While I'm not a respected Egyptologist, I am an archaeologist and can make some speculation in this regard.
I would first like to point out that the ancient Egyptians seemed to pretty poor at keeping secrets. They wrote everything down. The walls of their temples and tombs were covered in information from the formula for making beer to the sacred, and secret, spells for protecting the dead on their journey. Priests even noted when they were scamming people with fake cat and falcon mummies.
In investigating ancient Egyptian martial arts I have found references to possibly three.
The first is wrestling,
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The second is a fighting skill or competition now known as Tahtib or stick fencing.
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The third is a little contentious. It is something called Sebek-Kha. It literally means "Crocodile Soul" and the term has been used to describe some practices also depicted at Medinet Habu, in which the figures are punching and kicking.
There are, of course, many depictions of soldiers using various weapons like maces, spears, and the khopesh.
As far as something like Kuta is concerned there is nothing.
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Its a tough ask for something like a technical concept to survive through possibly three thousand years, especially when the region it originated in is in frequent violent turmoil. It wasn't written down so it must be passed through oral tradition. Seems very unlikely to have survived unchanged, but then again perhaps that is why it is rubbish.
Great information; thanks. Just out of curiosity, are the words "Kuta" or "Hikuta" consistent with ancient Egyptian languages?
I'm not at all surprised that there is evidence of the existence of Egyptian martial arts; I'd expect it. What I know of the ancient Egyptian culture is that it was relatively highly developed and organized, and kept lots of records. I'd expect that they would have some systemization of military and martial skills. And I'd expect them to include wrestling, striking, and weapons. After all, there just aren't but so many ways to use the human body to fight. You can grab and grapple with someone, you can punch or kick them, or you can use tools to make your body more effective -- weapons.
I can even buy the idea that some sort of family tradition would be passed along for several generations. That seems to be a pretty common human drive, too.
What I'm skeptical of is the idea of some secret approach being passed along for 2000 or so years. I'm skeptical of the idea of a secret, apparently mercenary force, that nobody heard about. And, if you're going to make such claims, I'm asking for evidence. I'm looking to see the style have something recognizably unique (if you'll forgive something of an oxymoron) about the style. What little I've seen doesn't look like anything unusual.
Now, I'm not saying that the current proponents of Hikuta (or any of the other martial arts with similar claims) are deliberately lying. They could easily be simply repeating what they were told. It's even possible that DOK Lee is doing this, too. But I don't buy the claims being made about Kuta or Hikuta.