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One example of a DLP based crypto system (or rather DDH based crypto system) where the public key parameters include two independent generators of the subgroup, is Cramer Shoup. Since the security proof of this scheme is based on DDH, it seems at least one of the two generators has to be uniformly chosen anew for each ...
I am developing a mostly-offline authorization system that authorizes a user using an deterministically generated AuthKey derived from a MasterKey derived from a high-entropy chunk of data (128 bits) and a low entropy data (for example, an email address). The total length of the concatenated data will never be more tha...
How to perform the Multiplicative Inverse Modulo in International Data Encryption Algorithm? I don't understand on how to perform it… For example, let's say I have a value of cf80 and the value that is appearing to me is this 3080. This is the thing that I've done based on my understanding: answer = 53120 % 65537 531...
Ars technica posted an article about the latest in cyber warfare from the stuxnet crew. They key stretched MD5 by iterating it 10,000 times. It seems to be effective at handicapping HashCat's gpu acceleration. Had I been them I would have used bcrypt instead. Why/When would someone prefer streched-md5 over bcrypt? Ho...
I know confusion is used in cryptography to make it more difficult to identify any relationship between the ciphertext and the symmetric key. I want to know when the confusion is applied during encryption, for example: Confusion applied on the symmetric key K1 to form the modified symmetric key K2, and then K2 is used...
Considering a scenario of anonymous data transfer between seller and buyer using proxies. If we consider that there are three proxies that are being selected by the buyer to obtain data from different sellers (like in emule), how can we make sure that these proxies do not collude with each other to obtain the copy that...
I'm designing a protocol for use into a VPN software. The VPN frames are encapsulated into AES-256 CBC encrypted frames. I understand that IVs must be uniquely used for each message encrypted with the same key. I believe that usually protocols generate those randomly and transmit them (in cleartext, since those are no ...
One of the distinguishing features of a good encryption algorithm, is that it is easy to encrypt, and hard to crack. Are there any that are easy enough for average folk to remember, and calculate by hand, and still stand up to brute force attacks on a computer. Imagine, a prisoner (with pen and paper) sending a message...
Reading Goldreich's Foundations of Cryptography II, I found this proof for the security of the common pseudorandom generator + XOR encryption scheme (Proposition 5.2.12 in the book): Assume you have a secure pseudorandom generator $g \colon \{ 0,1\}^n \to \{0,1\}^m$, and you construct your encryption scheme in the sta...
I'm currently migrating a project to use the bouncy castle GCM mode. I understand that using an authenticated mode removes the need for a HMAC, however I want to be sure that I'm using this mode correctly. When applying a HMAC to ciphertext, I know it's imperative to include in the HMAC'd data any params that could a...
We know Grover's algorithm speedup brute-force attacks two times faster in block ciphers (e.g brute-forcing 128-bit keys take $2^{64}$ operations, not $2^{128}$). That explains why we are using 256-bit keys to encrypt top secrets. But latest practical attack on AES shows brute-forcing AES-256 take $2^{100}$ operations....
I'm having trouble understanding the algorithm for finding the original message $m$, when there is a small public exponent. Here is the example I'm trying to follow (you can also read it in the 'Low exponent RSA paragraph' of this article- http://www.cims.nyu.edu/~regev/teaching/lattices_fall_2004/ln/rsa.pdf): Someone ...
"Linear Cramer-Shoup" is defined on pages 4 and 5 of $\:$ eprint.iacr.org/2007/074.pdf . Are the ciphertexts in Linear Cramer-Shoup computationally indistinguishable from uniform under a chosen-ciphertext attack? $\:$ (We can assume without loss of generality either that $M$ is the identity element or that $M$ is a a...
I'm reading a cryptanalysis on Blowfish, and I've come across something that I don't quite get. Let's denote $$\delta = a \oplus a'$$ where a and a' are bytes that cause a collision in some S-box when entered as the position to use in said box. The structure of Blowfish looks like this: with N+2 positions in P, where...
This is a rather trivial question, I just want to see if I get the concept right. In OFB you generate a pseudorandom stream based on some nonce, and you XOR that with the message to get the ciphertext. This sounds a lot like a one-time pad but I don't see the parallelism anywhere in literature. Am I missing something?
