Restrict API requests to only my own mobile app
Yes, It's public
This app will be distributed on Google Play and the Apple App Store so it should be implied that someone will have access to its binary and try to reverse engineer it.
From the moment its on the stores it's public, therefore anything sensitive on the app binary must be considered as potentially compromised.
The Difference Between WHO and WHAT is Accessing the API Server
Before I dive into your problem I would like to first clear a misconception about who and what is accessing an API server. I wrote a series of articles around API and Mobile security, and in the article Why Does Your Mobile App Need An Api Key? you can read in detail the difference between who and what is accessing your API server, but I will extract here the main takes from it:
The what is the thing making the request to the API server. Is it really a genuine instance of your mobile app, or is it a bot, an automated script or an attacker manually poking around your API server with a tool like Postman?
The who is the user of the mobile app that we can authenticate, authorize and identify in several ways, like using OpenID Connect or OAUTH2 flows.
Think about the who as the user your API server will be able to Authenticate and Authorize access to the data, and think about the what as the software making that request in behalf of the user.
So if you are not using user authentication in the app, then you are left with trying to attest what is doing the request.
Mobile Apps should be as much dumb as possible
The reason why is because I need to deliver data to the app based on data gathered by the phone sensors, and if people can pose as my own app and send data to my api that wasn't processed by my own algorithms, it defeats its purpose.
It sounds to me that you are saying that you have algorithms running on the phone to process data from the device sensors and then send them to the API server. If so then you should reconsider this approach and instead just collect the sensor values and send them to the API server and have it running the algorithm.
As I said anything inside your app binary is public, because as yourself said, it can be reverse engineered:
should be implied that someone will have access to its binary and try to reverse engineer it.
Keeping the algorithms in the backend will allow you to not reveal your business logic, and at same time you may reject requests with sensor readings that do not make sense(if is possible to do). This also brings you the benefit of not having to release a new version of the app each time you tweak the algorithm or fix a bug in it.
Runtime attacks
I was thinking something involving the app signatures, since every published app must be signed somehow, but I can't figure out how to do it in a secure way.
Anything you do at runtime to protect the request you are about to send to your API can be reverse engineered with tools like Frida:
Inject your own scripts into black box processes. Hook any function, spy on crypto APIs or trace private application code, no source code needed. Edit, hit save, and instantly see the results. All without compilation steps or program restarts.
Your Suggested Solutions
Security is all about layers of defense, thus you should add as many as you can afford and required by law(e.g GDPR in Europe), therefore any of your purposed solutions are one more layer the attacker needs to bypass, and depending on is skill-set and time is willing to spent on your mobile app it may prevent them to go any further, but in the end all of them can be bypassed.
Maybe a combination of getting the app signature, plus time-based hashes, plus app-generated key pairs and the good old security though obscurity?
Even when you use key pairs stored in the hardware trusted execution environment, all an attacker needs to do is to use an instrumentation framework to hook in the function of your code that uses the keys in order to extract or manipulate the parameters and return values of the function.
Android Hardware-backed Keystore
The availability of a trusted execution environment in a system on a chip (SoC) offers an opportunity for Android devices to provide hardware-backed, strong security services to the Android OS, to platform services, and even to third-party apps.
While it can be defeated I still recommend you to use it, because not all hackers have the skill set or are willing to spend the time on it, and I would recommend you to read this series of articles about Mobile API Security Techniques to learn about some complementary/similar techniques to the ones you described. This articles will teach you how API Keys, User Access Tokens, HMAC and TLS Pinning can be used to protect the API and how they can be bypassed.
Possible Better Solutions
Nowadays I see developers using Android SafetyNet to attest what is doing the request to the API server, but they fail to understand it's not intended to attest that the mobile app is what is doing the request, instead it's intended to attest the integrity of the device, and I go in more detail on my answer to the question Android equivalent of ios devicecheck. So should I use it? Yes you should, because it is one more layer of defense, that in this case tells you that your mobile app is not installed in a rooted device, unless SafetyNet has been bypassed.
Is there any way to restrict post requests to my REST API only to requests coming from my own mobile app binary?
You can allow the API server to have an high degree of confidence that is indeed accepting requests only from your genuine app binary by implementing the Mobile App Attestation concept, and I describe it in more detail on this answer I gave to the question How to secure an API REST for mobile app?, specially the sections Securing the API Server and A Possible Better Solution.
Do you want to go the Extra Mile?
In any response to a security question I always like to reference the excellent work from the OWASP foundation.
For APIS
OWASP API Security Top 10
The OWASP API Security Project seeks to provide value to software developers and security assessors by underscoring the potential risks in insecure APIs, and illustrating how these risks may be mitigated. In order to facilitate this goal, the OWASP API Security Project will create and maintain a Top 10 API Security Risks document, as well as a documentation portal for best practices when creating or assessing APIs.
For Mobile Apps
OWASP Mobile Security Project - Top 10 risks
The OWASP Mobile Security Project is a centralized resource intended to give developers and security teams the resources they need to build and maintain secure mobile applications. Through the project, our goal is to classify mobile security risks and provide developmental controls to reduce their impact or likelihood of exploitation.
OWASP - Mobile Security Testing Guide:
The Mobile Security Testing Guide (MSTG) is a comprehensive manual for mobile app security development, testing and reverse engineering.
Any credentials that are stored in the app can be exposed by the user. In the case of Android, they can completely decompile your app and easily retrieve them.
If the connection to the server does not utilize SSL, they can be easily sniffed off the network.
Seriously, anybody who wants the credentials will get them, so don't worry about concealing them. In essence, you have a public API.
There are some pitfalls and it takes extra time to manage a public API.
Many public APIs still track by IP address and implement tarpits to simply slow down requests from any IP address that seems to be abusing the system. This way, legitimate users from the same IP address can still carry on, albeit slower.
You have to be willing to shut off an IP address or IP address range despite the fact that you may be blocking innocent and upstanding users at the same time as the abusers. If your application is free, it may give you more freedom since there is no expected level of service and no contract, but you may want to guard yourself with a legal agreement.
In general, if your service is popular enough that someone wants to attack it, that's usually a good sign, so don't worry about it too much early on, but do stay ahead of it. You don't want the reason for your app's failure to be because users got tired of waiting on a slow server.
Your other option is to have the users register, so you can block by credentials rather than IP address when you spot abuse.