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Instructions

This assignment is part of the mandatory assessment of the COMP0025: Introduction to Cryptography

module and will count 25% towards your final overall mark.

Assignment submission is due via Moodle through the TurnItIn interface on January 04, 2023 at 16:00

UK time. Late submissions will be accepted with deductions according to UCL’s late submission policy.

Only PDF submissions that are typeset with LaTeX, e.g., via https://www.overleaf.com/edu/ucl,

will be accepted. Submissions must not include screenshots, e.g., of handwritten or drawn solutions,

unless explicitly permitted. Students with disability accommodations are excluded from this requirement.

Next to the PDF with your answers, you may be asked to hand in additional files, e.g., containing source

code, which should be submitted separately to the PDF. In particular, do not hand in your files via a zip

archive. For more details, please refer to the instructions in the question text.

This assignment is open note, open book, and open course resources. You must identify sources as

accurately and fully as possible. UCL plagiarism policies will be strictly enforced. For more details, see

http://www.ucl.ac.uk/current-students/guidelines/plagiarism.

You are not allowed to consult other people (outside of course staff) on this work. Each student has to

work on the assignment individually.

Your answers will be judged in terms of their quality, the depth of understanding, and also their brevity.

Explain your answers clearly, but succinctly. Partial credit may be awarded.

The assignment has a maximum of 100 marks allocated as follows:

Q1 Q2 Q3 Q4 Total

Marks 25 25 25 25 100 marks

1

Question 1: Cryptographic Software [25 marks]

This question is meant to prepare you for interacting with security software that you may not be familiar

with. In particular, you will use OpenSSL (https://www.openssl.org/), a widely used open source

cryptography library and tool, to perform certain cryptographic operations.

In many cases the OpenSSL tool comes already pre-installed with your operating system. If that’s not

the case, you can download it from their website or install it through your favorite package manager.

Make sure you are using at least version OpenSSL 3.0.7 from 1 November 2022.

Use OpenSSL to answer the questions below. You are expected and encouraged to read the documentation of the library, and use the help commands. Unless stated otherwise, copy the full commands as

well as any output from those commands to the provided q1answer.sh file and submit it together with

the PDF containing your answers to the remaining questions. Make sure that the script runs without

errors when executed in the same folder as the other provided files (see below). We also expect you to

cite your sources fully.

(a) Check the version of OpenSSL you are using. [1 mark]

(b) You are provided with two encrypted zip files: crypto1.zip.enc and crypto2.zip.enc. One of

those encrypted archives contains files required for the rest of the question. The SHA1 checksum

of the correct file is 917544d912dd14670c583f7be1954c68b4f47622. Verify the checksum of

the two files and identify the correct one. [4 marks]

(c) The archives were encrypted using the AES256 block cipher, with a key derived from the password

cryptorulez using the PBKDF2 key derivation function. Decrypt the encrypted zip file that

you identified previously as the correct one. [4 marks]

Inside the zip file (no need to provide a command for unzipping), you will find an RSA public

key public-key.pem, an associated timestamp file key ts.tsr, as well as three certificates

cert1.pem, cert2.pem, and cert3.pem.

(d) Use OpenSSL to output 16 bytes of random data as base64 and save it to a text file. [1 mark]

(e) Encrypt the random text file towards the provided RSA key. Output the result in base64. [4 marks]

Hint: You can use the pipe operator | and the OpenSSL base64 function after your encryption

command.

(f) Use OpenSSL to read the timestamp file. [1 mark]

(g) When was the timestamp created? What does that imply about the key file? (Answer as a

comment in the provided shell file) [2 marks]

(h) Verify that the timestamp belongs to the public key. [3 marks]

Note: The timestamp file was created using http://timestamp.digicert.com/. The timestamp server’s certificate and any intermediate certificates are included in the timestamp, but you

will need the root certificate to complete the trust chain. You can download the root certificate from: https://www.digicert.com/kb/digicert-root-certificates.htm. It is called

DigiCert Assured ID Root CA and has a SHA1 fingerprint of:

05:63:B8:63:0D:62:D7:5A:BB:C8:AB:1E:4B:DF:B5:A8:99:B2:4D:43

(i) One of the three certificates from the archive is valid; the other two would not be accepted by

a modern browser. Read the three certificates using OpenSSL. For each certificate, state if it is

valid or invalid. If invalid, state the reason. [5 marks]

2

Question 2: Hash Functions [25 marks]

Let H be a hash function that is based on the Merkle-Damg˚ard mode (with MD-finalization) and that

uses a Davies-Meyer compression function f parameterized with 3TDEA (which is basically 3DES and

specified in NIST Special Publication (SP) 800-67 Revision 2).