Elsewhere someone stated that he couldn't publish his crypto software on the Internet, because US Export Regulations require approval, if the key size is greater than 56 bits. I know that several decades ago there were such extremely rigorous restrictions. Could anyone kindly check whether that indeed remains so even t...
How we can measure key space in transposition algorithms? Should we specify the method, like rail fence?
I'm having trouble finding info in the docs about how to solve a system of congruences. The closest I can find is 'matsolvemod' in here: http://pari.math.u-bordeaux.fr/dochtml/html.stable/Vectors,_matrices,_linear_algebra_and_sets.html but I'm not sure if that's exactly what it's used for because I don't really underst...
How large should the prime $p$ and generator $g$ values be in a Diffie-Hellman handshake if the messages are encrypted. If the key that encrypted the Diffie-Hellman messages becomes compromised, then does that compromise the key generated using Diffie-Hellman?
Alice and Bob are going to follow the protocol below. Are there any crypto-constructions to help Bob verify the correctness of the answer he gets?: Alice encrypts a set of numbers using some cryptosystem and a secret key (e.g. Paillier crypto-system). Bob multiplies the ciphertexts to obtain an encryption of the sum ...
I am working on 224-bit elliptic curve cryptography. In this 224-bit * 224-bit multiplication results 448-bit output. I am reducing 448-bit into prime field range( prime number $2^{224}-2^{96}+1$) using modulus operation. Can any tell me efficient modulus operation for remainder calculation? I am working on VHDL.
If Alice (sender) sends a private message embedded with public key of the Bob(recipient) through onion network, so in this key exchange, does Bob remains anonymous to Alice as Alice knows about the public key of Bob. So if we want to achieve mutual anonymity, how can we do that? Can we achieve mutual anonymity through ...
Suppose I want to use the following simple hash function. For a mesage $m$, take some public $a$ and prime $p$ and raise $a^m \bmod p$ (never mind the computational expense of this operation). This hash function is secure because the discrete log problem is hard, but only if $p-1$ has a large prime factor (to avoid bei...
Suppose I have a file server running ZFS with file deduplication enabled, and I have two users that are sharing files: Alice and Carol. If Alice uploads huge_file.tgz to her www/ directory, and Carol uploads the same huge_file.tgz to her www/ directory, then they are only stored on disk once, and both files are availab...
Given current technology such as GPUs and GPU cracking software I was wondering if anyone has an idea on how long it would take to brute force the key used to derive an HMAC?
I asked a question on security.stackexchange, but was told it would be a better fit here: https://security.stackexchange.com/questions/32779/how-would-one-crack-a-weak-but-unknown-encryption-protocol My question is: Assume I'm an attacker trying to break a weak-but-unknown encryption algorithm. i.e., someone not too br...
A deterministic encryption scheme is a cryptosystem which always produces the same ciphertext for a given plaintext and key, even over separate executions of the encryption algorithm. Although we cannot achieve semantic security or indistinguishability for deterministic encryption due to lack of randomness in cipher te...
As stated in this page large documents hashed using md5 maybe vulnerable to collision attacks. My question is even passwords of 6-30 character are vulnerable to such hash collision attacks? If yes, is there any proof available online or any example of two password phrase giving same md5 hash? I googled but of no help!....
Please See Why Not Allow Special Characters In a Password on Security. Several websites I've come across limit the allowed special characters for a password. My thought was, it should all be hashed anyway so why does it matter? Is there an actual security reason to restrict this? Example error: The password needs to...
I want to know how rand() works (even when I don't provide any seed how it produces PRNs?) thanks!
How can I take a maximum 19-digits long BigInteger and encrypt it with the following rules: The result must be based on digits and lower-case English letters only. All outputs must have the same length to any input. The length must be between 11 to 16 characters, depending on your method, but should be consistent for ...
I have a project wherein I have to crack a given cipher text encrypted using RSA and have been given N and e. Can someone suggest an RSA attack using a very small exponent e(here e=3) and no padding?
Before I provide details, I want to clarify that I am not looking to implement this practically, but I'm only asking to get a better understanding. The way I currently understand it, we use slow hashes for passwords because there are too few possibilities in passwords. This means people could calculate all the potentia...
i am working ECC-224 bit. can any one tell me, how to calculate y value from ((y*y) mod prime) efficiently for large bit numbers.
