Introduction to Medical Computing

Transcription

1 CS 2125 Introduction to Medical Computing Stephen M. Watt The University of Western Ontario

2 Topic 3 Cryptography University of Western Ontario CS Stephen M. Watt

3 Cryptography Some things should be public, and some things should not be. Can secure data by physical means, e.g. locks, guards. Can secure data by access controls, e.g. passwords. Can secure data by encryption.

4 Types of Cryptography Secret Key Cryptography Public Key Cryptography Hash Functions

5 Some Vocabulary Plain text the original data in unencrypted form Cipher text the data in encrypted form Key a piece of data used to do the encryption, like a code word. Dramatis Personae: Alice and Bob: want to exchange secret info Eve: an evesdropper

6 Secret Key Cryptography Single key is used for both encryption and decryption. Encrypt(plain text, key) -> cipher text Decrypt(cipher text, key) -> plain text E.g. (simple) Encrypt(char, offset) -> (char + offset) mod 256 Decrypt(char, offset) -> (char offset) mod 256

7 Secret Key Cryptography Electronic Codebook (ECB) Data divided into blocks and each encrypted separately. Pro: Simple. Con: Same plaintext -> same ciphertext

8 Secret Key Cryptography Cipher Block Chaining (CBC) Invented at IBM in the 1970s Each block used to modify the input of the next

9 Secret Key Cryptography Cipher Block Chaining (CBC)

10 Many More

11 Chained Block Cypher Message The man The plan Panama! Block Size 4 Key Z Encryption Block No Text block The man The plan Pan ama! Numeric plaintext 177,877, ,192, ,877, ,663,022 68,432, ,222,049 Hexadecimal A9A32A0 DB87720 A9A32A0 E1B30EE 41430EE C3B70A1 Combine 895,864,159 1,193,289, ,564,191 1,135,438, ,014,115 1,266,870,842 Hexadecimal 3565CD5F F 3247C49F 43AD69F B82EA3A Encrypt Key Block Cypher Encryption Key Block Cypher Encryption Key Block Cypher Encryption Key Block Cypher Encryption Key Block Cypher Encryption Key Block Cypher Encryption Init Vector Numeric cyphertext 1,073,741,823 1,251,496, ,070,591 1,303,795,999 1,011,921,869 1,203,346, ,489,348 Hexadecimal 3FFFFFFF 4A98505F 38DDF63F 4DB6591F 3C50B3CD 47B99A9B 347B3384 Decryption Init Vector Numeric cyphertext 1,073,741,823 1,251,496, ,070,591 1,303,795,999 1,011,921,869 1,203,346, ,489,348 Hexadecimal 3FFFFFFF 4A98505F 38DDF63F 4DB6591F 3C50B3CD 47B99A9B 347B3384 Decrypt Key Block Cypher Decryption Key Block Cypher Decryption Key Block Cypher Decryption Key Block Cypher Decryption Key Block Cypher Decryption Key Block Cypher Decryption 895,864,159 1,193,289, ,564,191 1,135,438, ,014,115 1,266,870, CD5F F 3247C49F 43AD69F B82EA3A Decombine Numeric plaintext 177,877, ,192, ,877, ,663,022 68,432, ,222,049 Text block The man The plan Pan ama!

12 Resources

13 Public Key Cryptography Most significant advance in cryptography in hundreds of years. First described publicly by Stanford professor Martin Hellman and graduate student Whitfield Diffie in 1976.

14 Public Key Cryptography Uses idea of functions that are hard to invert, one way functions. E.g. Multiplication vs Factorization Multiplication is easy. Takes time proportional to b log b log log b to multiply two b bit numbers. Factorization is thought to be hard. Best known algorithm for a b bit numbers is

15 Public Key Cryptography Uses idea of functions that are hard to invert, one way functions. E.g. Exponentiation vs logarithms Easy to compute 3 6 to get 729 Hard to take 729 and find 3 and 6.

16 Public and Private Keys In both cases (multiplication, exponentiation) we have 2 pieces of information combining to give a result from which it is hard to find the pieces. Each participant can have a private number and reveal something publicly that does not give away the private number.

17 Original Diffie Hellman Multiplying integers mod p, a prime. Need g, a primitive root mod p. That is a number g such that { g 1 mod p, g 2 mod p, g 3 mod p,, g p-1 mod p } gives the values {1, 2,, p-1} in any order.

18 Original Diffie Hellman E.g. 3 is a primitive root mod 7 because 3 1 = 3 = 3 (mod 7) 3 2 = 9 = 2 (mod 7) 3 3 = 27 = 6 (mod 7) 3 4 = 81 = 4 (mod 7) 3 5 = 243 = 5 (mod 7) 3 6 = 729 = 1 (mod 7)

19 Original Diffie Hellman Alice and Bob agree to use a prime p and base g. Alice chooses secret a. Bob chooses secret b. Alice sends Bob A = g a (mod p). Bob sends Alice B = g b (mod p). Alice computes s = B a = g ab (mod p). Bob computes s = A b = g ab (mod p). Now Alice and Bob share a secret to use for Shared Key Crypto.

20 Example Alice computes s = B a mod p s = 19 6 mod 23 s = 47,045,881 mod 23 s = 2 Bob computes s = A b mod p s = 8 15 mod 23 s = 35,184,372,088,832 mod 23 s = 2

21 Example Alice and Bob now share a secret: s = 2. Somebody who had known both these private integers might also have calculated s as follows: s = 5 6*15 mod 23 s = 5 15*6 mod 23 s = 5 90 mod 23 s = 807,793,566,946,316,088,741,610,050,849,573,099,185,363,389,5 51,639,556,884,765,625 mod23 s = 2

22 Public Key Cryptography Can use Diffie-Hellman for public key crypto. Alice choses her private key a Alice publishes a public key (A = g a mod p, g, p) Bob chooses a random b and sends Alice (B = g b mod p, message encrypted with A b mod p)

23 RSA Ron Rivest, Adi Shamir, Leonard Adleman (1978)

24 RSA Cryptography Alice s key generation: Choose two distinct random primes of similar size p and q. Compute N = p q and φ = (p-1)(q-1) Compute e such that 1 < e < φ and gcd(e, φ) = 1. Compute d = 1/e mod φ. e is the public key exponent, d is the private key exponent. Alice s public key is (N, e). Communication: Bob sends Alice the message M by sending c = M e mod N. Alice decrypts the message by computing M = c d mod N.

25 Cryptography in Medicine Examples: Piotr Kasztelowicz, Marek Czubenko, Iwona Zięba, Security of Medical Data Transfer and Storage in Internet. Cryptography, Antiviral Security and Electronic Signature Problems, which Must Be Solved in Nearest Future in Practical Context Pol J Pathol 2003, 54, 3, Johannes Heurix, Thomas Neubauer, Privacy-Preserving Storage and Access of Medical Data through Pseudonymization and Encryption Y Zhou, K. Panetta, S. Agaian, A lossless encryption method for medical images using edge maps Engineering in Medicine and Biology Society, EMBC Annual International Conference of the IEEE, Sept 2009