Name 525 Homework 2 Block Ciphers 1 2 Pts Recap

Name 525 Homework 2 Block Ciphers1 2 Pts Reca

Name______________________ 525 Homework 2: Block ciphers 1. (2 pts) Recall that S-DES operates on a block size of 8 bits, producing a ciphertext that is also 8 bits long. a. In this scenario, how many different ciphertexts are there? b. How many different 8-bit block ciphers can there be? 2. (2 pts) Explain why it is impractical to use an ideal block cipher for real-world encryption. 3. (2 pts) To make a secure block cipher, why is diffusion not enough? That is, why is confusion also necessary? 4. (3 pts) Rewrite each bit string according to the given permutation. a. , P = (4, 1, 2, 7, 8, 5, 3, 6) b. , P = (5, 1, 7, 2, 8, 3) c. , P = (2, 5, 8, 5, 3, 1, 6, 6, 4, 7, 4, . (3 pts) Below is a substitution table (S-box) expressed in octal (base-8, digits are 0-7). Note that it replaces two digits with one digit, cutting the size of the resultant string in half: Use the table to replace each of the following 8-digit values with 4-digit values, by using the first digit of each pair to index the row and the second to index the column. a. b. c. . (5 pts) Suppose we use S-DES to encrypt m = with the key K = . Give the values of K1, IP(m), and the output at the end of the first round, that is, fK1(IP(m)). Show your work.

Paper For Above instruction

The assignment requires an exploration of fundamental concepts in block cipher cryptography, including understanding the total number of possible ciphertexts, the rationale behind the impracticality of using an ideal cipher in real-world situations, and the essential components that make a block cipher secure. Additionally, it involves applying permutation and substitution techniques to bit strings, as well as performing a sample encryption step in S-DES, which embodies the core principles of symmetric key cryptography.

Understanding the Number of Possible Ciphertexts and Cipher Variants

Firstly, in the context of the S-DES (Simplified Data Encryption Standard), which operates on 8-bit blocks, the total number of possible ciphertexts corresponds to the number of different 8-bit binary strings. Since each bit can be either 0 or 1, there are 2^8, or 256, possible unique ciphertexts. This implies that, for any given plaintext, the cipher can produce any of these 256 ciphertexts, depending on the encryption key and process involved.

Furthermore, the total number of possible 8-bit block ciphers refers to the number of distinct permutation or transformation functions that can be established over the set of 256 input patterns. Conceptually, if each input pattern could be mapped independently to any output pattern, the total number of possible unique functions (permutations) would be (256)! (factorial of 256). However, in practice, ciphers are designed with specific structures, and the number of feasible algorithms is constrained by their design principles. Nonetheless, the theoretical universe of 8-bit block cipher functions is astronomically large, emphasizing the vast potential for secure encryption schemes.

Practical Limitations of Ideal Block Ciphers

Using an ideal block cipher in real-world encryption scenarios, while desirable due to its perfect secrecy properties akin to a one-time pad, is highly impractical. An ideal cipher requires that each key corresponds to a completely random permutation of all possible plaintexts. Implementing such a scheme is computationally infeasible because it demands generating, storing, and managing a vast number of permutations—specifically (2^n)! for an n-bit block size, which rapidly becomes impossible as n grows. Additionally, key management becomes complex, and ensuring perfect randomness in deterministic algorithms is unattainable. Therefore, modern cryptography relies on building secure, computationally feasible algorithms that approximate ideal properties without the impracticalities of true randomness and perfect secrecy.

Diffusion Versus Confusion in Cryptography

Achieving a secure block cipher involves more than just diffusion, which aims to spread the influence of each plaintext bit over many ciphertext bits. While diffusion ensures that small changes in the plaintext alter the ciphertext substantially—a property essential to thwart certain cryptanalytic attacks—this alone does not suffice. Confusion, introduced by complex substitution layers such as S-boxes, obscures the relationship between the key and the ciphertext. Without confusion, an attacker could potentially deduce the key or predict ciphertexts through analysis of the structure. Thus, both diffusion and confusion operate synergistically to produce cipher systems that are resistant to cryptanalysis, with diffusion dispersing statistical structures and confusion complicating the key's influence on the ciphertext.

Permutation of Bit Strings

Given a permutation P, reordering bits within a string according to P rearranges their positions to either obscure or preserve certain patterns. For example, suppose P = (4, 1, 2, 7, 8, 5, 3, 6); applying this permutation to an 8-bit string rearranges its bits such that the bit originally in position 4 moves to position 1, the original in position 1 moves to position 2, and so on. Each permutation enhances security by mixing the positions of bits, contributing to the cipher’s diffusion property. Similar permutations are applied with different P vectors to achieve desired levels of scrambling within cryptographic rounds.

Substitution Using S-Boxes

In cryptography, substitution boxes (S-boxes) perform non-linear substitution, replacing input bits with different output bits to increase complexity. Given an S-box expressed in octal, we interpret the two octal digits as a row and a column index, respectively, to find the replacement value. For example, if the input is 8 bits, they can be divided into four pairs of 2 bits each, with each pair mapped through the S-box. The first digit of each pair specifies the row, and the second specifies the column. This operation ensures cryptographic non-linearity, making ciphertext weaker to linear and differential cryptanalysis.

Sample Encryption Process in S-DES

Suppose we have a plaintext message m and a key K. To perform encryption using S-DES, one would first generate subkeys, such as K1, from the main key K. The initial permutation IP(m) reorders bits of the plaintext before applying the function fK1, which combines the subkey with the permuted plaintext through substitution and permutation layers. Calculating IP(m) involves applying a fixed permutation table, while the function fK1 involves expanding, substituting via S-boxes, and permuting again. The output after the first round, fK1(IP(m)), forms the basis for further rounds, ultimately leading to the ciphertext.

Conclusion

This exploration highlights the essential facets of block cipher design, including the importance of large key spaces and permutations, the critical roles of diffusion and confusion, and the practical limitations faced in implementing truly ideal ciphers. Through understanding substitution, permutation, and cryptographic principles exemplified in algorithms like S-DES, one gains insight into establishing secure encryption mechanisms necessary for safeguarding confidential information in digital communications.

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