Is there a practical homomorphic encryption scheme that can give reasonable execution time results in computing a dot product: $$a_1*b_1 + a_2*b_2 +a_3*b_3 +\ldots+ a_n*b_n$$ I imagine the scheme will need to support additions and multiplications to achieve this, so I assume we will need something that is fully homomor...
As defined by Wikipedia a hash function is [...] any algorithm or subroutine that maps large data sets of variable length to smaller data sets of a fixed length. For example, a person's name, having a variable length, could be hashed to a single integer. The values returned by a hash function are called hash values, h...
What is the current state of the art zero knowledge proof compiler ? I need one that can minimally handle double exponentiation by a known value E.g. $$Pok\{(\alpha):h=g^{\alpha^b}\} $$ where b, g and h are public but $\alpha$ is secret. Preferably it should be able to handle double discrete log proofs E.g. $$Pok\...
Let's say Alice and Bob are playing a game where Bob is trying to guess a number Alice has chosen. Alice chooses a key $K$ and a number $N$ at random and performs $C=Commit(K, N)$ where $Commit(K, N)=h(h(K) \| h(N))$. $h()$ is a collision-resistant hash function and $K$ and $N$ can be of any length. Bob guesses $N'$ an...
Is it a multi-disciplinary, inter-disciplinary, cross-disciplinary, or maybe even trans-disciplinary? What are the basis to describe each kind? I mean, how can we infer by which disciplinary field we can describe the cryptography? I got an answer saying that it is multi-disciplinary because it contains multiple subjec...
i'm still don't understand about 'reduces the coefficients of a modulo 3' on NTRU tutorial $a = f*e = 3 - 7X - 10X^2 - 11X^3 + 10X^4 + 7X^5 + 6X^6 + 7X^7 + 5X^8 - 3X^9 -7X^{10} \pmod{32}.$ Note that when Bob reduces the coefficients of $f*e \bmod{32}$, he chooses values lying between -15 and 16, not between 0 and 3...
What is secure modulus length for modular exponentiation so that finding discrete log is computationaly impossible?
Why do I start the word buffers in MD5 with those values specifically? What in the world motivated the decision to use the numbers 1, 1, 2, 3 for the mix columns stagein AES? And what kind of witchcraft is going on in SHA when I call the sigma_x functions? What kind of math should I be starting with, assuming I only kn...
Suppose, for some security parameter $n$ you choose a prime $p$ such that $p = 2^n+c$ for some relatively small $|c| < 2^m << 2^n$. I have seen such primes being called Pseudo-Mersenne Primes (if $c < 0$), Barrett Primes or Crandall Primes. Does choosing such special primes $p$ as the group modulus for Discrete Logarit...
To get the highest performance, which asymmetric cipher provides the fastest encryption/decryption? ECC or DH? What key size is recommended today?
HMAC used to secure chosen cipher-text attacks. if we are encrypting a large file (video) and sending it over TLS for decryption, how server can check MAC for ciphertext when we didn't sent whole of ciphertext yet ?
I am doing a project which requires the encryption to be done using AES. Is it really possible (technically) to crack AES? If yes, please tell me: What is the probability of breaking AES? How does the round number influence this probability? Do more rounds really help in decreasing this probability? What is the cont...
What is the amount of decrease in the probability of decrypting a data encrypted using an AES algorithm? (That is by introducing each round out of ten whole rounds, how the probability of decryption is reduced?)
I know that inventing one's one crypto always sucks, but the problem is that hand ciphers are usually very insecure very slow. This is an attempt to make a relatively secure, keyable, and modular multi-round cipher by hand. In each round of encryption the following two steps are done: The text is permuted by conc...
I've come across an implementation of DH in Java that uses the Random class to generate the secret integer value $a$, as shown in in Wikipedia's description of the algorithm. As such, the seed of the RNG is only 32 bits and the output may be predictable. By observing DH exchanges only, would it be possible to predict o...
We have an application that requires cleartext passwords for user authentication because of the authentication mechanism in use (RADIUS/CHAP), which unfortunately we cannot change. Since we don't want to store cleartext passwords, an idea is to use asymmetric encryption to secure them as good as possible. After creatin...
some story first: Alice and Bob both have public/private key pairs. Now Bob wants Alice to sign his public key id. Alice agrees but only when Bob signs the public key id of her. Is this something that can be achieved? at the end, the signatures should only be valid or extractable if Alice and Bob did both made their si...
