Cryptography

Questions & Answers

Question 3 (total of 6 marks) Choose one of the modes of operation for block ciphers discussed in class. a. Describe in detail how the mode of operation you have chosen works, highlighting features and weaknesses (if any). (2 marks) b. Describe and reference a real-world use of the mode of operation you have chosen, stating and justifying any advantages or disadvantages of this mode of operation in this context. (4 marks) Your answer to part b of Question 3 should not exceed half a page of A4 paper.


(a) The ephemeral Diffie-Hellman Key Exchange (DHKE) protocol allows two parties to agree on keying material in the presence of an adversary. The protocol assumes the two parties already agreed on two primes p, q such that q divides p - 1 and a value> 1 such that of 1 mod n From this starting point, describe the remainder of the protocol, recalling that the ephemeral version of the protocol involves the exchange of fresh Diffie-Hellman values.[6 marks] (b) Assuming that the adversary is passive (i.e. acts only as an eavesdropper), identify the computational problem underlying the security of this protocol. How does it relate to the Discrete Logarithm Problem (DLP) in the given setting? [4 marks] (c) How large should p and q be so that the ephemeral DHKE protocol in your answer to Question 4(a) is secure against an adversary willing to expend an effort of 280 basic operations? What if the adversary is willing to expend an effort of 2128 basic operations? Justify your answers with reference to algorithms for solving the DLPin the given setting.[6 marks] (d) Discuss the security weaknesses of ephemeral DHKE in the situation where the adversary is an active party. (e) Explain how you might modify the ephemeral DHKE protocol to avoid the weak-nesses identified in your answer to Question 4(d).[4 marks] (f) The ElGamal Public key encryption (PKE) scheme is derived from the Diffie-Hellman key exchange algorithm. Describe the El Gamal algorithm, and the relationship be-tween DHKE and the EI Gamal PKE.[4 marks] (g) Public key encryption can also be used to establish keying material in the presence of an adversary. (i) Describe a simple protocol for achieving this. (ii) Compare and contrast the approaches based on PKE and ephemeral DHKE[4 marks]in terms of security and efficiency.


(a) An Authenticated Encryption (AE) scheme consists of a triple of algorithms,(KGen, Enc, Dec). Describe the function of each of these algorithms and explain what is meant by the correctness of an AE scheme.[4 marks] (b) Security for AE schemes is defined in terms of the combination of two security notions: indistinguishability under chosen plain text attacks (IND-CPA security), and integrity of ciphertexts (INT-CTXT security). Give informal descriptions of these two notions, using diagrams to illustrate your answer if you wish. For both notions, state what it means for an AE scheme to be[8 marks]secure. (c) AE schemes can be built using generic composition of symmetric encryption schemes and MAC schemes. There are three principal methods for doing so, known as EtM,MtE and E&M. Briefly describe each of these three methods, and comment on their AE security when instantiated using an IND-CPA secure encryption scheme and a strongly unforgeable MAC scheme. In each case, justify your answer.[12 marks] (d) In applications, we are often interested in simultaneously providing confidentiality and integrity for some data but only integrity for other, associated data. An Authenticated Encryption with Associated Data (AEAD) scheme meets this goal. Define the syntax of an AEAD scheme and show how to extend the generic EtM construction of an AE scheme to obtain an AEAD scheme. Use a diagram to illustrate the second part of your answer.[6 marks] (e) Nonce-based AEAD is a further extension of the AEAD paradigm. Explain what is meant by nonce-based AEAD and why it is a good primitive to offer to software developers.[4 marks]


