Secured polar code derived from random hopped frozen-bits

Modern communication systems, especially those used by military applications, need to be able to send information securely. In this regard, numerous cryptographic solutions have been introduced for secure data transmission. We propose a novel concept for integrating channel coding and cryptography into a single system block to create secure connections between the transmitters and receivers facing the problem of insecure communication. Polar code is our channel coding to execute the new strategy to create safe communication over unsecured channels [1].

The secure data transmission is critical in current communication systems to ensure secrecy among entities. Usually, the protection will depend on utilizing a separate cryptosystem to encrypt the transmitted data. But, gathering the cryptosystem with channel coding techniques in one block and reducing computational complexity is undoubtedly an exciting feature. In addition, our proposal includes a feature that embeds security into the channel coding block. This joint encryption-channel coding scheme has good security, low error performance, and simple implementation.

Arikan presented polar codes in 2009 [2]. It is frequently another error-correcting linear block code class with a It has been demonstrated that the polar codes can achieve Shannon's capacity for a terrific selection of channels with a simplified development process and shallow encoding and decoding complexity, including the dual symmetric channels and discrete memoryless channels. Several modifications to the fundamental structure of polar codes were suggested, demonstrating that this code is ideal for lossless and unfortunate source coding [3]. In this paper, we introduce a new crypto polar code scheme by reconstructing its input indices using a different strategy than the original Arikan's code. This new construction strategy is known only by the transmitter and receiver.

The hopping technologies provide excellent protection against eavesdropping and jamming in digital communication systems by introducing frequency diversity and a high ability for interference rejection to improve the system's resistance against interference and develop overall system security. For those reasons, hopping is an excellent technique for struggling with Rayleigh fading channel issues, cutting down on interleaving depth, and empowering productive frequency reuse in multiple-access communication systems. In this regard, we proposed another methodology for secured data transmission that relies on the random hopping technique, which is based on a Random Number Generators (RNG's) [4]. In this way, the idea of using a random hopping pattern will make it harder for different attacks to work.

The RNG techniques are classified into two categories: pseudo-random number generators (PRNGs) and true random number generators (TRNGs) [5]. These two classes generate a random-like sequence with a high degree of perceptible randomness. Therefore, it can achieve a random-like arrangement that is useful for certain security functions for simulating and cryptographic purposes.

One of the most widely used RNGs is Mersenne Twister (MT), which is a PRNG that meets all the security requirements to be a suitable RNG for cryptographic applications [6]. In a short amount of time, MT allows for creating exceedingly high-quality pseudo-random numbers with an extended period length. The encrypted form of MT is called Crypt MT, and it is unaffected by memory-trade-off attacks, and it is much quicker than other stream cyphers. Our security polar code system uses MT-PRNG to make random-looking values called "frozen bits" and their "hopping indices."

Recently, several techniques have been developed to ensure the reliability of the transmitted data over noisy wireless channels, especially over one-way noisy communication channels (NCC) and two-way NCC, such as mobile communication systems that allow a feedback channel from the receiver to the transmitter. For example, the design of image transmission systems that use feedback channels to maximize reliability has been addressed by using a combined source-channel coding with a feedback channel to adapt the source and channel coding rate according to the channel condition [7].

The use of chaotic signals to transmit information over a noisy channel with additive white Gaussian noise that has a normal noise distribution by giving an explicit formulation for the optimal receiver to minimize the bit-error-rate for the symbol-to-chip rate and the noise variance [8]. Moreover, one of the solutions to get a robust security communication scheme is to use a communication system with partial noisy feedback using non-malleable codes to detect and correct transmission errors [9]. Another protocol has been shown over a lossy bosonic channel, which is a change to the feedback-assisted private communication model [10].

Numerous systems rely on polar codes to ensure a reliable secured communication link, where all the bit channels are polarized completely when the code length reaches infinity. In the case of small code length, the privacy rate is degrading, where many bit channels do not consummately polarize. Thus, combining several coding schemes with polar codes, such as Repetition Coding (RC) and Exclusive Or Coding (XORC), improved the system's privacy and safety. So, the bad polarization issue has been resolved [11].

