Would these adjustments to the ranger archetype Beast Master help the animal companion be more useful?

I recently playtested a Beast Master ranger from level 1 to level 20 (I was playtesting a new homebrew archetype, which was my primary reason for doing so; the Beast Master ranger was just one of the other party members), but there were a few things I noticed regarding the relative power of the beast companion itself. For reference, the beast I went with was a wolf, which is probably a fairly standard choice.

Issues

Now, I know that Beast Master rangers are infamously weak, but I still wanted to see if I could try to improve what I felt were some of its weakest points during my playtesting. I was already using the popular houserule of letting the ranger tell the beast to attack using a bonus action instead of an action, but the other things that bothered me were:

  • Relatively low HP (as the first linked Q&A points out), although this was more of a problem during Tier 1/2, less so during Tier 3/4, at least during my playtesting;
  • Hardly any hit die, which is related to the above problem, since I remember having to spend a lot of healing resources to keep bringing the wolf’s health back up to full/close to full;
  • The DC for resisting the knocked prone secondary effect from the wolf’s Bite attack remains pathetically low at DC 11 for the whole game.
  • The lack of any saving throw proficiencies really screwed the wolf over during the big finale where it died to a meteor swarm, but with a decent DEX saving throw bonus, it would probably have made it.
  • I was sometimes hesitant to use the wolf, because it was dropped to 0 HP a few times at lower levels, unless I knew it would probably land the killing blow or could avoid an opportunity attack or otherwise being hit.

I will point out that at higher levels, the AC was fairly decent (for a wolf), and the HP wasn’t as bad as it was at earlier levels, and I was impressed with the damage output thanks to attack rolls and damage scaling with the ranger’s proficiency bonus. Its Stealth and Perception skill bonuses were also impressive. These things I don’t feel the need to change.

Changes

Here are the changes I propose, somewhat inspired by the UA Sidekick rules:

  • You get a new hit die whenever you take another level in ranger, so at level 3 your wolf starts off with two hit die, but at level 4 they would have three hit die … by level 20 they have 19 hit die. I doubt this would make their max HP better than four times the ranger level, so it would only really be for the purposes of short resting.

  • To improve the max HP a little, maybe something as simple as adding the beast’s CON modifier to that, so it’s now:

    \begin{align} \text{ (ranger level + beasts’s CON modifier)} \times 4 \end{align}

    This way, the animal’s toughness is also taken into an account; I feel like the wolf having 5 instead of 4 more HP each level would have been just enough to help, combined with more hie die to heal, but also from a flavour perspective, I feel like choosing a boar should end up tougher than a hawk, whereas RAW, they would both have the same HP. I would however, keep the minimum HP gained per level to 4, in case the beast somehow has a negative CON modifier, since I think taking HP away from the beast would be cruel, given how underpowered this whole archetype is.

  • Any DCs it has, such as the wolf’s ability to knock people prone, should scale with your proficiency bonus, like this AC and attack/damage rolls do, so rather than a measly 11, at level 3-4 it would be 13, and at level 5, it would be 14 … ending up at 18 at level 17+.

  • Unless it already has a "physical" saving throw proficiency (meaning STR, DEX or CON), it gains one of your choice at level 3, which of course would just mean a +2 (because that’s every valid animal companions’s proficiency bonus) but that also has your proficiency bonus added to it, like AC, etc. This would have certainly helped when it was hit by meteor swarm during our final level 20 showdown, it might have actually survived (even with its RAW hit points) had it made that DEX saving throw.

  • Finally, since I’m letting the beast be commanded as a bonus action, the first half of the 7th level ranger feature Exceptional Training is kinda wasted, so I was considering changing that to not only make the beast’s attacks magical, but also to effectively give the beast a rogue’s Cunning Action, which it can use if you command it to using the same bonus action you used to command it to attack (or do something else with its action). In short, you use one bonus action to tell it what to do with its turn, and it can now effectively use its action and bonus action to do something useful.

Question

Do the above changes seem reasonable, and do you foresee any balance issues coming from my proposed changes? My intention is for the Beast Master’s beast in particular to become more useful and survivable, without increasing its damage output (since I was happy with that), but not making it more powerful than I intended by overlooking something. I suppose also double checking whether there’s a problem with making some animals tougher than others based on their CON; does this unfairly favour tougher animals to the point where that’s a balance issue in and of itself?

Spear mastery or Polearm master feat for a Barbarian?

First off so no one branches too far out, the character I am making will be a human variant totem barbarian (bear all the way through). Also I will not take any more feats after the one I pick to start with so no polearm/ sentinel comboing. This are character choices that are set in stone for me.

I will either do spear mastery and switch between dual wielding spears and throwing one and then using two hands with the other spear, or I’ll go polearm master and flavor a glaive as a great spear but it will stay a glaive in all but name so no throwing range, no Versatile.

