Military radios generate a frequency-hopping pattern under the control of a secret Transmission Security Key (TRANSEC) that the sender and receiver share in advance. Using the SAMD51's TRNG (True Random Number Generator) propose a Frequency-hopping spread spectrum TRANSEC key system that allows Adafruit Metro M4 boards to generate and share TRANSEC keys over SPI, I2C, etc.

For creating a Frequency-hopping spread spectrum (FHSS) TRANSEC key system using SAMD51's TRNG, you can use the following approach:

1. **Key Generation**: Each Adafruit Metro M4 board will use the SAMD51's TRNG to generate a secure random TRANSEC key. This key will be used as a seed for the frequency-hopping pattern.

2. **Key Exchange**: Before communication can start, both the sender and receiver must exchange and store the TRANSEC keys. For this purpose, we can use a key exchange protocol. One method could be using asymmetric encryption (like RSA) to securely exchange keys over an insecure channel (SPI, I2C, etc.).

3. **Frequency Hopping Pattern Generation**: Use the TRANSEC key as a seed for a Pseudo Random Number Generator (PRNG). This PRNG will generate a sequence of numbers which will be used as the hopping pattern. The same seed (TRANSEC key) will produce the same hopping pattern on both boards.

4. **Communication**: During communication, both sender and receiver will change frequencies according to the hopping pattern generated in step 3.

Here's a simplified overview of what the code could look like:

```cpp
#include <Adafruit_ZeroTimer.h>
#include <SPI.h> // or <Wire.h> for I2C

// Constants
#define KEY_LENGTH 256  // Length of TRANSEC key in bits

// Variables
uint8_t TRANSECKey[KEY_LENGTH/8];

void setup() {
  Serial.begin(9600);
  while (!Serial); // wait for serial port to connect
  
  // Initialize TRNG
  MCLK->APBCMASK.bit.TRNG_ = 1;
  TRNG->CTRLA.bit.ENABLE = 1;

  // Generate TRANSEC key
  generateTRANSECKey();
  
  // Securely exchange TRANSEC keys with the other board
  // NOTE: This is a placeholder. Implementing secure key exchange
  //       typically involves using asymmetric encryption (like RSA).
  exchangeKeys();
  
  // Initialize communication hardware (SPI, I2C, etc.)
  // ...

  // Initialize timer for frequency hopping
  initTimer();
}

void loop() {
  // Communicate using the Frequency-hopping spread spectrum
}

void generateTRANSECKey() {
  for (int i = 0; i < KEY_LENGTH/8; i++) {
    TRANSECKey[i] = get_trng() & 0xFF;
  }
}

void exchangeKeys() {
  // Implement secure key exchange here
  // ...
}

void initTimer() {
  // Initialize a hardware timer that triggers frequency hopping
  // ...
}

uint32_t get_trng() {
  while ((TRNG->INTFLAG.reg & TRNG_INTFLAG_MASK) == 0);
  return (TRNG->DATA.reg);
}

void TimerHandler() {
  // Called periodically to change the frequency according to hopping pattern
  // ...
}
```

Please note that this is a simplified example and implementing secure communication is a complex task. In particular, the secure exchange of keys (in the `exchangeKeys` function) can be complex and should ideally be done with established cryptographic libraries and protocols.

List any remaining programs necessary to implement our Frequency-hopping spread spectrum (FHSS) TRANSEC key system and suggest file names for each.

To fully implement a Frequency-hopping spread spectrum (FHSS) TRANSEC key system, several additional programs/modules are necessary:

1. **Frequency Hopping Pattern Generator**: This program will use the TRANSEC key to generate a pseudorandom frequency hopping pattern. This is essential for the devices to know which frequencies to use at what times.

    Suggested file name: `FrequencyHoppingPatternGenerator.ino`

2. **RF Communication Module**: This is the program that will handle the actual communication over the airwaves. It should include functions to send and receive data packets, as well as change the transmission frequency according to the frequency hopping pattern.

    Suggested file name: `RFCommunicationModule.ino`

3. **Synchronization Module**: This program ensures that both the transmitter and receiver are in sync regarding the frequency hopping pattern. This might involve timestamping data packets and accounting for any delays.

    Suggested file name: `SynchronizationModule.ino`

4. **Error Handling Module**: This program handles any errors that might occur during data transmission, such as lost packets or interference. Techniques like Automatic Repeat reQuest (ARQ) can be used.

