## 这是一篇来自加拿大的关于解决下面多重加密，哈希函数，AES operations等几个问题的作业代写

1. Multiple encryption (Slide 2.51, 6 marks)

No collaboration. Although Problems #2—#5 on this assignment may be solved collaboratively, this problem must be solved independently under the “open book but not open Internet” academic integrity rules. You are permitted to use anything that is posted on the course LEARN,Crowdmark, and Piazza pages, and any notes that you have created yourself. However, you are not permitted to use the internet to search or ask for answers. You are also not permitted to communicate with anyone (except for the instructor and TA’s, who will provide only limited assistance on this question) in any form about this question until after the assignment deadline.

Let E be a block cipher with key space K = {0, 1} 64, plaintext space M = {0, 1} 64, and ciphertext space C = {0, 1} 64. The encryption scheme Four-E has key space K = {0, 1} 256. A plaintext m ∈ {0, 1} 64 is encrypted under key k = (k1, k2, k3, k4) (where k1, k2, k3, k4 R {0, 1} 64) as follows:

Enck(m) = Ek4 (Ek3 (Ek2 (Ek1 (m)))).

(a) Give a formula for the decryption process.

(b) Describe and analyze a known-plaintext attack on Four-E that has total running time approximately 2 128 E or E1 operations. (You can ignore the time to sort or store things in a large table.) Estimate the storage requirements of your attack. Justify why the number of known plaintext-ciphertext pairs you have is suffiffifficient to uniquely determine the secret key with high probability.

1. AES operations (Slide 2.69, 8 marks)

(a) Compute f7×aa in GF(28 ). Use the conventions on slide 2.59 for the representation of GF(28) elements. Show your work.

(b) Consider the AES column a = 00000000. Compute MixColumn(a). Show your work.

(c) Consider the AES column a = 01010101. Compute MixColumn(a). Show your work.

(d) Consider the AES column a = 01234567. Compute MixColumn(a). Show your work.

You may use the following tables for multiplication in GF(28 ) by 02, multiplication in GF(28 ) by 03, and xor of hexadecimal digits.

1. Hash functions #1 (Slide 3.15, 4 marks) Let F : {0, 1} 2n → {0, 1} n and G : {0, 1} 2n → {0, 1} n be two hash functions. Defifine the function H : {0, 1} 2n → {0, 1} n by H(x) = G(F(x), F(x)). (Here,the comma “,” denotes concatenation.) Prove that if F and G are collision resistant, then H is also collision resistant.

Note: As mentioned in lectures, such a statement is best proven using the contrapositive statement.

1. Hash functions #2 (Slide 3.21, 8 marks)

Suppose that f : {0, 1} 512 → {0, 1} 256 is a compression function that is preimage resistant. Defifine H : {0, 1} 1024 → {0, 1} 256 as follows. Given x ∈ {0, 1} 1024, write

x = xLk xwhere xL, xR ∈ {0, 1} 512 .

Then defifine

H(x) = f(xL xR).

(a) Prove or disprove: H is preimage resistant.

(b) Prove or disprove: H is second-preimage resistant.

(c) Prove or disprove: H is collision resistant.

1. MACs (Slide 4.12, 4 marks)

Suppose that EMAC (as described on slide 4.12 for authenticating variable-length messages) is used with one key (so k = s). Show that this variant of EMAC is insecure.