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Could anyone tell me what the difference is between a theoretical attack (Like the one done on SHA-1) and an actual, practical attack (Like the one done on SHA-0)? Is a theoretical attack a proposed method of attacking, that no one has ever done due to lack of resources (like computing power)?
What would be a good analogue with which to describe Message Authentication Codes to a person who has little to no understanding of cryptography? For instance, a vault is a reasonable analogue for symmetric authenticated encryption: anyone with the key can open the vault and inspect, alter, or replace the contents. Wit...
For testing purpose, I want to create a PKCS#12 file in which the private key doesn't match the certificate. Is there any tool which will help me do this? If I use openssl pkcs12 -export command line to create the p12 file using a cert and a private key which are mismatched, then openssl gives me an error No certifica...
Am I thinking about this correctly? Consider that the keyspace of the Playfair cipher is equivalent to all permutations of the alphabet, e.g. the number of distinct permutations of 25 objects, and is equal to $1.551 · 10^{25}$. (A lot!) 1) What is the keyspace of Four-Square, when considering both keys, as presente...
I'm using AES to encrypt the message. both sender and receiver know the passphrase, but only sender knows IV and salt I assume that receiver also needs to know IV and salt to decypher text; is it safe to pass IV and salt along with the cyphertext over an unsecured network, or will this compromise the security? I'm usin...
How would one go about selecting an appropriate bijective function for introducing permutations into a cipher or hash? For example, $f(x) = x+1 \space mod \space n$ is a bijective function, but isn't particularly good as a permutation, in that the outputs do not vary sufficiently when an input bit is changed. The same ...
I have an application for determining if a number is a prime or not, currently I'm getting a random number, then doing the Fermat primality testing to find out if the number is probably prime (so this can still be a Carmichael number), I then test with a Miller-Rabin algorithm to determine if the number is probably pri...
Homomorphic Encryption (HE) which supports any function on ciphertexts is known as Fully Homomorphic Encryption (FHE), while Partially Homomorphic Encryption (PHE) includes encryption schemes that have homomorphic properties, with respect to one operation (e.g., only addition or only multiplication, but not both). Are ...
I would like to encrypt a string asymmetrically. I would like the ciphertext to be as short as as plaintext. (or shorter) I've read about format preserving encryption functions - are there asymmetric variants of them?
In point 4 of the Algorithm 4.2 (Classic Merkle Tree Traversal) of *"Post_Quantum_Cryptography.pdf"* shows *"Stack_h.update(2)"* each stack receives two updates. I am trying understand to understand why there are "two updates" but I don't. There is an example on page 19 of *"Boris_Ederov.bachelor.Merkle-Tree-Traversal-...
I would like to know which area of Mathematics would be most beneficial to cryptography. Surely Algebraic Number Theory and maybe to a lesser extend, Elliptic Curves, are closely linked to Cryptography; but are there more fields that would apply neatly to Cryptography? What topics would you recommend to a Maths graduat...
I am a bit confused about the use of IV, and key hash. Is the key hash and IV stored inside the headers of the cipher text? Isn't it a security risk to store this header information? By key hash I mean the hash of key is stored inside the headers. It seems that when I use gpg or OpenSSL , the header of cipher text has...
From reading RFC 2015 (MIME security with PGP), and from looking at the raw format of (GnuPG) encrypted and digitally signed email messages, it seems that first a signature is calculated from the plain text message, and then both are encrypted. My question is why isn't this vulnerable to what Moxie Marlinspike dubs th...
We use MobilePASS at work but the latest version of the android client seems to be buggy so I wanted to have a go at implementing the algorithm myself. You can download the client to play with here: http://www.safenet-inc.com/support-downloads/mobilepass-download-page/ It looks to be based on the HMAC-SHA256 OTP algori...
I have a (long) text in two versions, encrypted and plaintext. I think it was encrypted using a substitution cipher method (I'm pretty sure, indeed). I'm not good in this matter, I know little about it, but I would know if there's a way to know the function/secret used for encrypt that text. I'm ready to study, anyway...
