Chapter 3 Encryption Algorithms & Systems (Part B)

csci5233 computer security & integrity (Chap. 3) 2 Outline

csci5233 computer security & integrity (Chap. 3) 3 Merkle-Hellman Knapsacks A public key cryptosystem The public key is the set of integers of a knapsack problem (the general knapsack); the private key is a corresponding superincreasing knapsack (or simple knapsack). A sample general knapsack: (17, 38, 73, 4, 11, 1) A sample superincreasing knapsack: (1, 4, 11, 17, 38, 73), where each item a k is greater than the sum of all the previous items. Merkle and Hellman provided an algorithm for the receiver to use a superincreasing knapsack (the private key) to decrypt the ciphertext.

csci5233 computer security & integrity (Chap. 3) 4 Merkle-Hellman Knapsacks Basic idea: To encode a binary message as a solution to a knapsack problem, reducing the ciphertext to the target sum obtained by adding terms corresponding to the 1s in the plaintext. Example: Fig. 3-5, p.84 Two kinds of knapsacks: Simple (superincreasing) knapsack Hard (general) knapsack

csci5233 computer security & integrity (Chap. 3) 5 Solving a simple knapsack problem Given a superincreasing knapsack S = (S 1, S 2, …, S n ) and a target sum T, find combination of S i that equals to T. Hint: No combination of terms less than a particular term can yield a sum as large as the term. That is, S i > S 1 + … + S i-1. Example: Given S = (1, 4, 11, 17, 38, 73) and T = 96, determine which terms in S correspond to the 1s of the plaintext. Solution: 1, 4, 17, 73 (See Fig. 3-6, p.85). That is, the plaintext was Exercise: S = (1, 2, 5, 9, 20, 43), T = 49.

csci5233 computer security & integrity (Chap. 3) 6 Deriving a Hard Knapsack from a Simple Knapsack An example on pages Choose the number (M) of items in a knapsack. Example: M = 4 2. Create a simple knapsack (S) with M items. Example: S = (1, 2, 4, 9). 3. Choose a multiplier w and a modulus n, where n > S 1 + … + S M and w is relprime to n. Note: Usually n is a prime, which is relprime to any number < n. Example: w = 15 and n = Replace every item in the simple knapsack with the term h i = w * s i mod n. Then H = (h 1, h 2, …, h M ) is the hard knapsack. Example: H = (15, 13, 9, 16).

csci5233 computer security & integrity (Chap. 3) 7 Encryption using Merkle-Hallman Knapsacks The plaintext is encrypted using the hard knapsack (the public key), while the simple knapsack, as well as w and n, are used as the private key. Example: Encrypt the plaintext ‘ ’ using the sample hard knapsack H = (15, 13, 9, 16). Ans: C 1 = 0100 * H = 13. C 2 = 1011 * H = = 40. C 3 = 1010 * H = = 24. C 4 = 0101 * H = = 29. So, ciphertext = (13, 40, 24, 29)

csci5233 computer security & integrity (Chap. 3) 8 The receiver of the ciphertext uses w and n to calculate the multiplicative inverse of w, w -1. w * w -1 mod n = 1. Example: w -1 = mod 17 = 8 (use algorithm on p.81 or the inverse.java program)inverse.java To decipher the ciphertext (C): Multiply each of the numbers in C by w -1. (w -1 * C = w -1 * H * P = w -1 * w * S * P = S* P )mod n Exercise: Decrypt the ciphertext from the previous exercise, i.e., 13, 40, 24, 29, by using the simple knapsack S = (1, 2, 4, 9). The answer: next slide Decryption using Merkle-Hallman Knapsacks

csci5233 computer security & integrity (Chap. 3) 9 Given: H = (19, 28, 76, 171, 293, 46, 130, 150). C = 13, 40, 24, 29 S = (1, 2, 4, 9) n = 17, w = 15, w -1 = To get the target T i : Multiply each C i by w -1 : T1 = (w -1 * C 1 ) mod n = 8 * 13 mod 17 = 2 T2 = (w -1 * C 1 ) mod n = 8 * 40 mod 17 = 14 T3 = (w -1 * C 1 ) mod n = 8 * 24 mod 17 = 5 T4 = (w -1 * C 1 ) mod n = 8 * 29 mod 17 = Given the target sum T i and the simple knapsack S, find the combination of items in S that produces T. e.g., The answer for T 1 is Decryption using Merkle-Hallman Knapsacks

