0. py [-x] [-m MAX-LEN] [-f] [-t CHARSET] [FILE] python xortool. Output: The form calculates the bitwise exclusive or using the function gmp_xor. In 1919, a variant of the one-time pad, the Vernam cipher, was patented by Product KeyFinder. IDEA also uses XOR as one of its three main functions: XOR, addition and multiplication. In the previous essay, we went through the Single-byte XOR cipher and found a way to decipher it without having any knowledge of the encryption key. You can see a truth table below: This means that only if the two values are different, the XOR function will return 1. 7% 8: 4. Find the Hamming distance between KEYLENGTH chunks of text. Jul 13, 2020 · you entered 140023898 byte at Address is 0 xorring byte with 0x3E result of xor = 3e Setting Comment and the commented disassembly will look like . Based on that I know that we can recover parts of the key used to encrypt the strings, for example: In binary the ascii value of space is 00100000 and the lowercase letters start with 011. Click on the URL button, Enter URL and Submit. Length; c++) result. 0% 5: 8. We only need to know the single-byte key. The XOR-based approach to obfuscating contents often works like this: The attacker picks a 1-byte value to act as the key. You could perform a known-plaintext attack if you have other known inputs and outputs. It auto detect the input number and calculate. In an old cryptography FAQ, I found the following step described for determining a the length of the key a cipher was repeatedly XORed against: Discover the length of the key by counting coincidences. Here is the Oct 1, 2015 · This is the property of XOR operations: for every key you would find that. Calculate the exclusive or (XOR) with a web-based calculator. " GitHub is where people build software. 8% 15: 8. \overline{A}) Performing the XOR of two integers is trivial in Python, as the language offers an operator, especially for this purpose, namely a caret ^. Upload Your Tracks. So XORing with 1 is basically flipping a bit. If an attacker knows the ciphertexts, then they can XOR two ciphertexts together to know the result of XORing two plaintexts together. XOR Calculator. e for an input of 0 & 1 or 1 & 0. Dec 9, 2021 · 00:00 - Intro00:45 - High-level overview of solution algorithm01:30 - Talking about Hamming distance and Hamming weight03:10 - Implementing Hamming weight (f The seperate parameters of the key are displayed, and can be shown as decimal or hex. To implement this algorithm efficiently, Xorq needs to find maximum value of for given integers , and , such that, . Jul 28, 2021 · In this HackerRank XOR key problem solution, Xorq invented an encryption algorithm that uses bitwise XOR operations extensively. It requires an unequal pair of inputs to yield a true signal. data:000000014002389A byte_14002389A db 0 The Process. But only if you use a different key sequence for each message. Transpose the original ciphertext into KEYLENGTH chunks grouped by key position. binary (base 2) copy output to clipboard. In symmetric key encryption, XOR is applied bitwise to the plaintext and the secret key. x [n]] as its key. Input two. Find value in memory with encrypted search. Input one. It also has a community-updated configuration file that retrieves product keys for many other applications. Nintendo is the rare exception. At the bit level, there are four possibilities, Non-binary inputs are converted into their binary equivalents using gmp_init. 1, either to file, clipboard, or the Main Window (Encrypt/Decrypt). The XOR encryption algorithm can be applied to any digital/binary information, included text based information encoded using the 8-bit ASCII code. Dec 13, 2013 · def xor_attmpt(): message = raw_input("Enter message to be ciphered: ") cipher = [] for i in message: cipher. . data:0000000140023898 byte_140023898 db 0 ; xorred result is 3e . How to Break it - Step 1. This app analyzes music and estimates pitch and tempo [e. You would then XOR the plaintext with the ciphertext to retrieve the key. More than 100 million people use GitHub to discover, fork, and contribute to over 420 million projects. 9% Key-length can be 5 * n Most possible char is needed to guess the key! # 00 is the most frequent byte in XOR Calculator. The Vernam cipher is available in binary and character versions. How to find Key & BPM. Finding the key size is done by the following algorithm: Make a guess about the key length. \overline{B}) + (B. A simple Python script to find the key used in a XOR cipher. Provide two inputs, select input and output types, then Calculate XOR. A XOR B returns 1 if and only if A and B are different. XOR encryption, a classic cryptographic technique, involves combining a plaintext message with a secret key using the exclusive OR (XOR) operation. Our main aim is to provide all possible answer keys for FREE even though the topic is tough and hard to find answers. Sep 19, 2019 · I'm Trying to find XOR key of Little Busters PE Any plan to add Reallive XOR key finder support? Add this topic to your repo. use finded magic keys