I'm looking at a datagram-based protocol that encrypts a payload. The payload is encrypted with AES or blowfish in CBC mode, and the result is then authenticated with a HMAC. To save space, the protocol uses a zero IV, but prepends a special prefix to the data before encryption. The prefix consists of 12 random bytes, ...
i'm working on an opensource project, A generator for static (html/js), password protected photo galleries. The generator AES 256 encrypts many images with the same key in PyCrypto, and decrypt them with CryptoJS. That's what my prototype does so far: General AES 256 mode=CBC, padding=Pkcs7 Encrypt many images with th...
I need some input on the use of HMAC-SHA256 in an open source project, any help appreciated. The problem I'm solving is session fixation. We have a username, for that username we'd like to generate a session ID that would only be valid for that particular username so that it cannot be reused across users. The session I...
Given a one-way compression function $h:\{0,1\}^n \rightarrow \{0,1\}^m$ and an attacker that picks $x_1 \ldots x_t \in \{0,1\}^n$ (uniformly distributed), I have to show that the probability to find a collision in the picked elements is smaller than $\epsilon$ if $m > log(\frac{t^2}{2 \epsilon})$. Assumption: $h$ is s...
I'm storing the salted hash of a credit card number in a database. What I'd like to be able to do is determine if two different entries in the same database correspond to the same credit card. Specifically, let $C_1$ and $C_2$ be two credit card numbers (which may or may not be the same). Let $S_1$ and $S_2$ be two dif...
I am invesigating the AES-SIV (rfc 5297) based block cipher. The construction of the S2V is lying on the AES-CMAC and dbl and XOR operation. Given a AAD the size of L and in the 128bit block operation, I have some concern that if the L<16bytes, the initial block with all 0s input is not more than an PRF function to ge...
It is possible to find an efficient algorithm for constructing a provably hard to factor number $N$, together with a witness that shows that it is indeed hard to factor. EDIT, since it was not clear: I also want to be sure that even the person that constructed the number cannot factor it. Of course, a random large enou...
E.g., in Python's hashlib, update() basically merges in string buffer functionality into the object. Several other digest APIs I've seen have this pattern. Considering this adds functionality that arguably doesn't belong here, why is this pattern popular?
I have a data set with 300 Million entries and every 5 minutes 4000 random entries in this table change. I need to calculate the merkle root on this data set to validate integrity multiple times every 5 minutes. Assuming sha224 hashes this would be 8 GB for just the leaf nodes and almost 16 GB of data would have to be...
I want to know if having a 8-byte hash value is better than 16-byte hash value for use within hash trees. My mind tells me that the shorter the hash value, the better. But doesn't a smaller hash value make it easier to invert? More strings would lead to the same hash value. I guess this is very opinion-based, but I'd l...
In continuation to Why are bitwise rotations used in cryptography?: what other operations are commonly used in symmetric cryptography? Are there particular operations on numbers that are good for building ciphers? Do common ciphers tend to use the same basic building blocks? Some similar questions: Why Addition Mod 3...
After coming back to try to finish more of my OpenPGP implementation, I refound one of my questions https://stackoverflow.com/questions/6021360/openpgp-tag-18-19-description-confusion that I am still having trouble with. This time however, I have been able to encrypt some data and get another program to decrypt it. Th...
Intel has an on-chip RdRand function which supposedly bypasses the normally used entropy pool for /dev/urandom and directly injects output. Now rumors are going on that Intel works together with the NSA... and knowing that PRNGs are important for cryptography is enough to get this news spreading. I personally don't bel...
I'm currently working with a secure transport protocol that defines the IV to be a counter (incremental nonce) to be encrypted with the same key. This is a followup to a protocol that did not provide any kind of IV. Could you please describe the possible attack vectors on CBC encryption using this encrypted nonce? Furt...
I've read about 20 papers about the criteria for choosing best $n \times n$ s-boxes. Now, I've got two problems: How to generate $2^{24}$ s-boxes, and how to test them considering the criteria. Any help is appreciated.
The NaCl ref implementation of Poly1305 performs modular multiplication to calculate a polynomial $\mod 2^{130} - 5$ using the following modular multiplication function: static void mulmod(unsigned int h[17],const unsigned int r[17]) { unsigned int hr[17]; unsigned int i, j, u; for (i = 0;i < 17;++i) { u =...
How do you prove that a cipher is resistant to differential cryptanalysis? It's said that Rijndael has been proven resistance to differential cryptanalysis. How do cryptographers do that?
Are there secure mechanical cryptosystems in use today? Not necessarily alphabetic either, ie digital but mechanical?
