Analysis of the BIKE post-quantum cryptographic protocols and the Legendre pseudorandom function

The field of post-quantum cryptography studies cryptographic systems that are secure against an adversary in possession of a quantum computer. In 2017, the National Institute of Standards and Technology (NIST) initiated a process to standardize quantum-resistant public-key cryptographic algorithms (NIST PQC Project). In this thesis we analyze the performance and security of the Bit-Flipping Key Encapsulation Mechanism (BIKE) -- one of the candidates in the NIST PQC project which advanced to the second round of the standardization process. BIKE is a code-based cryptographic system featuring three different variants of the protocol. In the first round of the NIST PQC project BIKE offered security only against chosen-plaintext attacks (CPA). In the second round, BIKE introduced three new variants that are claimed to be secure also against chosen-ciphertext attacks (CCA). Firstly, we build a secure implementation of the CCA protocol and show that its performance characteristics are only negligibly worse than the CPA variant. In the key decapsulation phase of the protocol BIKE uses a decoding algorithm which fails with some probability, called the Decoding Failure Rate (DFR). We analyze the DFR of two decoders used in BIKE, Back-Flip and Black-Gray, and propose a new decoder, called Black-Gray-Flip, that achieves the same DFR as the two previously used decoders while being almost twice as fast. Finally, we propose an algorithm for inversion of binary polynomials in a polynomial ring used in BIKE-2, the second variant of BIKE. Our implementation of the inversion significantly outperforms previously used algorithms. With this and the fact that the bandwidth requirement for BIKE-2 is the smallest among the three variants, BIKE-2 is positioned as the preferable variant of BIKE. The second part of this thesis studies the Legendre pseudorandom function (PRF) which is proposed to be used in the context of blockchains. We present a new algorithm for cryptanalysis of the Legendre PRF. The complexity of our algorithm is lower than the previous best known algorithm. Moreover, we show the results of breaking three Legendre PRF challenges posed by the Ethereum foundation. The most difficult challenge that we solved set the new record which is not broken so far.