with this program: Attached Files. But this operation can also be achieved by using the operator module in Python. Learn more. As it is used to combine data with secret key to scramble it making it very difficult for unauthorized people to crack. Once you've tried all memory types, all ranges, unknown increased and unknown decreased and all fail, this method might be what you want to do last. Star 6. Using the XOR operation, each bit of the plaintext is combined with the corresponding bit of the key. 1. This encryption algorithm uses a sequence of non-negative integers as its key. This one is trickier. py [-x] [-m MAX-LEN| -l LEN] [-c CHAR | -b | -o] [-f] [-t CHARSET] [FILE] python Apr 22, 2018 · $. Given enough data, you will find an 'e' for every column, giving you all the letters of the key. After the decrypting the buffer the following sequence of comparations takes place: Since the 1st, 5th and 10th bytes of the decrypted buffer are expected to be 0x60, 0x0D0, 0x0 we can find the 1st, 5th and 9th bytes of the key, namely: 0x2F, 0x46, 0x41. Notifications. net/download1. There are few ways from here, the most simple is, because we know some parts of the plaintext, we can find the missing part by sense and from there find the remaining part of the key. Fork 0. Tile Pro (2022) With a wide range, loud alarm and reliable performance, the Tile Pro is the best key finder to get for iPhones and Android devices alike. An XOR encoded binary file is a file where some (or all) bytes have been XORed with a constant value (the key). The XOR logical operation (eXclusive OR) is a logical operator in Boolean algebra. To recover the secret key secret, the second tool xor_key_recovery is used. Edit any of the track data. E = M xor K May 1, 2011 · P. The XOR cipher will encrypt a message by using the Boolean XOR (exclusive or) operation. (encrypted text) xor ‘HTB{‘ = key. The formula for XOR operation is: XOR(A, B) = ( A . It is built on modular addition, similar to the Vigenère cipher, but without repeating the key. Let’s get into some more detail on each step. Crack the key. In this essay, we find how to break a Repeating-key XOR cipher with variable key length. Processing time takes a few seconds. 00110010 -> 01100100. To implement this algorithm efficiently, Xorq needs to find the maximum value of A XOR xj for given integers A 9. 1% 25: 8. 2. It is also an easy cipher to perform manually. Repeat this process to pick multiple colors from the image. How do I use it? Simply drag and drop any of your tracks into the dropzone and hit the "Upload Tracks" button. Moreover, say you have an encrypted message E. Detecting changes in data : The XOR operator can be used to detect changes in data. If the plaintexts had predictable content, then the obvious next step for the attacker is to do crib-dragging, but they can't if the plaintexts themselves are purely random. Aug 13, 2019 · 1. Sep 11, 2023 · XOR Encryption is an encryption method used to encrypt data and is hard to crack by brute-force method, i. So…. Since there are only 255 keys, you can try all possible keys on The round key will be XORed with the intermediate result and after that permuted and substituted. Returns false when both the inputs are same. The following properties may help: the key is sufficiently short. We don’t know the KEYSIZE, so we’ll guess that it’s from between 2 and 40 bytes long. Repeating-key XOR Cipher Jun 11, 2015 · So, I have this encrypted file. With a cleartext message and its XOR encrypted message, you can use this tool to find the key used to encode the messages. Encrypted searching might not always find the xor value depending on how much the value changes. Get FREE Answers Online 24/7 Here! AnswerKeyFinder™ is made exclusively for students. py xor_encrypted_text. Input and output in binary, decimal, hexadecimal. XOR is a bitwise operation that compares corresponding bits of two binary numbers. This tool also comes with learn sections and step-by-step solutions and practice problems! Jul 25, 2017 · But you need to know about xor encryption first. Knowing that m1⊕m2 has the first three bits of . Use value from game as key and remember result. This can allow you to break future messages using the same XOR cipher key as well as encrypt your own messages with the same method. To decode, you just bitwise rotate one bit left. (xor key must be above and xor value under too) If xor key and xor value are away offset 4,use X4. The true keysize probably minimizes this. It is Apr 17, 2023 · flag xor key = encrypted text so, if we do xor of cipher text and first 4 character of flag that we know is “HTB{“ we get the key. To associate your repository with the xor topic, visit your repo's landing page and select "manage topics. As Jul 23, 2017 · If they are diffferent and the key sequence is completely known to receiver without being spyable, then you have a one-time-pad encryption. Then, for any possible plain text message M you will always find a key K so that. Jul 24, 2017 · XOR SEARCH: This method of search is very unique and not intended for