# Evaluating the Efficiency of Physical and Cryptographic Security Solutions for Quantum Immune IoT
## 3 Quantum Immune Security Solutions
### Implementation of Secrecy Coding Scheme  
It is shown in [45] that polar codes provide strong security for binary discrete channels, that is, channels with binary inputs and outputs. One takes advantage of the so-called bit-channel polarization phenomenon of polar codes. If we assume that [PERSON]'s message consists of \(n\) bits, then polar encoding operation can be seen as transmitting those \(n\) bits through \(n\) different parallel bit-channels. It can be shown that, as \(n\) grows, the bit-channels start polarizing: they approach either a noiseless channel or a pure-noise channel. We say that the noiseless bit-channels are good while the noise-pure channels are bad. Given the channel polarization phenomenon, the general idea of the secure communication scheme is quite simple. One transmits random bits over the bit channels that are good for [PERSON], message bits over the bit channels that are good for [PERSON] but bad for [PERSON], and zeros, or frozen bits, over the bit channels that are bad for both [PERSON] and [PERSON]. We implemented a polar-based secrecy-coding scheme shown schematically in Figure 4. We propose using a concatenated secrecy-coding scheme, where the outer polar code is concatenated with the inner forward-error correcting code, the scheme that is proposed in [20]. The inner low-density parity check (LDPC) code provides reliability, and the outer polar code provides secrecy. Figure 4: Concatenated low-density parity check (LDPC)-polar coding for multiple-input, multiple-output orthogonal frequency-division multiplexing (MIMO-OFDM) system. We use quadrature phase shift keying (QPSK) modulation with LDPC coding scheme from IEEE 802.16 standard with a code length of 1248 bits and a code rate 5/6, i.e., the payload is 1040 bits. We plot the bit error rate versus the signal-to-noise ratio that can be achieved with a given LDPC code in Figure 5. The rationale for selecting QPSK as modulation method is the fact that a communication channel between [PERSON] and [PERSON], including QPSK modulator and demodulator, is equivalent to binary symmetric channel with a crossover probability \(P_{[PERSON]}\)[46]. Similarly, a communication channel between [PERSON] and [PERSON] is equivalent to another binary symmetric channel with a crossover probability \(P_{[PERSON]}\). These equivalence properties do not necessarily hold for high-order quadrature amplitude modulations (QAM) such as 16-QAM or 64-QAM. Our design parameters for secrecy coding are as follows: We require [PERSON]'s bit-error-rate to be at the most \(P_{[PERSON]}=10^{-4}\) after LDPC decoding, which, in turn, implies that [PERSON]'s signal-to-noise ratio should be at least 6 dB, cf. Figure 5. Furthermore, we would like [PERSON]'s bit-error-rate to be at least \(P_{[PERSON]}=10^{-1}\) after LDPC decoding, which, in turn, implies that [PERSON]'s signal-to-noise ratio should be at most 2 dB, cf. Figure 5. The required difference in signal-to-noise ratios between [PERSON] and [PERSON] is commonly referred to as radio advantage. For the given parameters, the minimum required radio advantage is 4 dB. For the given values of [PERSON]'s and [PERSON]'s bit error rates, the secrecy capacity of a binary wiretap channel is  
\[C_{s}=h\left(P_{[PERSON]}\right)-h\left(P_{[PERSON]}\right)\approx 0.457588\text{ bits per channel use.} \tag{1}\]  
The symbol \(h(P)\) in (1) denotes the binary entropy function  
\[h(P)=-P\log_{2}P-(1-P)\log_{2}\left(1-P\right). \tag{2}\]  
We have used the design method presented in [45] to determine which polar bit-channels should be used to convey a secret message under the assumption that we use 1024-bit long polar code, \(P_{[PERSON]}=10^{-4}\), and \(P_{[PERSON]}=10^{-1}\). Namely, we identified 479 bit-channels that are good for [PERSON], 361 bit-channels that are good for [PERSON] but bad for [PERSON], and 184 bit-channels that are bad for both [PERSON] and [PERSON].