According to the primary polar code construction that is not secured, the assailant can build the generator matrix quickly, especially in the case of non-systematic polar codes. This matrix can be easily generated in the Binary-Discrete Memoryless Channel (B-DMC) case. So, the advantages of polar code have been utilized to use a secret key cryptosystem to keep the generator matrix secret and prevent the assailant from decoding the critical information [12]. In addition, to facilitate a secure transmission without relying on Channel-State-Information (CSI) between the allowed transmitter and the permitted receiver, this can be accomplished by using the Amplify-Forward (AF) relay protocol with polar codes over parallel relay channels [13]. This study achieved a sufficient level of security and the analyses revealed that these schemes resist common attacks on secret key cryptosystems and provides a high code rate with an acceptable error rate.

Following channel polarization, the resultant good channels will convey the user-transmitted data while the induced bad channels will aid in message reconstruction by sharing fixed-value frozen bits. Thus, if the frozen bits in bad channels are kept secret, the rival will have difficulty re-constructing client information in good channels without knowing the frozen bit values. As a result of this discovery, a secure data transmission technique was created by incorporating pre-post processing with polar codes that will secure the conveyed sub-blocks message [14].

The concept of encrypting the frozen bits to remain secret between the transmitter and receiver by using a conventional symmetric cryptography system will make it difficult for the adversary who knows the standard frozen bits to decode the entire message. So, the adversary's major problem becomes intractable, as assailants can correctly decode only fractional data. But the system's complexity increases. Additionally, a secure polar code has been developed based on a chaotic one-dimensional (1-D) map and using a secret pre-shared introductory condition as a Chaotic-pseudo-Random-Binary-Generator (CPRBG) to generate secret hidden values for frozen bits. This method makes decoding the data message by decoders that do not have the chaotic mystery starting condition extremely challenging. As a result, another 1-D chaotic map generated secret data bits to increase system security. That gives a single way to fix unsafe data channel links and the ability to keep information private and secure [15].

Also, an efficient and secure scheme has been constructed by joining the channel coding with encryption dependent on polar codes with the Rao-Nam (RN) cryptosystem. Compared to previous work, this scheme provides a sufficient level of security with a moderately smaller key size. The results show that the technique is simple to implement and benefits from a higher coding rate, which can be leveraged to approach channel capacity when using sufficiently big polar codes. The main advantage of this scheme is that, by increasing the block length of polar codes, we may get a higher coding rate and a higher level of safety without modifying the scheme's key size. The scheme's backward properties make it good for high-speed transmissions, like the ones used in deep space communication systems [16], 17, 18.

In this article, we contribute another secure channel coding plan over Additive White Gaussian Channel (AWGN) based on consolidating polar codes with MT PRNG. This scheme was designed to provide data security for communication systems against various attack techniques. Polar codes are recognized to account for channel polarization marvel (CPM), which generates bit channel indices for both K information bits and N-K frozen bits, where N is the code length. So, for the decoder to be able to decode all bits correctly, it needs to know the indices and values of each bit.

The contribution of this paper is to design a new efficient and secure polar coding scheme based on the use of MT with an arbitrary starting seed state to generate hopping polarized bit channel indices and random secure frozen bits values. So, the construction of our scheme makes it hard for the receiver to figure out both the value of frozen bits and its input indices without knowing the MT's pre-shared initial seed. Summarize the relevant work and presented its security concept with its used security system, as shown in Table 1.

The remaining sections of the article are organized as follows: offering a comprehensive overview of the RNGs' foundation, outlining the principles of polar codes in Sect. 2, and demonstrating the MT Theories in Sect. 3. Using MT-PRNG to generate the values of frozen bits and hopping polarized bit channel indices, Sect. 4 describes the novel secured polar coding technique. In Sect. 5, the framework model and numerical security analysis were presented. Section 6 finally marks the conclusion of this work.

Shannon demonstrated the possibility of a channel code that accomplishes the noisy channel capacity using random coding [19]. One of these codes is polar codes, which deliver channel capacity and explicitly build on low encoding and decoding complexity. This section gives a general overview of polar coding and the main idea behind how channel polarization encoding works.