This is just an idea I had while looking through some threads so take it as you will. Given a plaintext of any length and a key that a specific hash function is capable of taking as input(I mention this as I know bcrypt has issues with extremely large inputs), what would be the potential issues/attacks of hashing the ...
I noticed that non-deterministic primality testing algorithms are more commonly used in practice while there is a deterministic algorithm e.g., AKS which runs in polynomial time? I am right? if so, then why? is there any drawbacks of using AKS? Thanks,
Reading some papers on security amplification of weak-PRP written by Stefano Tessaro, I noticed the use of notion "epsilon-PRP" and the notion of "security amplification". These notions, in conjunction with the statement We consequently call a primitive satisfying this assumption an epsilon-PRP: Clearly, AES is much m...
Reading through some documentation on a piece of software, it claims that a 4-6 digit number is created using a 256-bit encryption process for the purposes of access control. The whole process works as follows: A 4-6 digit "start code" (key?) is chosen by the user and programmed into the client software and into the m...
The RSA algorithm is used for encryption (plain text to cipher text) and decryption (cipher text to plain text) of data. What is the need for using public keys in these sites; why is data is encrypted as everything is viewed by everyone? Why do we need to use public key cryptography in Google; how does it provide secur...
In terms of the usual suspects. The problem I have is that Alice wants to send Bob a message M, which gives Bob permission to perform some actions. However Alice wants to verify, from time to time, that the Bob has not changed the message in anyway, allowing himself more privileges than Alice originally intended, wi...
I have two questions here. What does the cipher-text overhead means? Why we were suggested not to encrypts the plain-text itself instead just encrypts only the key (i.e random number)
(significantly reworded this question to fit better into crypto.se). I have static RSA keypairs for both sides and an encrypted transport that needs symmetric keys, and need to exchange session keys. The library I base this on is OpenSSL, which doesn't seem to have some pre-canned AKC protocol for me to use, so I have ...
What I am trying to do is generate a large (4096bit) random number in JavaScript that is cryptographically safe to use. My approach is the following: I am creating a Uint8Array with the desired length. I am using Math.random() to prefill it. I am showing a a 256x256pixel box, and show a message asking for moving the m...
I am reading about the birthday attack in Wikipedia: We consider the following experiment. From a set of $H$ values we choose $n$ values uniformly at random thereby allowing repetitions. Let $p(n; H)$ be the probability that during this experiment at least one value is chosen more than once. This probability can...
I recently read Improved Generic Algorithms for 3-Collisions by Joux and Lucks (Asiacrypt 2009), available as http://eprint.iacr.org/2009/305.pdf. I was wondering about applications of this technique (excluding of course the applications already given in appendix). So my question is: Are there interesting applications ...
I would ask you some questions about users' password storage in a login system. What algorithm do you suggest me to use? bcrypt or PBKDF2? I was looking for a way to implement them client-side and i thought this way: The client computes 10.000 iteractions of user's password with PBKDF2 and username (or email) as SALT...
I'm attempting to send a message to a group of recipients, through an untrusted intermediary. Each of the recipients will have been invited to the group by someone who already has a trusted public key from them. The best solution I've come up with so far is that a shared secret is created by the first member of the gro...
DH key agreement protocols require the participation of both parties, so are only suitable for synchronous connections. Is it possible to implement PFS in a fashion usable for asynchronous protocols, like e-mail, or storage? I'm intuitively thinking not, but I haven't be able to find conclusive proof one way or the o...
Suppose I am building an encrypted document storage system. Suppose each user has a "user key": a symmetric key. Suppose each document is encrypted with a "document key". So, in order to give a user access to a document, we store a tuple of (user_id, document_id, encrypted_doc_key) where the encrypted document key i...
So this might sound like a crazy question but bear with me for a minute. I can't find any info on the internet and so am here, although this might have been a good place to start. I've recently developed an encryption engine using the .net's AES managed classes. I use a 256 bit key generated by the Rfc2898DeriveBytes f...
My friend came up with the following idea: assuming we agreed on curve parameters, use the following algorithm for key stretching/derivation from user-entered password. Pad the ascii representation of the password to n bits or use any simple hash function, e.g. MD5 to allow passwords of arbitrary length Compute the re...
