Set which is easy to sample, but difficult to sample from its complement

Given a set $ S \subseteq \{0,1\}^*$ , the algorithm $ A$ is a generator for $ S$ if given $ n$ random bits $ x \in \{0,1\}^n$ , $ A$ generates an element of $ S$ of size $ n$ , and $ A$ can generate at least $ \frac{2}{3}$ members of $ S$ of size $ n$ (for all $ n$ ). $ A$ does not have to be uniform.

Is there a set $ S$ such that there exists an efficient algorithm $ A$ such that for all $ n$ , $ A$ generates at least $ \frac{2}{3}$ members of $ S$ (of size $ n$ ), but any efficient algorithm for $ S^C$ can only generate at most $ \frac{1}{3}$ elements from $ S^C$ of size $ n$ (under complexity asuumptions)?

Would a difficult to access “Key” be an option to securely solve the Apple vs. FBI problem?

In recent times, there has been an escalating demand by legislators in the US and the world around to be able to decrypt phones that come pre-configured with strong encryption. Key escrow is commonly suggested as a solution, with the risk seeming to arise out of the escrow agent misusing or not appropriately securing the keys — allowing for remote, illegal, or surreptitious access to the secured data.

Could a system secure from remote attack be devised by adding an offline tamper-evident key to the device? This could be an unconnected WLCSP flash chip or a barcode within the device with the plaintext of a decryption key.

I recognize the evil maid attack, but presume a tamper seal could be made sufficiently challenging to thwart all but the most motivated attackers from surreptitious access to the data.

What would be lost in this scheme relative to the current security afforded by a consumer-grade pre-encrypted device (cf. iPhone)? Bitcoin, Subpoena efficacy, and other scenarios that seem fine with “smash and grab” tactics come to mind.

Does NP $\cap$ coNP less difficult than NP-complete?

I am taking a complexity class now, and I struggle to understand the concept of “hardness”:
Assume that $ L \in \textit{NP } \cap \textit{ coNP}$ . In means that under the assumption $ NP \neq coNP$ , $ L$ cannot be NP-complete. The formal meaning is that not all languages in NP can be reduced to $ L$ , but does it mean that $ L$ is easier to solve than NP-complete language (in the sense that it is more likely to have non-exponential algorithm which decides it)?
Does is plausible that the optimal algorithm for $ L$ is exponential? (For 3-SAT there is a known assumption, ETH, which as far as I understand states that the optimal algorithm for it has to be exponential).

Is jumping farther than STR feet possible, how difficult is it, and does it take an Action when performed in combat?

I am slightly confused about how far a PC can jump in combat. On page 182, the PHB defines the mechanics of the long jump:

When you make a long jump, you cover a number of feet up to your Strength score if you move at least 10 feet on foot immedialely before the jump. When you make a standing long jump, you can leap only half that distance. Either way, each foot you clear on the jump costs a foot of movement.

The subsequent description of the high jump is essentially analoguous with the Strength score replaced with the Strength modifier. However, it also features the following addition:

In some circumstances, your DM might allow you to make a Strength (Athletics) check to jump higher than you normally can.

Since this is explictly spelled out for the high jump and no similar mechanic is mentioned in the context of the long jump, I’d be inclined to infer that being able to jump farther by making a successful Strength check is not intended by the game designers. However, on page 175, the PHB explictily lists the following as an example of an Athletics check:

You try to jump an unusually long distance

Questions:

  1. Is it possible to exceed your normal maximum jump length by passing a Strength (Athletics) check?
  2. If so, is there a guideline for the DC of jumping a given number of feet farther than one’s Strength score?
  3. Would such an unusually long jump still be simply a part of one’s Movement, or would it consume an Action? Would it maybe cost extra feet of Movement?

Context:

When playing on a grid, the relevant jump lengths are usually multiples of 5 feet. If for example a PC has a Strength score of 8, that effectively means they can jump only 5 feet. In my opinion, letting them jump 10 feet instead should obviously not come for free, but it also shouldn’t be too big a deal. If someone could refer me to an official source that offers clarification on that matter, I’d be very happy.

How to find a context-free grammar from a difficult language?

Some Languages are trivial to find their respective context-free grammar. Like for example $ L= \{a^nb^n: n \geqslant 0\}$ . However some are really difficult to solve. I would like to have some advice on how I can tackle them.

For example I have the following language that I have been trying to solve for a while :enter image description here

I tried to divide the problem into three cases as follow:

case i: na $ \le$ nb

case ii: nb $ \le$ na $ <$ 2nb

case iii: na $ \ge$ 2nb

The first case was easy to solve however I am stuck in case ii. At this point I don’t even know if the procedure that I chose is the correct one.

Is the illusion created by Invoke Duplicity affected by difficult terrain?

Is the illusion created by the Trickery Domain Cleric affected by difficult terrain when moving?

As a DM, I’m assuming that it is not affected by difficult terrain. Is that correct?

I assume if a player does move the illusion at normal pace on difficult terrain you would give the NPC / monster some form of saving throw to realise it’s an illusion.

Deleted Ubuntu, Windows bootloader is difficult to access

I recently had a separate drive that I tried to install Ubuntu on. Some things happened, and as a result I had to uninstall Ubuntu. Now when I try to boot up my computer, I have to bring up the boot menu as the computer is booting up. From here, I am able to select the Windows boot loader and load up windows. If I don’t press f11 in time, the Ubuntu menu comes up and says its not found. When I go into the BIOS, I try to rearrange the boot order to what I think the correct loader is, it says “please select a proper boot medium” or something to that effect. Is there something I can do so I don’t have to be present to reboot my computer?

Thanks for any help -Dakota

How difficult to bruteforce KeePass KDBX4 master password?

How difficult to crack a KeePass file which use KDBX4 file format if someone only obtain the file without knowing any hint of the master password?

With assumption the password length is equal/more than 20 character, uses lower cases, upper case, number & non-alphanumeric character on QWERTY keyboard.

P.S. i know there’s similar question at How difficult to crack keepass master password?, but it was created before KDBX4 released

How do ball bearings and difficult terrain stack?

There is already this Q/A on whether difficult terrains stack (they do not). And this Q/A on whether ball bearings and caltrops stack (they do not). But when looking at the description for ball bearings I realized it states:

A creature moving through the area at half speed doesn’t need to make the saving throw…

And when looking for other references to “half speed” all that existed were caltrops (with a similar description) and this bit from the “Difficult Terrain” section:

You move at half speed in difficult terrain–moving 1 foot in difficult terrain costs 2 feet of speed–so you can cover only half the normal distance in a minute, an hour, or a day…

I had always assumed you could choose to move at half speed through ball bearings to avoid the saving throw but perhaps that is not the case…

Regardless, it is unclear to me what happens if ball bearings are thrown onto difficult terrain. You would (outside of features such as the Ranger’s Land’s Stride) already be moving at half speed due to the difficult terrain but do ball bearings require it to be halved again, or is putting ball bearings on difficult terrain useless unless a creature that ignores difficult terrain?