exponent d is the private exponent n, the modulus, is common to both pairs.

  31. RSA Concepts – Steps to Generate RSA Keys 1.) Select two prime numbers: p, q. 2.) Find the modulus n = pq. This will be the n in our public and private key pairs. 3.) Find e, the public exponent. e must be relatively primeto (p-1)(q-1). 4.) Find d, the private exponent. d must be chosen so (ed-1)/(p-1)(q-1) is an integer. If d satisfies this equation, then d will be the multiplicative inverse of e. 5.) Discard p and q, only the public key (e, n) and private key (d, n) pairs are needed now.

  32. RSA Concepts - Encryption If Bob wanted to send Alice a message that only Alice can read, he would do the following: • Obtain Alice’s public key pair (e, n) • Encrypt a message using Alice’s public key to create the ciphertext. Symbolic Function: P(M)  C Mathematical Function: C = Me mod n • Bob sends this ciphertext to Alice. The original message is protected during transmission.

  33. RSA Concepts - Decryption Now that Alice has received the ciphertext, she needs to decrypt it to get the original message back: • Use the secret key pair (d, n) known only to Alice to convert the Ciphertext back to the original message. Symbolic Function:S(C)  M Mathematical Function:M = Cd mod n • Now Alice has retrieved the original message.

  34. RSA Example – Key Generation 1.) Select two primes p, q:p = 3, q = 5 2.) Find modulus n = pq: n = 15 3.) Find e relatively prime to(p-1)(q-1): (3-1)(5-1) = 8. Factors of 8 are 8, 4, 2, 1. Let e be 3 – gcd(8, 3) = 1, so this works. 4.) Find d where the equation (ed-1)/(p-1)(q-1) is an integer:(3d-1)/8. Let d = 11 since (3*11-1)/8 = 4. 5.) Discard p, q. Our pairs are: Public:(3, 15)Private:(11, 15)

  35. RSA Example – Encryption If Alice wants to send a message to Bob that can only be viewed by Bob, she needs to encrypt the message using Bob’s public key. This message can only be decrypted using Bob’s private key, so he will be the only one able to view it. Public:(3, 15) The message she wants to encrypt is “GO”. We take this one character at a time representing each as a number based on it’s position in the alphabet. M = 7, since the letter ‘G’ is mapped to the number 7 Using the equation C = Me mod n we create the ciphertext C. C = 73 mod 15 C = 13 Alice sends C to Bob without worrying about it being intercepted.

  36. RSA Example – Decryption Bob has received this ciphertext and wishes to decrypt it. He applies the RSA function to the Ciphertext C using his private keys to get the original message back. Private:(11, 15) C = 13 M = 1311 mod 15 M = 7 = “G” In reality, we would use ASCII values to represent characters in the message, but this would require much larger primes since the largest value a message can have is the value of n, the modulus, minus 1 (in this case 14 is the max value).

  37. RSA Concepts – Digital Signatures Alice wants to send Bob a message in a way that Bob can be certain that M came from Alice and is authentic. • Alice takes the message M, and creates a signature S by encrypting the message using her secret key pair. S(M)  S • She then sends Bob S and M (note M is not encrypted). • Bob uses Alice’s public key pair to decrypt the signature S and verifies that this matches M. P(S)  M • If the decrypted signature matches the message, it must have come from Alice and be unaltered, since only Alice could have created S using her secret key pair.

  38. RSA - Efficiency To maximize efficiency, public exponent e chosen should be small to make computing the ciphertext easier. This makes encryption faster than decryption, and verification faster than signing in the case of digital signatures. Using typical modular exponentiation algorithms, here are the efficiencies (where k = the number of bits in the modulus n): Public Key Operations:O(k2) steps Private Key Operations:O(k3) steps Key generation:O(k4) steps

  39. RSA – Advantages/Disadvantages Advantages: • Increased security – Keys never need to be transferred. Transferring keys is a security risk. • Digital Signatures – RSA can be used for authentication and verification of message integrity. Disadvantages: • Speed – Most secret key (symetric) encryption methods are faster than public key cryptosystems. In software, DES is generally 100 times faster than RSA, in hardware DES is between 1,000 and 10,000 times as fast.

  40. RSA – Breaking RSA General Methods – ways of finding the secret key d • Method 1: Factor modulus, n, into prime factors to find p, q - Integrity of RSA depends on factoring n being difficult. - Protect against by using proper key lengths. - Improvements in hardware will not make this easier if long keys are used. - It is much easier to multiply two numbers than to determine which numbers create a given product. EXAMPLE - 6 x 4 = 24 What are the two numbers multiplied together that gave you 24? 1 x 24 = 24 2 x 12 = 24 3 x 8 = 24 4 x 6 = 24

  41. RSA – Breaking RSA General Methods – ways of finding the secret key d • Method 2: Compute the eth roots mod n. - Since C = Me mod n, the eth root of c mod n is the message. - There is no algorithm to accomplish this. Key lengths are critical to strength of RSA - In 1997 it was estimated that 512-bit RSA keys could be factored for less than $1,000,000 in cost and eight months of effort. - RSA Laboratories reccoments 1024 bit key length for corporate use, and 2048 bit key length for extremely important uses.

  42. RSA – Breaking RSA Single Message Attacks – finds the contents of a particular message but does not expose keys. • Guessed plaintext attack • The attacker guesses the contents of the message, and encrypts it using the public key. If it matches the intercepted cyphertext, the attacker knows he is right. • Can be defended against by appending some random bits to the message.

  43. DES Background • DES was originally developed as Lucifer in the early 1970s • by IBM • Its original purpose was to provide a standard method for • protecting sensitive commercial and unclassified data • NSA (National Security Agency) and NIST (National Institute of • Standards andTechnology ) created a modified version of Lucifer • and named it DES • DES officially became a federal standard in November of 1976

  44. DES Overview • DES is a private or symmetric key cryptographic method • Algorithm depends on the secrecy of the key • There are 72 quadrillion or more potential keys that may be used • DES is a block cipher--meaning it operates on plaintext blocks of a given size and returns cipher text blocks of the same size

  45. DES Steps 1. Subkey generation 2. Encrypting the message 3. Applying the final permutation

  46. Components of DES • XOR • Left Shift/Rotation • Permutations-reorder bits • Substitutions- S boxes, have never been made public • The f Function

  47. Components of DES • XOR operator: • 0 False, 1 True • 0 0 = • 0 1 = • 1 0 = • 1 1 = 0 1 1 0

  48. Components of DES Example of XOR operator: 1 0 1 0 1 0 1 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 1 0 1 0 1

  49. Components of DES • DES uses a circular left shift for its key generation • A one bit shift would yield the following results: Left Shift/ Rotations: 1 1 0 0 1 1 1 0 1 0 0 1 1 1 0 1

  50. Components of DES Permutations: Permutation Table 3 5 2 7 4 6 8 1 Initial Message 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 0 0 1 1 0 0 1 1 Permuted Message Permuted Message

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