csci5233 computer security & integrity (Chap. 3) 10 Another Example Step 1: Derive a Hard Knapsack from a Simple Knapsack 1. Choose the number (M) of items in a knapsack. Example: M can be 8 when the plaintext is in ascii. 2. Create a simple knapsack (S) with M items. Example: S = (1, 2, 4, 9, 17, 34, 70, 150). 3. Choose a multiplier w and a modulus n, where n > S 1 + … + S M and w is relprime to n. Note: Usually n is a prime, which is relprime to any number < n. Example: w = 19 and n = Replace every item in the simple knapsack with the term h i = w * s i mod n. Then H = (h 1, h 2, …, h M ) is the hard knapsack. Example: H = (19, 38, 76, 171, 20, 40, 118, 123).

csci5233 computer security & integrity (Chap. 3) 11 Another Example Step 2: Encrypt a plaintext using the hard knapsack Encrypt the plaintext ‘PEACE’ using the sample hard knapsack H, H = (19, 38, 76, 171, 20, 40, 118, 123). ‘P’ = 50 h ‘E’ = 45 h ‘A’ = 41 h ‘C’ = 43 h Ans: C 1 = 50 h * H = * H = = 209. C 2 = 45 h * H = * H = = 201. C 3 = 41 h * H = * H = = 161. C 4 = 43 h * H = * H = = 279. C 5 = C 2 = 228. So, ciphertext (‘PEACE’) =

csci5233 computer security & integrity (Chap. 3) 12 Step 3: Decrypt using C, w, n and S The receiver of the ciphertext uses w and n to calculate the multiplicative inverse of w, w -1. w * w -1 mod n = 1. Example: w -1 = mod 303 = 16 (use algorithm on p.81 or the inverse.java program)inverse.java To decipher the ciphertext (C): Multiply each of the numbers in C by w -1. (w -1 * C = w -1 * H * P = w -1 * w * S * P = S* P )mod n Exercise: Decrypt the ciphertext from the previous exercise, i.e., , by using the simple knapsack S = (1, 2, 4, 9, 17, 34, 70, 150). The answer: next slide Another Example

csci5233 computer security & integrity (Chap. 3) 13 Given: H = (19, 28, 76, 171, 293, 46, 130, 150). C = , n = 303, w = 19, w -1 = 16. S = (1, 2, 4, 9, 17, 34, 70, 150). 1. To get the target T i : Multiply each C i by w -1 T1 = (w -1 * C 1 ) mod n = 16 * 209 mod 303 = 11 T2 = (w -1 * C 1 ) mod n = 16 * 201 mod 303 = 186 T3 = (w -1 * C 1 ) mod n = 16 * 161 mod 303 = 152 T4 = (w -1 * C 1 ) mod n = 16 * 279 mod 303 = Given the target sum T i and the simple knapsack S, find the combination of items in S that produces T. The answer for T 1 is , which is 11. S = (1, 2, 4, 9, 17, 34, 70, 150) Ans. for T 1 = ( )  50 h Another Example

csci5233 computer security & integrity (Chap. 3) 14 p.90 The modulus n can be guessed. S, the secret key, and the multiplier, w, are then exposed. 1980: Shamir found that if the value of the modulus n is known, it may be possible to determine the simple knapsack S. 1982: Shamir came up with an approach to deduce w and n from H, the hard knapsack, alone. Weakness of the Merkle-Hellman Algorithm

csci5233 computer security & integrity (Chap. 3) 15 Summary Merkle-Hellman is a public-key cryptosystem based on the knapsack problem. The encryption can be broken, mainly due to its use of simple knapsacks as the secret keys. Next: –RSA Encryption –El Gamal Algorithms –DES –Hashing Algorithms –Key Escrow & Clipper