UNIVERSITY OF WEST ATTICA
SCHOOL OF ENGINEERING
DEPARTMENT OF COMPUTER ENGINEERING AND INFORMATICS

---

Information Technology Security

Cryptography

Vasileios Evangelos Athanasiou
Student ID: 19390005

GitHub · LinkedIn

---

Supervision

Supervisor: Ioanna Kantzavelou, Associate Professor

UNIWA Profile · LinkedIn

Co-supervisor: Angelos Georgoulas, Assistant Professor

UNIWA Profile · LinkedIn

Athens, May 2023

---

Project Overview

This project explores fundamental cryptographic operations using the OpenSSL BIGNUM (BN) library. It focuses on the practical implementation of RSA cryptography, including key generation, encryption, decryption, digital signatures, and certificate verification.

All implementations are written in C and utilize the OpenSSL BN library to handle large integer arithmetic beyond standard machine limits.

---

Table of Contents

Section	Path / File	Description
1	assign/	Official laboratory exercise specifications
1.1	assign/Exercise 2 (Cryptography)_2023.pdf	Assignment description (English)
1.2	assign/Ασκηση 2 (Cryptography)_2023.pdf	Assignment description (Greek)
2	docs/	Project reports and theoretical analysis
2.1	docs/Cryptography.pdf	Technical report (English)
2.2	docs/Κρυπτογραφία.pdf	Technical report (Greek)
3	src/	Cryptographic implementations and verification tools
3.1	src/private_key.c	RSA private key generation
3.2	src/public_key	RSA public key extraction
3.3	src/crypto_msg.c	Message encryption using public key
3.4	src/decrypto_msg.c	Message decryption using private key
3.5	src/sign_msg.c	Digital signature generation
3.6	src/verify_sign.c	Digital signature verification
3.7	src/verify_sign_cert.c	Signature verification using X.509 certificates
3.8	src/bn_sample.c	Big number (BN) operations example
3.9	src/c0.pem, src/c1.pem	RSA certificates
3.10	src/c0_body.bin	Binary certificate body
3.11	src/cert_info.txt	Certificate metadata and inspection output
4	screens/	Experimental evidence and execution results
4.1	screens/Activity1–6/	Step-by-step screenshots for each cryptographic activity
5	README.md	Repository overview and usage instructions

Project Structure

The laboratory work is divided into independent activities, each targeting a specific cryptographic concept:

• Private Key Generation (private_key.c)
  Calculates the RSA private key d using prime numbers p, q, and the public exponent e.

• Message Encryption (crypto_msg.c)
  Encrypts a plaintext message (e.g., a student's name) using the RSA encryption formula:

  $$ C = M^e \pmod{N} $$

• Message Decryption (decrypto_msg.c)
  Decrypts an encrypted hexadecimal ciphertext back into readable text using:

  $$ D = C^d \pmod{N} $$

• Digital Signature Creation (sign_msg.c)
  Generates a digital signature for a given message using the RSA private key.

• Signature Verification (verify_sign_msg.c)
  Verifies message authenticity by comparing the decrypted signature with the original message.

• X.509 Certificate Verification (verify_sign_cert.c)
  Manually extracts and verifies the digital signature of a real-world web server certificate
  (e.g., apachefriends.org).

---

Key Mathematical Formulas Used

• Modulo (RSA modulus):

$$ N = p \times q $$

• Euler’s Totient Function:

$$ \phi(N) = (p-1)(q-1) $$

• Private Key Calculation:

$$ e \cdot d \equiv 1 \pmod{\phi(N)} $$

• Encryption:

$$ C = M^e \pmod{N} $$

• Decryption:

$$ D = C^d \pmod{N} $$

---

Key Observations

Sensitivity

Even a single-bit change in a message or signature produces a completely different result, ensuring data integrity.

Efficiency

The OpenSSL BN library efficiently manages large integers required for secure RSA operations.

Real-world Application

The same RSA principles implemented in this lab are used in X.509 certificate verification for secure web communications.

---

Conclusion

This laboratory project demonstrates how theoretical cryptographic principles are applied in practice using professional-grade libraries. It bridges the gap between academic RSA concepts and their real-world security applications, such as encrypted communication and digital certificate validation.

---

Installation & Setup Guide

This guide explains how to install prerequisites, compile, and execute the Cryptography laboratory project, which implements RSA cryptographic operations using the OpenSSL BIGNUM (BN) library in C.

The project is intended for academic and laboratory use within the context of the Information Technology Security course at the University of West Attica (UNIWA).

---

Prerequisites

1. Operating System

• Linux-based OS (recommended)
  • Ubuntu 16.04 / 18.04 / 20.04
  • SEED Ubuntu VM (fully compatible)

2. Required Software

C Compiler

• GCC

Verify installation:

gcc --version

If not installed:

sudo apt update
sudo apt install -y build-essential

OpenSSL Development Library (Mandatory)

The project relies on the OpenSSL crypto (BN) library. Install OpenSSL development headers:

sudo apt install -y libssl-dev

Verify installation:

openssl version

Python (Optional but Recommended)

Used for hexadecimal encoding/decoding and verification steps.

python3 --version

Install if missing:

sudo apt install -y python3

---

Installation

1. Clone the Repository

git clone https://github.com/Information-Technology-Security/Cryptography.git
cd Cryptography/src

---

Compilation Instructions

All programs must be linked against the OpenSSL crypto library -lcrypto.

Compile each file individually, as each source file represents a separate cryptographic activity.

1. Big Number Example (BN Library)

gcc bn_sample.c -o bn_sample -lcrypto
./bn_sample

2. RSA Private Key Generation

gcc private_key.c -o private_key -lcrypto
./private_key

This program:

• Computes RSA modulus N
• Calculates Euler’s totient φ(N)
• Derives the private exponent d

3. RSA Public Key Extraction

gcc public_key.c -o public_key -lcrypto
./public_key

4. Message Encryption (Public Key)

gcc crypto_msg.c -o crypto_msg -lcrypto
./crypto_msg

Encrypts a plaintext message using:

$$ C = M^e \pmod{N} $$

5. Message Decryption (Private Key)

gcc decrypto_msg.c -o decrypto_msg -lcrypto
./decrypto_msg

Decrypts ciphertext using:

$$ D = C^d \pmod{N} $$

6. Digital Signature Generation

gcc sign_msg.c -o sign_msg -lcrypto
./sign_msg

Produces a digital signature using the RSA private key.

7. Digital Signature Verification

gcc verify_sign.c -o verify_sign -lcrypto
./verify_sign

Validates message authenticity by comparing hashes.

8. X.509 Certificate Signature Verification

gcc verify_sign_cert.c -o verify_sign_cert -lcrypto
./verify_sign_cert

This program:

• Extracts certificate fields
• Verifies the digital signature of a real-world X.509 certificate
• Confirms authenticity using RSA public key parameters

---

Troubleshooting

Issue	Cause	Solution
openssl/bn.h not found	Missing OpenSSL headers	Install libssl-dev
Undefined reference to BN_*	Missing crypto library	Add -lcrypto to linker flags
Compilation fails	Old GCC	Update build tools
Incorrect output	Wrong key parameters	Verify p, q, e values

---

Open the Documentation

1. Navigate to the docs/ directory
2. Open the report corresponding to your preferred language:
   • English: Cryptography.pdf
   • Greek: Κρυπτογραφία.pdf