(a) Give a formal definition of a block cipher. Your answer should make reference to the block size n and the key size k.. (b) Explain why block ciphers are usually used in modes of operation. (c) Describe the Electronic Code Book (ECB) mode of operation, using a diagram to illustrate your answer if you wish.[2 marks] (d) Explain what the main security limitation of ECB mode is. Illustrate your answer with reference to a specific example.[4 marks] (i) Draw a diagram to illustrate the encryption algorithm of this mode. (ii) Write down equations, similar to those above for encryption, that describe thedecryption operation for CBC mode.[2 marks] (f) The XML encryption scheme operates on byte-oriented data and uses the following padding method for CBC mode: at least 1 byte of padding is always appended to the raw message M, and at most one complete block of padding is appended; if s bytes of padding are needed for some s > 1, then appends - 1 random bytes to M followed by the byte encoding of integer s. So, for example, if s = 1, then the padding appended to the message M is just 0x01, while if s = 2, then a random byte followed by 0x02 is appended, and so on. (i) Explain in detail, using pseudo-code, how a typical implementation would re-move the padding from a plain text to recover M for this padding scheme. Your pseudo-code should work for general block sizes, and you should use a variable name block size in your pseudo-code to reflect this. Your pseudo-code should also generate an error message "padding error" if the padding is in-valid in some way. (Hint: your pseudo-code should check that the number of padding bytes does not exceed the maximum possible value that is implied by there being at most one complete block of padding.)[4 marks] (ii) A simplified version of XML permits bytes of any value to occur in messages M except for 0x00. After CBC mode decryption and padding removal, a simplified XML implementation checks whether the resulting message M contains byte value 0x00. If a byte with this value is found in any position of M, then the implementation returns an error string "parsing error", otherwise normal XML processing continues (and no error is returned).Now suppose you have a target block of cipher text C that is known to correspond to a full message block P* (that is, P* is a plain text block entirely from M, not containing any padding). Suppose you also know C+₁, the cipher textblock preceding C".-15 (iii) By modifying delta in other positions, or otherwise, show how to recover the value of the last byte of plain text (the padding byte) in the cipher text IV*, C* that you obtained in the previous part of the question. (Hint: consider modifying IV* by all possible XOR offsets in each position and requesting decryptions. One offset will produce 0x00 in the plaintext in position j. Does this cause a parsing error or not?) (iv) By further modifying the ciphertext IV*,C", or otherwise, sketch how you would recover the complete plaintext block P. (Hint: consider modifying IV*so that the padding byte now contains 0x01, and then doing a similar analysis to that in the previous part.)


(a) What security service is a Message Authentication Code (MAC) intended to provide? (b) A MAC scheme consists of a pair of algorithms (Tag, Vrf). Describe the function of each of these algorithms and explain what is meant by the correctness of a MACscheme.[4 marks] (c) The standard notion of security for a MAC scheme is existential unforgeability under chosen message attack, or EUF-CMA security for short. Give a description of this security notion. You may use a diagram to illustrate your answer.[6 marks] ==(d) Suppose (Tag, Vrf) is an EUF-CMA-secure MAC scheme. Consider the MAC scheme (Tag', Vrf') in which Tag (m) = Tag (m)||m, where || denotes concate-nation of strings, mo denotes the first bit of m, and in which Vrf' is defined in the obvious way. (i) Provide a brief argument to show that (Tag', Vrf') is also EUF-CMA-secure. (ii) What can you conclude about the confidentiality that is provided by an EUF-CMA-secure MAC scheme for its input messages m?[1 marks] (iii) Consider combining this MAC scheme with an IND-CPA-secure symmetric encryption scheme in an "Encrypt & MAC" construction to obtain an Authenticated Encryption (AE) scheme. Describe the ciphertexts in this scheme. What can you say about the security offered by this construction? What can you conclude about the generic security of the "Encrypt & MAC"approach to building AE?[3 marks] (e) Consider the following MAC scheme, based on an n-bit hash function H: let the key K also have n bits; define Tag (m) := +K; Vrf' is defined in the obviousway. (i) Show that this MAC algorithm is vulnerable to a key recovery attack. State the resources that your attack consumes, in terms of calls to the oracles in the EUF-CMA security definition.[3 marks] (ii) In general, how does a key recovery attack on a MAC scheme relate to its EUF-CMA security? (f) Recall the Merkle-Damgård construction of an n-bit hash function from a compression function h : {0,1}* × {0,1}" → {0,1}", a padding scheme pad(-) such that the bit-length of m' = pad(m) is a multiple of k, and an n-bit value IV: Compute m' = pad(m). • Break m' up into blocks m,...,m, each of bit-length k. Set ho = IV and compute h; | = h(m², h¡) for i = 0, … ..‚† — 1. Output h₁. (i) Draw a diagram to illustrate the hashing operation for this construction. [3 marks] (ii) Suppose that the padding scheme pad(-) adds a length field and additional padding bits to its input messages, so that pad(m) = m len(m)||padding. Show that the hash function is vulnerable to length extension attacks. That is, show that if H (m) is known for some (possibly unknown) bit-string m, then H(m||1en(m)||padding||s) can be computed, for any chosen bit-string 8. (iii) Now consider the following MAC scheme, based on such a hash function H:let the key K have k bits and define Tag, (m) H(Km); algorithm Vrfis defined in the obvious way. Show that this MAC scheme is vulnerable to a forgery attack, by exploiting the length extension weakness of H. What resources does your attack consume, in terms of calls to the oracles in the EUF-CMA security definition?[4 marks]