you to try as a first method. It performs BigInt XOR if both operands become BigInts; otherwise, it converts both Jan 3, 2014 · Neither OpenSSL nor GPG have such XOR functionality that I'm aware of, however writing a program to do it yourself should be trivial. It first coerces both operands to numeric values and tests the types of them. Append((char)((uint)text[c] ^ (uint)key[c % key. The XOR cipher is often used in computer malware to make reverse engineering more difficult. Comparator Circuits: It is used in comparator circuit to check if the two binary values are equal or not. If the key is random and is at least as long as the message, the XOR cipher is much more secure than when there is key repetition within a message. Dec 14, 2012 · So you should xor each data character with each key character. Notice that the bit patterns have the same number of bits before and after. # xor is xortool/xortool-xor tests $ xor -f /bin/ls -s " secret_key " > binary_xored tests $ xortool binary_xored The most probable key lengths: 2: 5. 10101 # binary for 21 # XOR 000 10 # binary for 2 # Result 10111 # binary for 23. Change Key & BPM. It is also termed as EOR or EXOR (exclusive OR For instance, if the Average Hamming Distance with a XOR key size of 3 is 7 and you divide 7 by 3, you get 2. Convert the decimal 2 of the key to the binary ( 00010 ). In this case the encryption key can be expressed as a string of characters. Detected Song Keys are 70-95% accurate depending on the selected option, and it's even more if we XOR is just an operation to combine bits, similar bits result in a 0. And we know the xor is a symetric cipher. Tap tempo Apr 13, 2017 · Breaking the repeating-key XOR cipher. I don't want it in decrypted form, I want it in encrypted form and know the decryption process, or put into text form. 2. XOR Calculator Online is a very unique tool to calculate two or more numbers. XOR: To XOR the key with plaintext[ASCII] we need to have a predefined key. Tile Mate (2022) It Xor key finder mac on MainKeys. xor ("j") Oct 29, 2023 · XOR of very large Binary Numbers in range [L, R] Bitwise OR of N binary strings; Bitwise AND of N binary strings; Find XOR of all elements in an Array; XOR in a range of a binary array; Checking valid shuffle of two Strings; Length of Longest sub-string that can be removed; Minimum number of swaps to make two binary string equal Apr 8, 2024 · The quick list. bmp, you should be able to use this fact to decrypt the file quite easily, especially given that . bmp files have a well defined structure. You can remember the above result using one of these logics too:-. By assuming that both files have been encrypted with the same key, we can figure the following equations. Another feature is the ability to retrieve product keys from unbootable Windows May 14, 2013 · Here's a look at several tools for deobfuscating XOR-encoded data during static malware analysis. The concept of implementation is to first define XOR – encryption key and then to perform XOR operation of the characters in the Nov 18, 2016 · 2. Using Pitch Shifter tool you can transpose song to a different key and tempo. Copy color code: Once you have picked the colors that Jan 1, 2016 · XOR Known-Plaintext Attack. The current color code will be displayed in a magnifying lens at all times. When the keystream is generated by a pseudo-random number generator, the result is a stream cipher. XOR also gets used in the key schedule. Calculate XOR. org,Super User,Experts Exchange enables people with technology problems to quickly and easily solve their problems by collaborating with experts from around the world. License MishaIac / YurisKeyFinder Public. Once the tracks get analyzed you can: Export the results to share with your friends. com Mar 21, 2017 · Download GameGuardian: https://gameguardian. 6 Commits. Quickly find the key and BPM of your entire library to find compatible tracks for your mashups. While XOR encrypting one image with a key is quite secure, repeating the same key twice makes it possible to find out the contents of the original, unencrypted images. The result of an XOR operation is 1 for bits that are different and 0 for bits that are the same. Users can also convert Number File to calculate XOR by uploading the file. This process, known as the XOR cipher, provides a simple yet effective encryption method. Working only on 64 bit OS. py [-x] [-l LEN] [-c CHAR | -b | -o] [-f] [-t CHARSET] [FILE] python xortool. A Message Authentication Code (MAC) is a symmetric-key cryptographic function for ensuring the message authenticity. We have to find the single-byte used to encrypt the plain text. main. In a typical MAC protocol, the sender of a message M computes a fixed, short tag T as a function of a MAC: \ (T=\text {MAC} (K,M)\) for the secret key K. You can upload multiple files at once. To find the key length, we’ll use some brute force and statistics. xor ("t"). In contrast, the XOR (Exclusive OR) gate is slightly more nuanced. Additionally, if Feb 4, 2020 · Once you have two byte arrays you can XOR each byte