Aumasson, Bernstein and Bosslet did an excellent job demonstrating fatal weaknesses in MurmurHash and CityHash, but what about SpookyHash? My first uneducated guess is that it would be much harder to create collisions, but not impossible. My second uneducated guess is that it would be impossible to create the massive k...
I am sorry, but I need to introduce some concepts which are not directly related to cryptography to make myself clear, I hope I won't stun you with this ... (I'd rather explain it here than redirect the reader to an even more stunning external link ;) ) UBI Properties I am working on Flash memory devices. These device...
I'm reading through a DH implementation, and I think I found a potential hole. Public $p$ and $g$ values are properly selected. A candidate secret value $\bar a$ is pseudo-randomly selected such that $0 \leq \bar a < 2^{|p|}$, where $|p|$ is the number of bits required to store $p$. If $\bar a \geq p$, discard it and ...
We are using the encryption built into Solaris 11 ZFS, which offers the choice between CCM (CBC counter mode) and GCM (Galois counter mode). What are the pros and cons of choosing each of these cipher modes?
In ASN.1, the X and Y values for a 256-bit elliptic curve key are stored as a single 66-byte ASN.1 BITSTRING. Are the values just the first and second half of this bitstring? The private key is an OCTET STRING at the beginning of the DER private key and I know what to do with it; the public key is a BITSTRING for some ...
Suppose I have two users who are issued two different UProve IDs. The Issuer has guaranteed that one UProve token bearer will never have more than one UProve token ID. How can I use UProve to guarantee that two users are "different" without disclosing their actual private ID? Here is a scenario: An online voting sy...
In one-time pads we always say do not use same key twice to encrypt ASCII messages, but if we use a random key to XOR a random binary message and then reusing the same key for XORing new random messages will compromise our key ? Or is attacking reused keys for random messages infeasible?
I know that other bit sizes are possible, e.g. this HTTPS server seems to have a 9000 bit key https://www.ssllabs.com/ssltest/analyze.html?d=qqq.gg, but it's very rare that one sees a key not of size 1024, 2048, 4096, etc. bits in common usage - what is the reason for this? Is it just customary, or is there a cryptogra...
In a paper about predicate encryption or attribute based encryption, the setup function is mentioned with the $setup(1^n)$ or $setup(1^l)$. I want to know what is meant here. Is it multiples of ones or representing something else? Also, why is this used here?
The owner of the RSA key can't store the key, so they need to hand the key to an untrusted entity for storage. Before they hand it over they encrypt it using AES with a multi-word phrase they can remember. The idea is that I could generate a private key and store it outside the entity in a non-secure location but not ...
If I have the following: E1 = RC4(M1, K); E2 = RC4(M2, K); If two different messages M1 and M2 are encrypted with the same key, K, can the key be recovered by doing: K = E1 XOR E2 I tried XORing the two streams and could not get the key. Does RC4 not work this way?
Will HMAC resist Grover's algorithm? If not, which MAC scheme will provide post-quantum security?
I'm writing a simple password manager program that will encrypt/decrypt a single file (it's size will most likely stay under a few K). This is my initial file format design: 1. hmac plaintext 2. hmacSalt plaintext 3. keySalt plaintext 4. iv ...
I've been researching how to implement a post-quantum SSL-like connection authentication, especially correct identification&authentization of the server/client. Because good post-quantum digital signature algorithms are generally unavailable (McEliece-based CFS is slow, NTRUsign is impractical, MQ schemes are generally...
I'm working on software where multiple components will communicate with each other using SSL. There would be one central component acting as a server, which would also require the clients to present a certificate. So there is mutual trust between the clients and server. The software will be deployed internally within o...
How to make a message into a polynomial on NTRU encryption? for example how we can conver 'hello world' to polynomial... i have read the converting operation on IEEE Std 1363.1- 2008, but i can't understand the process without example, can anyone show for the small example?
I am not sure if I missed some subtlety in the definition of Attribute Hiding found in this paper on Predicate Encryption, but isn't it possible to construct a scheme for Predicate Encryption supporting disjunctions that is both a lot less convoluted, and still supports Attribute Hiding? (I realize that "convoluted" i...
From Wikipedia: A trapdoor function is a function that is easy to compute in one direction, yet believed to be difficult to compute in the opposite direction (finding its inverse) without special information, called the "trapdoor". What's the difference between this and encryption functions like AES? Are all en...