There are many algorithms to create random numbers that embrace producing a series of numbers with no recognizable patterns or regularities to appear random. Each of these algorithms has its own degree of randomness in a stream of numbers. MT is one of PRNG, which utilizes a fragmented array to determine the period of a Mersenne prime and consider an alteration of the Twisted Generalized Feedback Shift Register (TGFSR). Additionally, it employs a strenuous decimation strategy to analyze the permittivity of the characteristic polynomial of a direct repeat with a computational complexity of \(O\left( {L^{2} } \right)\) where \(L\) is the level of the polynomial function.

The MT has a period of 219,937–1 and a dimensional equidistributional of 623 with a precision of nearly 32 bits. As a result, it is a quick and efficient in-memory use, as the working region requires only 624 words. Additionally, it avoids multiplications, divisions, and employment at the cash and pipeline points of interest. Additionally, it generates free long-term correlations.

It is noted that MT PRNG has many limitations parameters for initializing every secret seed. These restrictions are shown in Table 2. As a result, no search technique can recognize the secret key parameter values. The MT mechanism is divided into two sections; the repeating shaping portion, which works similarly to the Linear Feedback Shift Register, is the first (LFSR), recursion is used to retrieve every bit of the state. Each bit of the output likewise satisfies the repeating bits. Furthermore, the other is a balancing act. The shift register has 624 cells in total, for a total of 19,937 cells. Every element has a 32-bit length apart from the first, which has just one bit due to bit discarding. In Fig. 3, a high-level block diagram of MT is depicted.

The MT has excellent output for S-distribution tests to its random stream output. It generates a series of identical randomly generated integers between 0 and \(2^{D} - 1\), where the dimension of row vectors is in the finite binary field. As such, it is a significant type of PRNG. It is based on the recurrence of the following equation:where: \(v\) is the level of recurrence \(r,1 \le r \le v\) and R is a regular \(D \times D\) matrix as given in.

\({\mathbf{R}} = \left[ {\begin{array}{*{20}c} 0 & 1 & {} & {} \\ 0 & 0 & \ddots & {} \\ \vdots & {} & {} & 1 \\ {{\text{a}}_{D} - 1} & \cdots & \cdots & {{\text{a}}_{0} } \\ \end{array} } \right]\)

Noting that \(T_{v}\) is the initial seed, with a word-sized \(D\) row vector generated with S equal to 0. Where, \({\text{S}} = 0,1,2, \ldots a\) lso,\({ }T_{0} ,T_{1} , \ldots ,T_{n - 1}\) are initial seeds,\({ }T_{{{\text{S}} + 1}}^{l}\) and \(T_{{\text{S}}}^{u}\) are the lower or rightmost bits of \(T_{{{\text{S}} + 1}}\) and the upper or leftmost bits of \(T_{{\text{S}}}\) respectively, \(\underline{ \vee }\) denotes a bitwise XOR, | denotes a concatenating operation and \(\left( {T_{s}^{u} |T_{s + 1}^{l} } \right)\) is the concatenation vector which gotten by Concatenating the Leftmost bit of \(T_{{\text{S }}}\) and the rightmost bits of \(T_{{{\text{S}} + 1}}^{l}\). Then, from the right, the matrix R is multiplied by this vector. Finally, a bitwise addition \(\underline{ \vee }\) is used to add \(T_{{{\text{S}} + r}}\) and then create the following vector \(T_{{{\text{S}} + v}}\). Finally, the multiplication \(\left( {T_{S}^{u} |T_{S + 1}^{l} } \right)R\) executed utilizing straightforward bit shift operations as exposed below.where \(T\) is \(\left( {T_{D - 1} ,T_{D - 2} , \ldots ,T_{0} } \right)\) It is the last vector of the matrix \({\mathbf{R}}\) and \(\gg\) is a bitwise right shift. Thus, the bit shift,

EXCLUSIVE-OR, and bitwise OR operations will be used to compute recurrence. We develop the S-distribution. The last counterfeit-random number is delivered by a technique labeled hardening. The most significant value of such S has been used to signify the equidistributional dimension. We develop the S-distribution by the raw successions formed by the recursion, which is V-bit precise. Every created word is multiplied by \(D \times D\) invertible matrix E from the right output due to tempering matrix X into \(z = X \times E\). The matrix E is picked like a dual operation which can be proceeded as:

Informing that u, s, t, and l are tempering bit shifts b and c are tempering bitmasks, (\(\ll\)) bitwise left shift and \(\wedge\) bitwise AND operation.