I have been recently going through a question: https://security.stackexchange.com/questions/15767/what-do-key-size-and-block-size-mean-in-cryptography and it created in me a series of doubts. Referring to the 2nd answer you may find the information that there is 50 percent chance of repetition of a duplicate block if w...
I have the following set of equations: $$M_{1}=\frac{y_1-y_0}{x_1-x_0}$$ $$M_{2}=\frac{y_2-y_0}{x_2-x_0}$$ $M_1, M_2, x_1, y_1, x_2, y_2,$ are known and they are chosen from a $GF(2^m)$. I want to find $x_0,y_0$ Does the previous set of equations is solvable? And more... If I have the following set of equations: $$M...
I read ampersand's question "Necessity for finite field arithmetic and the prime number p in Shamir's Secret Sharing Scheme", where he asked why Shamir's Secret Sharing Scheme uses arithmetic in a finite field of prime order. The answers to that question explain that a prime field is not necessary for Shamir's scheme, ...
When changing a (small) part of the plaintext of a CBC-encrypted file, how much of the file will have to be rewritten? My suspicion is that everything after the modified block will have to be decrypted and reencrypted as well, due to the changes of the ciphertext blocks cascading through the rest of the file.
I'm currently reading the paper (Leveled) fully homomorphic encryption without bootstrapping , and the following paragraph was near the start: What is meant by the symbol used? Is it merely to represent polynomial time or something else entirely?
Is there a significantly advantage to these data structures, or is it simply the status-quo and the easiest to use for describing constructions?
According to Wikipedia, the IND-CPA game is: The challenger generates a key pair PK, SK based on some security parameter k (e.g., a key size in bits), and publishes PK to the adversary. The challenger retains SK. The adversary may perform a polynomially bounded number of encryptions or other operations. Eventu...
Here's a scenario I want to handle with a secret sharing scheme: Alice wants to share the secret $S$ with Bob, Charlie and Dave. Alice generates private shares based on $S$ for Bob, Charlie and Dave, and communicates them to their intended recipients via a secure channel. Alice generates a parameter $P$ based on $S$ ...
I am reading the "Biclique cryptanalysis of the full AES" paper. What do they mean by "sharing active S-boxes"? How can this concept can be advantageous to make a bicycle? If there is someone who reads this paper, please explain it to me.
Real-world encryption systems are often not broken algorithmically but because the implementation has a bug. Sometimes the bug only happens in circumstances that could not have been foreseen, such as weak random numbers, or an error in a library causing e.g. predictable IVs. So I wonder, are there workable methods of e...
Assume a simple protocol for authentication - one side sends a randomly generated challenge string and encrypts it with RSA. The receiver decrypts it and replies with a hash of the challenge, proving that he owns the private key (and nothing more). My question: is it safe to also use parts of the same random challenge ...
Regarding HKDF (RFC 5869) - HKDF has two steps, extract and expand. Some answers on this site suggest that you can extract once, and then generate multiple keys from the resulting PRK by multiple invocations of HKDF-expand (with different info strings). Other answers say this is not how HKDF should be used, and possibl...
I had the idea of converting map locations into passwords. Basically Jumblar (github.com/micheal-swiggs/jumblar) takes a hash of the user's location and stores this as a comment on the PGP network. The 'stored-hash'(Vague-Hash) currently is only 2-bytes long. Each time the user wishes to use that location to generate p...
Assume for the sake of the question that I have two variable-length bit strings, each with 128 bit cryptographic randomness, and I want to extract two 128 bit keys via HKDF-SHA256. Which alternative is better (if any), and why? Use a single HKDF-extract on the concatenation of the two strings, and two HKDF-expands wit...
Let's say I pick a 3000+ bit prime $p$ for which I also know that $q = (p - 1) / 2$ is a prime. If I've understood correct, then I can pick pretty much any $g$. Is generating $g^x$ now inevitably slow for any large $x$ regardless of choice of $g$? Do I still only select some $2n$ bit value for $x$ - where $n$ is the nu...
I'm writing a small encryption library which will be required to use the shortest possible amount of overhead for describing ciphertexts. I want to use CTR mode with a 6-byte random nonce. Normally I would never disregard best practices like this, but I think in this context it is acceptable because the encryption key...
