Dead By Daylight

Question

This puzzle is a reference to a game, Dead By Daylight. Though you don't need a knowledge about the game. Consequently, there is no "video-games" tag.

Story

The Entity wants to have the killers' names concealed to the survivors, so he is going to encrypt the killers' names!

Encryption

$$ \begin{array}{c|c} \text{Plaintext} & \text{Encrypted} \\ \hline \text{trapper} & \text{mayysvvsgpamdjpgjd} \\ \text{wraith} & \text{yimisrpidijrgiviar} \\ \text{hillbilly} & \text{tnehqbbwnqzkkewzht} \\ \text{nurse} & \text{witfhun kxzlerboqc} \\ \text{shape} & \text{vdvjmyvvvampvmvsmg} \\ \text{hag} & \text{pkpegzptpnghpbpwgq} \\ \text{doctor} & \text{uwlwcwuwlwcwuwlwcw} \\ \text{huntress} & \text{pogcpipfgup pxglpr} \\ \text{cannibal} & \text{bjkgkvbakykmbskpkd} \\ \text{nightmare} & \text{nzhzbhwzqzkhezzzth} \\ \text{pig} & \text{ehbwhtwztnzknqkeqb} \\ \text{clown} & \text{miafgud sxylvrjopc} \\ \text{spirit} & \text{jpyammaypjdvsggsvd} \\ \text{legion} & \text{h wobczrnftuqiexkl} \\ \text{plague} & \text{hbhehzhkhnhhhthwhq} \\ \text{ghost face}& \text{cu oxulrrfolciixf} \\ \text{demogorgon}& \text{ejeawjejeawjejeawj} \\ \text{oni} & \text{kyzaembgqjwvtphsnd} \\ \text{deathslinger}& \text{nkktqkekbthkwkttzk} \\ \text{executioner}& \text{dkmwvzdtmevhdbmnvq} \end{array} $$

Objective

Explain how the Entity encrypted the names.

CSV Version

Plaintext   ,Encrypted
trapper     ,mayysvvsgpamdjpgjd
wraith      ,yimisrpidijrgiviar
hillbilly   ,tnehqbbwnqzkkewzht
nurse       ,witfhun kxzlerboqc
shape       ,vdvjmyvvvampvmvsmg
hag         ,pkpegzptpnghpbpwgq
doctor      ,uwlwcwuwlwcwuwlwcw
huntress    ,pogcpipfgup pxglpr
cannibal    ,bjkgkvbakykmbskpkd
nightmare   ,nzhzbhwzqzkhezzzth
pig         ,ehbwhtwztnzknqkeqb
clown       ,miafgud sxylvrjopc
spirit      ,jpyammaypjdvsggsvd
legion      ,h wobczrnftuqiexkl
plague      ,hbhehzhkhnhhhthwhq
ghost face  ,cu  oxulrrfolciixf
demogorgon  ,ejeawjejeawjejeawj
oni         ,kyzaembgqjwvtphsnd
deathslinger,nkktqkekbthkwkttzk
executioner ,dkmwvzdtmevhdbmnvq

Note that the Ghost Face's encrypted name has two spaces.

Subtle Hint

The Entity encrypted them by his name. What does that mean?

Moderate Hint

I think the Entity's real name is Joseph F______.

Decisive Hint

$2^{18} ≡ 1 \pmod{26+1}$

Answer 1

This may be a partial answer, because I don't see how to reverse the "encryption", and I'm not enough of a mathematician to confidently say this transform is not reversible.

Explain how the entity encrypted the names.

The entity's name is Joseph Fourier. The "encryption" method is a generalization of the Discrete Fourier Transform called the Number-Theoretic Transform, which substitutes complex arithmetic with arithmetic on a finite field.

Source code

Solve path:

I put the hint "Joseph F" and the tag "number theory" into a Google search, which turned up Joseph Fourier and the Fourier transform. Next, I Googled up "Fourier transform for sequence of integers" and found out about the NTT. I inferred from the other hint that the 18th primitive root of unity mod 27 was 2, which gave me the parameters for the transform ($\omega=2, M=27, n=18$), and then implemented the transform in a handful of lines of Python, transforming the killers' names into their "encryptions".

Before attacking it as a riddle, I tried using old-school cryptographic methods to analyze it. The most significant thing I could find was that by dividing each "encryption" into its even and odd positions, I saw that the same letters kept appearing in groups.

 ghost.face.. , c.ourflix u.xlrocif
 legion...... , hwbzntqek .ocrfuixl
 clown....... , magdsyvjp ifu.xlroc
 huntress.... , pgppgppgp ocifu.xlr
 nurse....... , wthnkzebq ifu.xlroc

 spirit...... , jymapdsgv pamyjvgsd
 trapper..... , mysvgadpj ayvspmjgd

 oni......... , kzebqwthn yamgjvpsd
 wraith...... , ymspdjgva iiriiriir

 hillbilly... , teqbnzkwh nhbwqkezt
 pig......... , ebhwtznkq hwtznkqeb
 deathslinger , nkqebhwtz ktkktkktk
 nightmare... , nhbwqkezt zzhzzhzzh
 hag......... , ppgppgppg keztnhbwq

 shape....... , vvmvvmvvm djyvapmsg

 cannibal.... , bkkbkkbkk jgvaymspd
 plague...... , hhhhhhhhh bezknhtwq
 demogorgon.. , eeweeweew jajjajjaj
 executioner. , dmvdmvdmv kwztehbnq

 doctor...... , ulculculc wwwwwwwww
 

Note the line for doctor which appears to have very little information in it, barely enough to encode the word. Also note groups like ayvspmjgd and .ocrfuixl where the distance between each pair of adjacent letters (ay, yv, vs...) is identical when counted along a 27-letter alphabet which includes a space. These groups must also encode very little information.