It is noted that the MT PRNG has its limitations parameters for initializing every secret key (K₁ or K₂). These restrictions are shown in Table 2. It is noticed that there are many limitations parameters. Therefore, the searching techniques for recognizing the secret keys parameters values will fail.

The main goal of the proposed scheme is to provide a secure channel coding scheme based on the polar code architecture. This new scheme relies on generating random frozen bits values and randomizing its indices by using MT PRNG in the polar code construction as mentioned previously in Sect. 2, utilizing the AWGN as a channel model, with a variance \(\sigma^{2} = N_{0} /2\) and zero mean, and binary phase-shift keying as a modulation type (BPSK).

The MT PRNG is utilized to generate a secure frozen bit’s values and indices. These are applied as the input frozen bits values and channel polarization indices for the polar code encoder construction. According to the random generation of MT PRNG, the receiver's complexity which has no idea about MT PRNG initial seed will rise. Additionally, it makes it more difficult for attackers to decode and get the message accurately.

The novelty of the proposed security approach for crypto-polar codes is to combine the advantages of cryptographic processes that apply mutation and permutation by duplicating the use of MT PRNG. Therefore, two groups of MT PRNG random outputs are created based on its secret primary state seed, one random output is used for generating random frozen bits values with a pre-shared key of 116 bits in length, and the other random output is used for providing random indices for the encoder input codelength N with a pre-shared key of 116 bits in length. According to the use of two secret pre-shared keys, the system's overall secret key is duplicated to be 232 bits in length. The utilization of MT PRNG enables the polar code encoder to perform mutation and shuffling operations. This imposes the difficulty to predict the random generated frozen bit values and N codelength indices.

The proposed MT-polar code block diagram is shown in Fig. 4. It utilized two MT PRNG initiated by its two initial seeds, which are K1 and K2. Each of these keys has a length of 116 bits. One is used to generate the random values of frozen bits and the second is used to randomize the output channel polarization indices, which are the input of polar code encoder indices. According to the code rate R = K/N, the encoder divided the input codelength (N) indices into K information indices and N-K frozen indices. Subsequently, one MT PRNG with its key generates N-K random frozen bits’ values and the other MT PRNG is used to randomize the N polarized indices. After generating the random N-K frozen bit values and randomizing all of the codelength indices, all of the input codelength bits (N) are interleaved and the K information and N-K random frozen bits are located based on their shuffling indices. As a result, two difficulties will face any receiver trying to decode the polar encoded bits. The first one is how to know the primary locations of the data sequence of the encoded N bit codeword. The second difficulty is knowing the exact values for the frozen bits in the codeword. The previous difficulties cannot be solved until the receiver knows the 232 bits of the two primary keys, k1 and k2, that are used by MT PRNG.

Since it is excessively inspected in PRNG and simplicity in realization, MT has been presented to accomplish better speed security for cryptographic employments. It depends on (9). Therefore, the output sequences of MT PRNG are irregular, which makes it unpredictable nearer to random. Also, concealing the two initial seeds or keys of the used MT PRNG will lead to stash the generated binary random output sequences for frozen bits values and indices, causing augmentation to the receiver’s complexity, the difficulty for the Eavesdropper who attempts to extract information about the system and the transmitted data, and amplifies the system immunity. Consequently, providing higher security than other previous schemes. Corresponding to original polar codes construction that locates the value of frozen bits equal to.

zero, and selected K indices polarized bits channels which have good mutual information \(I\left( H \right)\) or Bhattacharyya parameter \({\text{Z}}\left( H \right)\) to transmit information bits and selected the rest of N Polarized bits channels N-K indices to be frozen bits which have less \(I\left( H \right)\) or \({\text{B}}\left( {\text{H}} \right)\).