Which one, given these restrictions, is the better choice for damage and battle field control?

Clarification of the proof involving the regularity condition in Master Theorem

I was going the text Introduction to Algorithms by Cormen Et. al. Where I came across the following statement in the proof of the third case of the Master’s Theorem.

(Master theorem) Let $ a \geqslant 1$ and $ b > 1$ be constants, let $ f(n)$ be a function, and let $ T (n)$ be defined on the nonnegative integers by the recurrence( the recursion divides a problem of size $ n$ into $ a$ problems of size $ n/b$ each and takes $ f(n)$ for the divide and combine)

$ T(n) = aT(n/b)+ f (n)$ ;

where we interpret $ n/b$ to mean either $ \lceil b/n \rceil$ or $ \lfloor b/n \rfloor$ . Then $ T(n)$ has the following asymptotic bounds:

  1. If $ f(n)=O (n^{log_ba – \epsilon})$ for some constant $ \epsilon > 0$ , then $ T(n)=\Theta (n^{log_ba})$ .

  2. If $ f(n)=\Theta (n^{log_ba})$ , then $ T(n)=\Theta (n^{log_ba}lg n)$

  3. If $ f(n)=\Omega (n^{log_ba + \epsilon})$ for some constant $ \epsilon > 0$ , and if $ af(n/b) \leqslant cf(n)$ for some constant $ c < 1$ and all sufficiently large n, then $ T(n)=\Theta (f(n))$ .

Now in the proof of Master’s Theorem with $ n$ as exact power of $ b$ the expression for $ T(n)$ reduces to :

$ $ T(n)=\Theta(n^{log_ba})+\sum_{j=0}^{log_bn -1} a^jf(n/b^j)$ $

Let us assume,

$ $ g(n)=\sum_{j=0}^{log_bn -1} a^jf(n/b^j)$ $

Then for the proof of the 3rd case of the Master’s Theorem the authors prove that,

If $ a.f(n/b)\leqslant c.f(n)$ for some constant $ c<1$ and for all $ n\geqslant b$ then $ g(n)=\Theta(f(n))$

They say that as, $ a.f(n/b)\leqslant c.f(n) \implies f(n/b)\leqslant (c/a).f(n)$ then interating $ j$ times yeilds, $ f(n/b^j)\leqslant (c/a)^j.f(n)$

I could not quite get the mathematics used behind iterating $ j$ times.

Moreover I could not quite get the logic behind the assumption of $ n\geqslant b$ for the situation that $ n$ should be sufficiently large.(As the third case of the Master’s Theorem says).

Moreover in the similar proof for the third case of the general master theorem( not assuming $ n$ as exact powers of $ b$ ) there again the book assumes that $ n\geqslant b+b/(b-1)$ to satisfy the situation of sufficiently large $ n$ .

I do not quite understand what the specific value has to do and why such is assumed as sufficiently large $ n$

(I did not give the details of the second situation as I feel that it shall be something similar to the first situation)

Master of Many Styles and feat prerequisites


Master of Many Styles

At 1st level, 2nd level, and every four levels thereafter, a master of many styles may select a bonus style feat or the Elemental Fist feat. He does not need to meet the prerequisites of that feat, except the Elemental Fist feat.

Does this mean that a MoMS could take a style feat later in the feat chain and make use of it (i.e. enter the associated style), without getting the first feat in the chain? For example, could I get and use Crane Wing and/or Crane Riposte without ever acquiring Crane Style?

Perhaps I’ve misunderstood the Master of Many Styles text, and “bonus style feat” refers only to the first feat in the style feat chain. If so, then it’s not clearly worded as such.

Manual TLS decryption with master secret

Assuming I have the master secret from SSLKEYLOGFILE client random, and server random, can I decrypt any tls traffic captured? I’ve started from Golang’s TLS implementation, pulled the connection stuff out, had it generate the keys and iv from the values above (https://github.com/golang/go/blob/cd18da451faedc4218a5fd0e38f9b3d13aa5da01/src/crypto/tls/prf.go#L121), but still can’t decrypt.

Thoughts? Is one able to generally decrypt any TLS (given correct version and cipher) with one instance implementation, like Golang’s?

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Relaxing hypotheses of Master Theorem

This question is related to Master Theorem on oscillating function.

Consider a recurrence of the form

$ T(n) = a T(n/b) + f(n)$

Master Theorem’s regularity condition excludes some cases (for example, when $ f(n)$ is oscillating).

Suppose, however, that $ f(n)=\Theta(g(n))$ for a function $ g(n)$ that does not violate the regularity condition, so that the Master Theorem is applicable if $ g(n)$ is used instead of $ f(n)$ . Consider then the following recurrence:

$ T'(n) = a T'(n/b) + g(n)$

and assume that the master theorem gives the solution $ T'(n)=\Theta(g(n))$ .