I am looking for a proof-of-work scheme which cannot be effectively parallelized. For example, in hashcash (and by extension bitcoin) you have some collision-resistant hash function $f()$, a target $T$ and some constant $C$. You obtain the proof of work by running $P=f(C, N)$ for some nonce $N$ which gets incremented w...
How can one show how to reduce the quadratic residuosity problem to an integer factorization?
I've read something to the effect that the HMAC construct is able to lessen the problem of collisions in the underlying hash. Does that mean that something like HMAC-MD5 still might be considered safe for authenticating encrypted data?
Most cryptographic libraries I've encountered uses some variant of processPlaintext(...) and then doFinal() to produce the ciphertext and resets the state. In the context of packet communications, it feels like it would be optimal to setup key and IV once and then use that single initialisation in some way for all pack...
Say that I have $$ C_1 = AES_{k_1}(M_1) $$ How difficult would it be to find a key, K2 and plaintext M2 $$ C_2 = AES_{k_2}(M_2) $$ such that $$ C_1 == C_2 $$ How would using a block cipher algorithm other than AES effect this problem?
I'm trying to write a cloud storage application where everything the user uploads is unreadable by the server, encrypted client-side before being sent. Currently, the flow looks like this: User inputs password Password is salted and a 256-bit SHA-3 hash is taken A new true-random key is generated The password hash is ...
I would like a user to choose a point on an image which is represented by (x,y). This is meant to be a security token. When a user uses this location to login they are unlikely to to choose the exact same (x,y), rather they will choose a point nearby (x_g, y_g). The user may also be required to enter a password. ATM I'...
After much deliberation, I've decided to use a normal Encrypt-then-MAC scheme instead of an authenticated encryption mode such as GCM as authenticated encryption primitive. This is due to the lack of good, clean and reliable implementation across all the platforms I intend to support. Background This is for a multiplay...
Let say that I wrote a 26 letter alphabet, each letter of my alphabet represent a letter from the latin alphabet. I'm writing in 3 languages, only I know which languages. Grammar is the one from my native language. Each time I need to write a word I randomly chooses in which language I'm going to write it. Can you actu...
Why do people say that using hash twice or more is bad? yes there is key derivation that do that job, but in a low bandwidth network countries, the solution i see is to use a simple hashing system on the server to gain one second or two. Is rainbow table effective when using more than one hash? because on internet, i ...
A paper I found (http://eprint.iacr.org/2012/064.pdf, which was submitted to a cryptography conference about two years ago), is based on the analysis of some 7.1 million 1024-bit RSA keys published online. By subjecting the modulus of each public key to an algorithm first postulated more than 2,000 years ago by the Gre...
What is the difference between the keylength and the length of a password for AES-256?
WEP is now considered as insecure. But how about the 256-bit WEP? Could someone explain why a 256-bit encryption isn't enough? 'A 256-bit WEP system is available from some vendors. As with the other WEP-variants 24 bits of that is for the IV, leaving 232 bits for actual protection. These 232 bits are typically entered ...
Could you explain the 'Breaking 104 bit WEP in less than 60 seconds' paper, without skipping too much technical details. Could you an answer questions, like: how is '1.24/256' a significant difference compared to '1/256'? And how this seemingly small difference could lead a '104-bit WEP key' to a reduced required comp...
Suppose you got a prime $p = 2\mathbb\Pi_{i=0}^{n-1}q_i+1$, where $2^{k-1} \lt q_i \lt 2^k$ for some $k$ and all $0 \le i \lt n$, and that you also got a generator $g$ of one of the prime order sub groups of $\mathbb Z_p^*$. Also assume that the factorization of $\frac{p-1}{2}$ is unknown. Question #1: Are there ways t...
I've been trying to implement the new SIMON block cipher in Ruby as a personal challenge. The spec didn't include a pseudocode version of the decryption but I thought I had it figured out. The code I wrote is located at https://github.com/Rami114/Personal/blob/master/Simon/Simon.rb I've based this on the spec available...
I read about how is an input message prepared to be hashed by MD4,MD5 or SHA-1: Step1 Append padding bits The input message is "padded" (extended) so that its length (in bits) equals to 448 mod 512. Padding is always performed, even if the length of the message is already 448 mod 512. Padding is performed as fol...
I'd like to test and see if certain software uses P-521 ECC curves, or if it uses another variant. Without having access to the sourcecode, or the specification, is there any way for me to test which ECC curve a given implementation of software uses? Take a x509 certificate as an example, aside from ASN.1 information t...