Our proposed security scheme changed indices selection of the channel polarization by locating both K information bits and N-K random frozen bits (that generated by MT PRNG) on the channel polarization indices even if it has good or bad \(I\left( H \right)\) or \({\text{B}}\left( {\text{H}} \right)\) randomly. Therefore, utilized MT PRNG to randomize the selection of all indices and vary its primary state seed \(T_{v}\) without pre-shared it with the receiver, caused the receiver to decode and obtain the correct transmitted data, consequently increasing the data Secrecy. Therefore, our new secured polar codes system will be encoded according to:where \({\text{U}}^{^{\prime}}\) denote the fresh random values of frozen bits generated by the MT PRNG, and \(\left( {C^{\prime}} \right) \triangleq \left\{ {0,1, \ldots ,N - 1} \right\}\) is the indices for \({\text{U}}\) information bits, and \(\left( {C^{{C^{\prime}}} } \right) \triangleq \left\{ {0,1, \ldots ,N - 1} \right\}\) are the indices of the frozen bits \({\text{U}}^{^{\prime}}\).

Furthermore, we present complete pseudo-code implementations of the proposed secured polar code scheme in pseudo-codes 1 and 2, indicating how we implement the MT PRNG. Finally, Fig. 5 depicts the suggested scheme's block diagram. That MT PRNG will be used to generate N-K secured frozen bits values, locate them in the interleaved indices, and locate the remaining of the code length K information bits in the remaining of the polarized channels interleaved indices.

In the original non-secured polar coding (NS) scheme, the AWGN data channel is divided into good channel and bad channel by using channel polarization.

The determination of good channels or bad channel depends on the calculated Bhattacharyya parameter for each polarized channel and the code rate. The polarized channel is arranged in ascending order according to Bhattacharyya parameter value and the encoding scheme select the lowest (K) values for the good channel (transmits the data) and the highest (N-K) values for the bad channel (transmits frozen bit values which are usually zeros).

At the receiver, the data rate is known the number of polarized channels accordingly with the locations (indices) of data bits and frozen bits. This leads the receiver decoder to ignore the frozen bit values and indices while utilizing tis functionality to recover the data values correctly.

At this case, the locations of the good channels bits and bad channels bits is known clearly to all the parties and there is no way to secure the data but utilizing a good encryption algorithm (shuffling and mutation) before coding it which is out of the scope of this work.

In our proposed secured polar coding scheme, the main target is to hide the data bits’ locations (indices) inside the whole codeword (not just the good channels to maximize the secrecy of the proposed algorithm while sacrificing the BER performance) utilizing the MT PRNG. This generator is used to randomize the location (indices) of the (N-K) frozen bits, and the rest of the codeword is used for the data bits over all the polarized channels. With the prior knowledge of data rate, if we fix the good channel indices for the data bits and bad channel frozen bits, there will be no secrecy in the algorithm as the polar code decoder typically ignore the frozen bit values and indices. Therefore, it is critical to notice that the key to our proposed security algorithm is to randomize both the data and frozen bit indices over all the codeword, which is significate to the receiver to recover the data bits correctly.

All the experiments on the proposed work were carried out using the MATLAB software version 2020 with a processor type of Intel(R) Core (TM) i7-10750H CPU with clock speed of 2.60 GHz.

In the original polar code non-secured scheme (NS), the following procedure is followed.

The following test procedure is followed to generate the results for our proposed secured polar coding scheme (SNS) which runs on the previously mentioned machine.

In the case of secured but without known sequence (SNNS), the same above procedure is followed but without the knowledge of the seed (Key) at the receiver at the Successive cancelation decoding step (Step no. 5).

An authentic cryptosystem has been introduced in this paper to construct a safe polar code system. This cryptosystem uses a mixed polar coding scheme with MT PRNG. The presented scheme concept is to randomize the polar code encoder's input indices (N bits), random select N-K indices for locating the randomly generated frozen bits, and random assign the data bits (K bits) to the remaining N random indices. The proposed scheme has a mysterious beginning state resulting from the pre-shared MT PRNG initial seed, which is known only to the transmitter and receiver. The MT PRNG is used to generate these random N indices and random frozen bits values. Therefore, unless the decoder recognizes that pre-shared initial seed, the data recovery is disabled. The simulation showed that if the receiver's decoding cypher key (initial seed) differed by one bit from the original encoding cypher key (initial seed), the receiver had a BER probability of roughly 0.5, indicating that without prior knowledge of the 232-bit pre-shared seed, the receiver cannot recover the original transmitted data. Moreover, the system's security can be improved by using PRBG with an extended seed to boost system secrecy and decode obscurity.