When i was dealing with an assistant of my bank through mobile call, she asked me to insert my bank login code after a beep. I was wondering if this procedure is secure against a Man In the Middle attack. In this case, EVE would be listening to the call and get the sensitive data. In Literature i found that GSM stream...
I want to provide both confidentiality and integrity for data at rest (many large files stored on disk). I plan to encrypt the data using AES, which will cover the confidentiality requirement. So the issue is how to provide integrity: Instead of AES, use AES-GCM-SHA256 (authenticated cipher). My understanding is that ...
According to this answer, "SHA-256d" was proposed in one of the Ferguson/Schneier books like so: SHA-256d(x) = SHA-256(SHA-256(x)) Apparently, the motivation for this construction is to avoid length extension attacks. Incidentally, SHA-256d is the hash function forming the core of Bitcoin. According to the comment on ...
I need to make a Skein hash's, specifically, Skein-1024; however, I only have access currently to Skein-512. Is there any way to utilize a Skein-512 function to get the result of a Skein-1024 hash? What's the difference between the two?
I understand how Mega's encryption works. For a quick summary of all those in the future looking for an answer on this... here is how it works: Upon first signing up for an account you make a username and password. It also generates a symmetric key which is used to encrypt your files and a RSA key pair to share secrets...
I understand the needs that lead to the development of cryptography and I am quite familiar with the uses we make of the cryptographic tools. But, as a programmer, I am conditioned to see them as "black boxes" with specific properties. To me, SHA-X (with X being 1, 2 or 3) is some dark magic, even though I understand w...
For homework, I'm asked to find the private key, $x$, in a DSA digital signature scheme. In the particular instance, we are given the parameters $p$, $q$ and $g$, the public key $g^x$, 3 messages $(m_1, m_2, m_3)$, their hash values $(H(m_1), H(m_2), H(m_3))$ and the three signatures $([r_1, s_1], [r_2,s_3], [r_3,s_3])...
Suppose that we are given DSA parameters $p$, $q$, $g$, a public key $y = g^x$, and two signatures $(r_1,s_1)$ and $(r_2,s_2)$. We are told that $(r_1,s_1)$ and $(r_2,s_2)$ were produced by related nonces $k_1 = k$ and $k_2 = k+1$, but we do not know the value of $k$, and we do not know the messages. (In practical term...
If we assume that the support for an irreducible binary Goppa code $\gamma_1, ..., \gamma_n$ is publicly known, when is it possible to efficiently decode the code? I know it's possible if one knows the generator polynomial $g(x)$, and also, if one can obtain a parity check matrix $H$ of the form $XYZ$, where $ X = \lef...
I'm trying to prove something and if I can show that there is a simple way to calculate $(g^a \bmod p)^k$ if I know both $g^k \bmod p$ and $g^a \bmod p$, then (I think) it will help me prove it, but I'm not able to see an obvious way to use the two expressions to calculate the other. Is there a property in modular arit...
I know that you can create a ciphertext c' using c and then find the corresponding m' for c' which you can use to find m for c. So, doesn't this mean that it is not semantically secure? But I also read that it is semantically secure when it meets the DDH assumption. Is there some terminology I'm misunderstanding becaus...
I need your help with a very basic concept in cryptography which I can't understand/prove on my own. I'm trying to prove and understand why, under "indistinguishability in the presence of an eavesdropper" encrpytion, it is required that the lengths of the messages output by the adversary satisfy $|m_0| = |m_1|$. Just t...
When can a decryption function detect that the ECC key I use for decryption is incorrect? Is it possible to do that during initialization, or does the complete message have to be decrypted to do that?
I am signing a short message using the base64 of an HMAC, like this (python): import hashlib import hmac import base64 raw = hmac.new("key", msg="secret", digestmod=hashlib.sha1).digest() base64.urlsafe_b64encode(raw) That last line returns 'ux5F7Ye-rpwvysbzUBLu-wGaYXA=', which I use as the signature. To verify a sig...
Is there a preferred way on how to transmit the initiliazation vector with witch AES encryption was made? I could imagine that, if the encrypted data is stored to a file, the IV is preppended at the file's beginning before the actual encrypted data (or somehting similar). Is there such a convention?