Can I then safely conclude that $ T(n)=\Theta(g(n))$ ? Or there are some reasonable conditions on $ g(n)$ we can add (I suppose that if $ g(n)$ is polinomially bounded then maybe Akra-Bazzi method will apply, even if one have to swich integration and $ \Theta$ , and I’m not sure this is sound)?

Notice that from $ f(n)=\Theta(g(n))$ we cannot deduce that $ f(n)$ is always less than or equal to $ g(n)$ or to a $ d\cdot g(n)$ for some fixed $ d$ . So I cannot prove by induction that $ T(n)<T'(n)$ for all $ n$ and, anyway, I only want to prove $ T(n)=\Theta(T'(n))$ .

To give some context, this is the case I want to apply the result to: consider the recurrence

$ T(n)=2\cdot T(n/2)+f(n)$

where I only know that $ f(n)=\Theta(n\sqrt{n})$ . Then I can unfold the recurrence obtaining \begin{equation*} \begin{split} T&(n)=2T\left(\frac{n}{2}\right)+\Theta(n\sqrt{n})= 2\left(2T\left(\frac{n}{4}\right)+\Theta\left(\frac{n}{2}\sqrt{\frac{n}{2}}\right)\right)+\Theta(n\sqrt{n}) =\ &=\ldots=2\left(2\left(2\ldots \left(2T\left(\frac{n}{2^h}\right)+\Theta\left(\frac{n}{2^{h-1}}\sqrt{\frac{n}{2^{h-1}}}\right)\right)\ldots\right)\right)+\Theta(n\sqrt{n}) \end{split} \end{equation*}

until $ 2^h=n$ , i.e. $ h=\log(n)$ . Then $ T(n)=\sum_{i=0}^{\log(n)-1} 2^i\cdot\Theta\left(\frac{n}{2^i}\sqrt{\frac{n}{2^i}}\right)$ .

Performing the calculations I get \begin{equation*} \begin{split} T(n)&=\sum_{i=0}^{\log(n)-1} 2^i\cdot\Theta\left(\frac{n}{2^i}\sqrt{\frac{n}{2^i}}\right)=\Theta\left(\sum_{i=0}^{\log(n)-1} 2^i\cdot\frac{n}{2^i}\sqrt{\frac{n}{2^i}}\right)=\ &=\Theta\left(n\sqrt{n}\cdot\sum_{i=0}^{\log(n)-1} \frac{1}{\sqrt{2^i}}\right) \end{split} \end{equation*} The series $ \sum_{i=0}^{+\infty} \frac{1}{\sqrt{2^i}}$ is convergent, so I can conclude (I hope correctly) that $ T(n)=\Theta(n\sqrt{n})$ .

On the other hand, I cannot directly apply Master Theorem, as from $ f(n)=\Theta(n\sqrt{n})$ I cannot conclude $ 2f\left(\frac{n}{2}\right)<cf(n)$ for some $ c<1$ . Indeed, if definitively we have $ c_1 n\sqrt{n}\leq f(n)\leq c_2 n\sqrt{n}$ , then

$ f(n)\geq c_1 n\sqrt{n}= 2\sqrt{2}c_1 \frac{n}{2}\sqrt{\frac{n}{2}}\geq\frac{2\sqrt{2}c_1}{c_2} f\left(\frac{n}{2}\right)=\frac{\sqrt{2}c_1}{c_2} 2f\left(\frac{n}{2}\right)$

and so

$ 2f\left(\frac{n}{2}\right)\leq \frac{c_2}{\sqrt{2}c_1} f(n)$

but $ \frac{c_2}{\sqrt{2}c_1}$ is not necessarily less than 1.

What is the best option between “half-plate armor+dexterity+Medium Armor Master feat” and “plate armor”

I want to make a melee fighter, and I’m strugling on the choise of my armor for the build. I put my stats on my character when I’ll define all I want to do for my build, so let’s suppose that I can have the stat and race you want to define which armor is better.

I hesitate between taking an half plate and a plate armor whenever I can (which will be replace with their magic version whenever I can, my DM allowing us to buy magic items with gold).

On one hand, Plate give the best base AC, 18, and have a strength and stealth restriction that force me to play only in one way, but have the advantage that the only cost on the character sheet to have some point in strength.

On the other hand, half-plate give 15+dex mod (max 2) AC, which is less and with the same stealth restriction, but with the Medium Armor Master feat I can have the same amount of armor, and I can play stealthy if I need to. Plus investing in dexterity seams to me a better choice regarding that dexterity have a far more usefull saving throw, far more skills affected, and give bonus on damage and attack rolls with some weapons.

On my point of view half-plate look far better, but on most of the online build that I found, they use Plate armor and not half-plate, so I think I miss something crucial.