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First of all, I've seen a question marked as off topic because OP doesn't cover any issue related to cryptographic work, but the legal aspects, but I'm unsure where to attend to ask the following questions.

I've never thought I was going to ask such a question, since I always thought that cryptography should be an open field, because the majority of information I know is on the public domain. But advices received from partners and relatives encouraged me to experiment with the idea. I know that some schemes were patented in their beginning: DH, RSA, DES, NTRU, some schemes on ECC.

Imagine that I own various schemes that are based on existing ideas, but the construction of these schemes are different from the original description, resulting in a different scheme that is believed to have better security or it is more efficient (optimized). I haven't studied any subject related to patents since I started my career so I'm not very familiarized with the scenarios that can arise:

Can you patent a cryptographic scheme and publish a paper opening your method entirely, which is descriptive enough, so its security can be analyzed by the cryptographic community? Could I provide an implementation of the scheme?

Can you patent a cryptographic scheme based on an scheme/idea that was previously patented but lacks of patent nowadays, considering that this scheme is a modification of the original one?

I just don't want to hit a barrier considering this option. I'd like everyone to read and study the presented schemes, so I can still contribute onward with the community I've always thought that patenting make your idea less interesting or restrictive because you can't disclose any technical secret related to it, but maybe I'm wrong because I'm lacking of information. Thanks in advance.

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    $\begingroup$ These sound like questions for a lawyer, or your companies lawyer(s) if applicable. What we can tell you is that it is probable that a patented algorithm will be at a disadvantage when trying to get people to actually use it (and thus pay for it). There are free, widely available, heavily studied, standardized algorithms that it would be competing with. This applies even if your algorithm manages to gain attention from the community at large despite being patented. Unless the people that were advocating patents are knowledgeable about cryptography, I would be cautious with their advice... $\endgroup$ – Ella Rose May 29 at 17:10
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    $\begingroup$ FYI, the idea that you can't disclose technical secrets related to patented inventions is a common misconception. When you patent something, the complete patent application becomes publicly available. You can protect your invention with a patent or with secrecy, but not both. $\endgroup$ – bta May 30 at 1:44
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    $\begingroup$ @bta is correct. You are confusing patents and trade secrets. They are entirely separate bodies of law. I suspect the confusion is rooted in the deceptive and misleading term "intellectual property," which has no actual legal meaning. I suggest reading the article Did You Say “Intellectual Property”? It's a Seductive Mirage $\endgroup$ – Wildcard May 30 at 1:49
  • $\begingroup$ I had a missconception on both terms, thanks to both of you for providing those links, have been helpful. As I said I had always related secrecy with patenting and that claim is simply wrong. $\endgroup$ – kub0x May 30 at 14:17
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    $\begingroup$ While legitimate algorithms can and have been patented, most people in the field would regard taking out a patent on an algorithm as evidence that the algorithm is more likely than not snake oil $\endgroup$ – John Coleman May 30 at 14:41
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But advices received from partners and relatives encouraged me to experiment with the idea. I know that some schemes were patented in their beginning: DH, RSA, DES, NTRU, some schemes on ECC.

Note that there is considerable reluctance in significant parts of the community to use any algorithm with any intellectual property claims. If you patent your algorithm, that pretty much guarantees that no one will use it, unless (or even if) it has very significant advantages over unburdened algorithms. For example, (EC)DSA has extra operations in it to avoid a (now expired) patent. DSA itself was introduced partly due to concerns over patents on RSA. OCB is virtually unused largely because of patents which led to the adoption of the slower CCM. Patents delayed the adoption of elliptic-curve cryptography.

However, to address your questions (and while I am not a lawyer, I do have some familiarity with how patent law works):

Can you patent a cryptographic scheme and publish a paper opening your method entirely, which is descriptive enough, so its security can be analyzed by the cryptographic community? Could I provide an implementation of the scheme?

Absolutely; a patent (or a pending patent) does not rely on the secrecy of the invention (unlike, say, a trade secret); in fact, the entire point of the patent process is to give you an incentive to publish your invention (by giving you a monopoly on it for a limited period of time). On the other hand, you have to do things in the correct order. In the US, you have a year after publishing your invention to file for a patent; in Europe (and, AFAIK, the rest of the world), you don't have any grace period - once you've published, you can't file for a patent.

Can you patent a cryptographic scheme based on an scheme/idea that was previously patented but lacks of patent nowadays, considering that this scheme is a modification of the original one?

You can, if the modification doesn't itself have prior art, and is sufficiently nonobvious to one "skilled in the art" (and, yes, that is a rather nebulous criteria).

I've always thought that patenting make your idea less interesting or restrictive because you can't disclose any technical secret related to it, but maybe I'm wrong because I'm lacking of information.

No, after you file for your patent, you can disclose any details you want. What makes your idea less interesting to other people is that they're concerned about the legal ramifications of using your idea (and you possibly taking them to court for patent infringement). Now, if you release your patent for general use, that's no longer an issue for them; however it begs the question "why did you file for a patent in the first place?"

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  • $\begingroup$ for not being a lawyer, dead on. I just went through this for a non-cryptographic item. $\endgroup$ – b degnan May 29 at 17:23
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    $\begingroup$ Even very flexible patents can severely limit an algorithm's acceptance, like OCB which, despite having a very liberal license, was rejected for 802.11 and replaced with the vastly inferior CCM. $\endgroup$ – forest May 30 at 0:32
  • $\begingroup$ @forest Sounds unlikely. Due to anti-trust laws, IEEE standards cannot exclude a patented technology if the patent owners agree to FRAND license terms. More generally, the process is heavily restricted from discussing commercial terms or costs. 802.11 is heavily patented. Same conditions apply to other bodies like ISO (see MPEG). $\endgroup$ – user71659 May 30 at 4:31
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    $\begingroup$ @user71659 Well, that's why CCM was created. OCB has a weird license which is a bit vague and made people uneasy. For example, it can't be used by any organization that "cooperates with" or "benefits from" the military (and several other government bodies). OCB is twice as fast as CCM but still had to be dropped. $\endgroup$ – forest May 30 at 4:38
  • $\begingroup$ Can't emphasize enough how much patents have held cryptography back for so many years: RSA, elliptic curves, lattices. To patent a cryptosystem is to sentence it to decades in prison—and to damage the security of everyone trying to work around your patent. $\endgroup$ – Squeamish Ossifrage May 31 at 5:52

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