If I have received an AES encrypted message, and if I do know the key with which it was encrypted, do I also need to know the mode of operation with which it was encrypted in order to decrypt it? My understanding is that this is indeed the case. If so, how is the mode of operation transmitted along with the encrypted f...
I'm taking a cryptography introduction course, and we're covering the definition of a secure PRF. I understand the test goes as follows: A challenger picks a function $f$ such that $f \leftarrow \left\{ \begin{array}{lr} b=1: k\leftarrow K, f\leftarrow \left(k,\cdot \right ) \\ b=0: Funs \left(X,Y \r...
I was reading RFC 2898 and something is not clear to me. When I use PBKDF2 with SHA-256 and I want a derived key with length 32 bytes (the same length as my hash function output), your derived key only exists out 1 block (l = 1) of size hLen = 32 bytes. This seems trivially, but i want to be very sure! Is this correct?...
I am trying to implement IDEA algorithm in C#, just to learn how it works. I have taken a 128 bit binary key and generated the 52 encryption keys using the following code: static ushort[] getKeys(string binaryKey) { ushort[] keys = new ushort[52]; int index = 0; while (true) { ...
I was thinking about this today and thought I should ask. I think I understand IV's enough to say that they are basically the same thing as Salts when talking about hashes. They are there to improve randomness between messages. If the IV is simple appended, or prepended to the plain text, why does AES need to know the ...
There is a problem with using DES as the block cipher in OFB mode, eg: the feedback that goes back into the next round will be encrypted with the same key $k$ resulting back into the plaintext IV used in the first round, and this goes on and on until OFB is done. So the output of the block cipher is deterministic and ...
Given an entrophy of 1.5 bits per character, how long does a passphrase have the be to guarantee a key strength of a 128 bit random key? Assuming an alphabet of 26, one character takes log2(26)=4.7 bits to be represented. Given this entrophy of 1.5 bits per character, can one simply answer this by: (128/4.7) * 1.5 = 40...
The protection scheme I faced recently seems so weak nowadays that a simple exhaustive key search would be enough to recover the user key in an acceptable amount of time (it's OK, since almost no practical benefit could be gained from knowing the plaintext), but I'm just curious whether the key recovery could be accomp...
I am currently researching into a small scale home automation system, aiming for cost. The system architecture is basically one master and several slaves which are connected in parallel. Recently i've bumped into the natural question of system security. I'm mainly concerned with authentication, that is, I need the slav...
Theoretically, when using a symmetric block cipher in CBC mode, the current block is dependent on the previous block. Suppose one plaintext is encrypted using CBC, and then one bit of it is changed, let's say somewhere in the middle. Then the new ciphertext will change from the middle onward. Would it be better if I us...
I had an idea earlier: Secure passwords are a) long, and b) unpredictable. A hash is both of these. Would it be safe to reuse a key between sites, and include the site's name as a salt? For example: SHA256("kFnU7Ra%P1_stackexchange") SHA256("kFnU7Ra%P1_google") And so on. As long as nobody discovers your key, it seems...
In my project we would like to encrypt and authenticate the the communication channel between our server and our Arduino nodes, which relies on an underlying TCP channel. We have chosen AES in CTR mode as the cryptographic cipher of our choice, transmitting the initialization vector and the message number in clear such...
What's the state of the art attack to get the first pre-image on MD4? Is it still $2^{102}$ as described by Gaëtan Leurent in "MD4 is Not One-Way" ?
The Wikipedia article on SHA-3 says that Keccak has been chosen but the standard hasn't been issued. Does this matter? That is, can we just assume it's going to be as specified by the submitters (specifically for SHA-3-256 it's the Keccak-f-1600 permutation, in a sponge construction with r=1088 and c=512 and the pad10...
I have found an encryption algorithm named ECKS-PS (published in the paper Efficient conjunctive keyword search on encrypted data storage system) that allows an user to search in encrypted data. All the steps are described here: And I have written them a little bit more clear here: I have tried to find a calculation...
My securities instructor recently posed this question to the class and I am bending my brain in half trying to figure it out. Suppose $y_1 = AES(K, x_1)$ and $y_2 = AES(K,x_2)$, and suppose you know the pairs $(x_1,y_1)$ and $(x_2,y_2)$. Without knowing $K$, how can you construct a new pair $(x_3,y_3)$ such that $y_3...