(a) The ephemeral Diffie-Hellman Key Exchange (DHKE) protocol allows two parties to agree on keying material in the presence of an adversary. The protocol assumes the two parties already agreed on two primes p, q such that q divides p - 1 and a value >1 such that gº = 1 med p.0 From this starting point, describe the remainder of the protocol, recalling that theephemeral version of the protocol involves the exchange of fresh Diffie-Hellmanvalues.[6 marks] (b) Assuming that the adversary is passive (i.e. acts only as an eavesdropper), identify the computational problem underlying the security of this protocol. How does it relate to the Discrete Logarithm Problem (DLP) in the given setting? [4 marks] (c) How large should p and q be so that the ephemeral DHKE protocol in your answer to Question 4(a) is secure against an adversary willing to expend an effort of 280 basic operations? What if the adversary is willing to expend an effort of 2128 basic operations? Justify your answers with reference to algorithms for solving the DLP in the given setting.[6 marks] (d) Discuss the security weaknesses of ephemeral DHKE in the situation where the adversary is an active party. (e) Explain how you might modify the ephemeral DHKE protocol to avoid the weaknesses identified in your answer to Question 4(d).[4 marks] (f) The ElGamal Public key encryption (PKE) scheme is derived from the Diffie-Hellman key exchange algorithm. Describe the EIGamal algorithm, and the relationship be-tween DHKE and the EI Gamal PKE.[4 marks] (g) Public key encryption can also be used to establish keying material in the presence of an adversary. (i) Describe a simple protocol for achieving this. (ii) Compare and contrast the approaches based on PKE and ephemeral DHKE in terms of security and efficiency.


Question 3 (total of 6 marks) Choose one of the modes of operation for block ciphers discussed in class. a. Describe in detail how the mode of operation you have chosen works, highlighting features and weaknesses (if any). (2 marks) b. Describe and reference a real-world use of the mode of operation you have chosen, stating and justifying any advantages or disadvantages of this mode of operation in this context. (4 marks) Your answer to part b of Question 3 should not exceed half a page of A4 paper.


Question 1 Explain what hybrid encryption is and discuss why it is a commonly used approach in cryptography.


(a) Show that (1, 1) ⊕(1, 1) = (5,6). Write out your work. (b) Show that (5, 1) ⊕(5, 1) = (5, 6). Write out your work. (c) Show that (1, 1) ⊕ (3,0) = (5, 1). Write out your work. (d) Show that (1, 1) ⊕(5, 6) = (3,0). Write out your work. (e) Show that (1,6) ⊕(5, 1) = (3,0). Write out your work.


Question 1 (total of 6 marks) Critique the following statement: We achieve security by obscurity: by keeping our algorithms secret we obtain the best guarantee of security. Your answer should cover roughly half a page of A4 paper. (6 marks) (ps: You are strongly encouraged to answer this question using a word processing / typesetting program, e.g., LaTex, Word. Please avoid submitting it handwritten.)


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