Input: A={a1, a2, … an} – public key, S - ciphertext

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Presentation transcript:

Ciphertext Only Cryptanalytic Attack on Merkle-Hellman Knapsack: Dynamic Programming Algorithm Input: A={a1, a2, … an} – public key, S - ciphertext Output: The binary array B – plaintext Algorithm: Let P[i, j] be TRUE if there is a subset of first i elements of A that sums to j, 0 ≤ i ≤ n , 0 ≤ j ≤ S Step 1: Computation of P P[0][0]  TRUE for j = 1 to S do: P[0][j]  FALSE for i = 1 to n do: for j = 0 to S do: if (j – A[i] < 0): P[i][j] = P[i-1][j] else: P[i][j] = P[i-1][j-A[i]] or P[i-1][j]

Step 2: Backtracking Let B be an array of n + 1 elements initialized to 0 i  n, j  S while i > 0: if (j – A[i]) ≥ 0): if (P[i-1][j-A[i]] is True): B[i]  B[i] + 1 j  j – A[i] i  i – 1 else: i  i – 1 Output: array B, elements of B that equal to 1 construct a desired subset of A that sums to S

P[i-1][j-A[i]] or P[i-1][j] EXAMPLE Input: A={1, 4, 5, 2}, S =3 j = 0 j = 1 j = 2 j = 3 i = 0 TRUE FALSE i = 1 A[1] =1 Element is taken i = 2 A[2] = 4 i = 3 A[3] = 5 i = 4 A[4] = 2 P[i-1][j-A[i]] or P[i-1][j]

Merkle-Hellman Multiplicative Knapsack Cryptosystem Alice: Chooses set of relatively prime numbers P = {p1, …pn} – private (easy) knapsack Chooses prime M > p1* …* pn Chooses primitive root b mod M Computes the public (hard) knapsack A = {a1, ….an}, where ai is discrete logarithm of pi to base b: 1  ai < M, such that: Private Key: P, M, b Public Key: A

Merkle-Hellman Multiplicative Knapsack Cryptosystem- Encryption Binary Plaintext T breaks up into sets of n elements long: T = {T1, …Tk} For each set Ti compute Ci is the ciphertext that corresponds to plaintext Ti C = {C1, …Ck) is ciphertext that corresponds to the plaintext T C is sent to Alice

Merkle-Hellman Multiplicative Knapsack Cryptosystem- Decryption For each Ci computes Si is a subset product of the easy knapsack: Tij = 1 if and only if pj divides Si

Merkle-Hellman Multiplicative Knapsack Example Easy (Private) Knapsack: P = {2, 3, 5, 7} M = 211, b = 17 Hard (Public) Knapsack: A= {19, 187, 198, 121} 2  1719(mod 211), 3  17187(mod 211), 5  17198(mod 211), 7  17121(mod 211) Plaintext: T = 1101 Ciphertext: C = 327 = 19 + 187 + 121 Decryption: S = 42 = 17327(mod 211) 42 = 21 * 31 *50 * 71 Plaintext: 1101