I need to take a potentially weak user password and derive a strong key for AES encryption in JavaScript. How can I do this? I expect the hardest problem is generating a good random salt - perhaps an SSL request could be used to ask a server to provide half of the salt. This would work but I'd rather keep it simple if ...
Suppose ECDSA is used without hashing the message, but directly using a short message (say 10-bytes long) as the value z (using the definitions at http://en.wikipedia.org/wiki/Elliptic_Curve_DSA) because we need extremely fast signing with ECDSA in a specific embedded platform. We also pre-compute kG and k^-1 values. A...
I'm looking for a toy encryption system that has a specific property that when decrypting it with a key that is close enough to the key used in the encryption, it will produce some of the original plain text. The idea is to use it as a game/teaching system where kids can try to break a cyphertext but instead of using c...
My question is based on the following situation: Bob and Alice work for a company that expects them to exchange data encrypted. They could do so using regular public-key cryptography, but Joe (their boss) wants to be part of the procedure to ensure nothing is encrypted or decrypted without his knowledge. Therefore, Joe...
I was reading Eric Bach paper entitles Discrete logarithms and factoring, in which he states the following reductions: solving the integer factorization problem suffices to solve the discrete logarithm problem and vice versa. I could not completely understand the explanation. Could anyone explain it to me or refer me t...
Let's say I create my PGP keys with my appropriate name and email address. Will these personal information be exposed if I publish my public key? When I encrypt a message for sending it to my communication partner, if someone intercepts this encrypted message, will he or she get to know about my personal information? ...
In the family of the believed (experts haven't come yet with a solution) intractable problems in cryptography belongs the Polynomial Reconstruction problem in which an attacker is given $n$ points in which at least $t$ belong to the evaluation of a polynomial $p$ of degree $k$ and the attacker wishes to reconstruct $p...
Apparently it is not possible to compute all MD5 values from 0x00000000000000000000000000000000 to 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF. As far as I understand this operation is hard because cryptographic hash functions make it difficult to calculate a message for a given digest (Pre-image resistance). Storing all MD5 ...
I found this security problem, for example: Alice has an RSA public key: $e = 37$ and $\phi$ (totient, $N_A = (pa-1)(qa-1)$) is $55$, so the key is $(37, 55)$. Bob has an RSA public key: $e = 37$ and $\phi$ (totient, $N_B$ = $(p_B-1) (q_B-1)$) is $77$, so the key is $(37, 77)$. So I see that $\gcd(55, 77)$ is $11$. B...
I've just skimmed over most of the paper "On Message Integrity in Symmetric Encryption". I've included (the last) part of the abstract below: We show that generic compositions of confidentiality-secure encryption schemes with Manipulation Detection Code (MDC) functions for authenticated encryption are, in general,...
What is an approximate length for a digital signature (SHA) given the following? Using RFID tags, size ranging from 100 to 2000 bytes (8000 bits), our project is setting aside a portion for a digital signature against its fixed data, which includes an unique 8-byte serial number and with 32 to 64 bytes of factory data....
I don't get the utility to chain and reduce in generating rainbow tables. Hash functions are made to not have collisions, or at least chain only 2 times so we don't return to the concept of the hash dictionary. So why do we bother the machine to hash and reduce thousand of times to only store the latest value?
According to this Q&A-discussion it is possible to remove all personal data (name and mail address) that is attached to a public key. What steps do I have to follow in order to remove all personal data? Will that modified public key still allow me to decrypt data with my regular passphrase?
I have implemented the Fortuna random number generator as described in chapter 10 of "Practical Cryptography" (Ferguson and Schneier, Wiley, 2003), the result can be found at http://www.seehuhn.de/pages/fortuna . My question: How can I test my implementation? Or more specifically: Are there any known-correct test vec...
My bank uses a smartcard reader (separate module not connected to the computer) as depicted below: When I need to log into my account it provides me with a challenge and I need to provide some details: I need to provide the 8 digits, enter my pin and then it returns an 8 number sequence which is then to be entered a...
This answer to another question describes the different values chosen as private key and public key by various Ed25516 implementations. What are the advantages / disadvantages of each? (Table cropped from diagram in the linked answer, originally taken from "How do Ed5519 keys work?" by Brian Warner.)