(a) What are the sizes of the IV, message blocks, and fingerprint of H and why? How are the

mathematical maps for H and f defined (in terms of domain(s) and codomain(s))? [4 marks]

(b) Describe two different attacks on H and their asymptotic complexity in terms of O notation, and

argue whether they are practically feasible. [4 marks]

(c) How would you change H to improve its security against the previously described attacks? How

does the attack complexity change? [2 marks]

Furthermore, consider the following questions:

(d) Let E : {0, 1}

λ × {0, 1}

b → {0, 1}

b be a block cipher. Assume λ = b. Consider the following

compression function

f(x, y) = E

−1

(x, x ⊕ y ⊕ 10b−2

1) ⊕ x .

where 10b−21 is the b-bit sized bit string that starts with bit 1 followed by b − 2 0 bits and ends

with bit 1. Is f collision resistant? Justify your answer either by constructing an attack or by

providing a security proof. [7 marks]

(e) Let F : {0, 1}

∗ → {0, 1}

λ

and G : {0, 1}

∗ → {0, 1}

λ be hash functions, one of which is secondpreimage-resistant, and let

H(x) = F(x ∥ G(x)) ∥ G(x ∥ F(x)) .

Is H second-preimage-resistant? Justify your answer either by constructing an attack or by

providing a security proof. [8 marks]

3

Question 3: Digital Signatures [25 marks]

Consider the textbook RSA signature scheme with N = 99301 and e = 5.

(a) What are ϕ(N) and d and how did you compute them? [2 marks]

(b) What is the message for signature σ = 2022? [2 marks]

(c) On the example of the textbook RSA signature scheme, describe how you can speed up the

signing process using the Chinese Remainder Theorem (CRT). [4 marks]

(d) Given the above parameters, compute the signature for message m = 2022 using the CRT

method. Provide all intermediate computations and results. [6 marks]

Consider the Schnorr signature scheme. Recall that a Schnorr signature σ on a message m for private

key x and public key y = g

x

is of the form σ = (s, c) with c = H(r ∥ m) and s = k + cx where

r = g

k

and k being a random value freshly chosen by the signer. Signature σ is considered valid if

c = H(y

−c

· g

s ∥ m).

Assume that the signer’s hardware randomness generator is faulty and provides a fresh random value

only after every second call. After returning a fresh random value k, the next call may return the value

ak + b for some constants a and b with a certain probability p.

(e) Show that an adversary can extract the secret key x if they are able to obtain two signatures

σ0 and σ1 on two different messages m0 and m1 where the random values are connected as

described above and assuming the adversary knows a, b, and y. [6 marks]

(f) Assuming the adversary collected n consecutive signatures from the signer with the faulty randomness generator (with n being even). What is the probability in terms of p that the adversary

can extract x? [3 marks]

(g) Assuming p = 10−4

, how many signatures would the adversary have to collect to extract x with

a probability of at least 20%? [2 marks]

4

Question 4: Existential Unforgeability [25 marks]

Let λ denote a security parameter. Let MAC1 = (Gen1,Tag1

, Verify1

) and MAC2 = (Gen2,Tag2

, Verify2

)

be two (deterministic) message authentication codes for which it is know that at least one of them is

EUF-CMA secure with respect to λ but it is unknown which one.

(a) Build a correct and EUF-CMA secure message authentication code MAC = (Gen,Tag, Verify) by

combining MAC1 and MAC2 somehow. Provide specifications for Gen, Tag, and Verify and show

that MAC is correct. [5 marks]

(b) Prove that MAC is EUF-CMA secure with respect to λ. [20 marks

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