    Suggested file name: `ErrorHandlingModule.ino`

5. **Encryption Module**: This program ensures that the data being sent is encrypted. While the TRANSEC key itself is a part of the encryption, the data payload also needs to be encrypted for additional security.

    Suggested file name: `EncryptionModule.ino`

6. **Device Authentication Module**: This program ensures that only authorized devices can communicate with each other by implementing an authentication process when they first establish a connection.

    Suggested file name: `DeviceAuthenticationModule.ino`

7. **User Interface**: Finally, there should be some user interface, either through serial communication or a physical interface (buttons and LCD), allowing the user to enter commands, view status, and control the system.

    Suggested file name: `UserInterface.ino`

Please note that in a real-world implementation, these modules would likely not be stand-alone programs but rather parts of a larger, integrated codebase. Furthermore, this kind of communication system should be developed with a thorough understanding of the legal and regulatory requirements for radio frequency communications in the relevant jurisdictions, as well as best practices for security.

Further work to be done: Generate a detailed proposal for an ESP32 firmware that interfaces with our SAMD51 chipset and satisfies our Frequency-hopping spread spectrum (FHSS) TRANSEC key system requirements.

An inverse multiplexer (often abbreviated to inverse MUX or IMUX) allows a data stream to be broken into multiple lower data rate communication links. An inverse multiplexer differs from a demultiplexer because the multiple output streams from the former stay inter-related, whereas those from the latter are unrelated. An inverse multiplexer is the opposite of a multiplexer in that it divides one high-speed link into multiple low-speed links, whereas a multiplexer combines multiple low-speed links into one high-speed link.

This provides an end to end connection of several times the data rate available on each of the low rate data links. Note that, as with multiplexers, links are often used in bi-directional pairs and, at either end of the link, an inverse multiplexer will be combined with its reverse (an inverse demultiplexer) and still be called an inverse MUX. How might we utilize inverse multiplexing to our Frequency-hopping spread spectrum (FHSS) TRANSEC key system to improve the latter's performance?

Utilizing inverse multiplexing can enhance the performance of the Frequency-hopping spread spectrum (FHSS) TRANSEC key system in several ways:

1. **Increased Throughput**: By splitting a high-data-rate stream into several lower-data-rate streams that can be transmitted in parallel over multiple frequency channels, the effective throughput can be significantly increased. This can be beneficial in applications where higher data rates are required than what a single channel can provide.

2. **Improved Reliability and Redundancy**: By distributing the data over multiple frequency channels, the system can achieve redundancy. If one of the channels experiences interference or noise, the data can still be retrieved from the other channels. This leads to a more robust and reliable communication, which is especially important in a security-related system like TRANSEC.

3. **Enhanced Security**: Distributing data across multiple channels can also enhance security. A would-be eavesdropper would need to intercept and correctly reassemble data from all channels, which is more difficult than intercepting a single channel. When combined with frequency hopping, this makes unauthorized interception even more challenging.

4. **Load Balancing and Channel Utilization**: Inverse multiplexing can enable better load balancing across multiple channels. This is especially useful in FHSS systems where some frequency channels might be more congested or have higher noise levels than others. Distributing the data over multiple channels ensures that no single channel becomes a bottleneck.

5. **Adaptability to Channel Conditions**: The system can adapt the number of channels used for inverse multiplexing based on the channel conditions. For instance, if the channels are relatively noise-free, it can use more channels to maximize throughput. In contrast, if the environment is noisy, it can use fewer channels with more robust error-correcting codes.

To implement inverse multiplexing in the FHSS TRANSEC key system, the following steps can be taken:

1. **Data Splitting**: At the transmitter side, implement an inverse multiplexer module that splits the outbound data into multiple lower data rate streams.

2. **Channel Allocation**: Dynamically allocate multiple frequency channels for transmitting the split data streams. Coordinate this with the frequency hopping algorithm to ensure that the channels are available.

3. **Transmission over Multiple Channels**: Transmit each lower data rate stream over a separate frequency channel.

4. **Data Reassembly**: At the receiver side, implement an inverse demultiplexer module that reassembles the split data streams into the original high-data-rate stream.

5. **Synchronization**: Ensure synchronization between the transmitter and receiver so that the receiver knows how to correctly reassemble the data from the multiple channels.

6. **Error Handling**: Implement error correction and handling mechanisms to manage any data corruption that might occur due to channel noise or interference.

In conclusion, implementing inverse multiplexing in an FHSS TRANSEC key system can significantly enhance throughput, reliability, security, and overall performance, making it well-suited for high-data-rate, secure communication applications.