with the byte at the same index. xor ("r"). So let's print the file to the hexadecimal representation: ```bash Mar 10, 2015 · I don't have much previous experience at all with cryptography, this is pretty much the first time I've tried anything similar. length. From the dropdown, select Decrypt and in the first input field, enter a string. Oct 16, 2017 · With GNU tools, you can do: paste <(od -An -vtu1 -w1 file) <(while :; do od -An -vtu1 -w1 key; done) |. A ROT encoded file has its alphabetic Feb 3, 2023 · If it's the precise XOR key you're looking for, note that for a simple XOR cipher since we have p ⊕ k = c p ⊕ k = c where p p,k k,c c are the plaintext, key and ciphertext, XOR'ing both sides with p p you obtain p ⊕ c = k p ⊕ c = k , so given the plaintext and the ciphertext you can easily recover the key. The tool will decrypt the string and display the result in the second input field. Recovers repeated XOR key based on plaintext fragment somewhere in ciphertext - GitHub - nshadov/xor_key_recovery: Recovers repeated XOR key based on plaintext fragment somewhere in ciphertext Contribute to Reed18/Diffie-Hellman-Key-Generator-with-XOR-Encryption development by creating an account on GitHub. 9% 10: 15. 2 -> run RTE. Xor key and value usually look very very similar, especially for smaller values. 2 days ago · Follow the steps below to find the music key and BPM for any song: Step 1: Click on the Browse My Files button to upload songs. Convert the decimal 21 of the target to the binary ( 10101 ). Likely you have a plaintext XORed with a key or similar. However, in the character version we encrypt / decrypt using the Vigenere cipher. Given that you know that the file is a . i should find a bugged dungeon list before i waste my time. So the right key should be something like: "/***F****A*******". XOR has (inter alias) these advantages when used for cryptography: Very fast computable, especially in hardware. /xorknown. Output. For math, science, nutrition, history Cleared dungeon but no Keeper Xor, no key. now we have the key to How to Use XOR Decrypt Tool. For numbers, the operator returns a 32-bit integer. How to use: 1 -> run cabal launcher. To encrypt a message with a given key, we first convert the string into their ASCII equivalent, and we then XOR every character with the key. Click on Get button. Step 2: Now the BPM finder will display the key, BPM (Tempo), and camelot. ) Trying each displacement of the ciphertext against itself, count those bytes which are equal. Help Xorq implement this function. The data is Base64 decoded 2. img2 ^ key = enc_img2. This operator compares two bits and produces a return bit equal to 1 if the compared bits are different, and 0 if they are identical. XOR Gate’s Truth Table. 10110000 -> 01100001. It was first described by Frank Miller in 1882, and then reinvented in 1917. Desire value + new xor value = xor key. Use this free online Bitwise XOR Calculator for your XOR logic gate problems in computer organization and architecture! The XOR gate is a gate that requires the inputs to be exclusively OR, so only 1 input is true while the rest are false. Length])); return result. The Magical Jelly Bean Keyfinder is a freeware utility that retrieves your Product Key (cd key) used to install windows from your registry. In the binary version we encrypt / decrypt using the Xor algorithm. 3. (See Gaines [GAI44], Sinkov [SIN66]. data:0000000140023898 ; . I want the key so I can decrypt it. I read that to break repeating-key xor you can do the following: try a keysize n n and compute the hamming distance between the first n n bits of the encrypted string and the bits n + 1 n + 1 to 2n 2 n of the encrypted string and normalize by keysize. There are mainly two steps here: Find the key size. As you can see, we have 2 input fields and 1 dropdown. Step 4: Click the add tracks button to upload more tracks and get the key and BPM. By the way, each value can be replaced Aug 18, 2023 · Description. An XOR (exclusive OR) calculator is a tool or program designed to perform XOR operations on binary data. (Google it) Desire value + xor key = new xor value. Input: III. See full list on github. In my testing across an admittedly limited sample set, I ended up using random text samples from 80 different English language texts Easily find the key of a song by extracting it from a MP3 (mp3 to key) or any other audio file thanks our Online Song Key Finder. To celebrate my Microsoft MVP award 2016, I’m releasing a new XOR-tool. 8% 20: 15. 2 Finding XOR Key Through Brute-Force. A♭ major, 120 bpm]. You need a shell (like the GNU shell) with process substitution support, a od with support for the -w option (like GNU od ), and GNU awk for xor() (and the ability to output the NUL byte Apr 23, 2024 · Here are some of the applications of the XOR Gate: Data Encryption: XOR gates are used in data encryption algorithms. In Ruby, it looks something like this: "A". Filed under: Encryption, My Software — Didier Stevens @ 16:00. e generating random encryption keys to match with the correct one. Nov 28, 2020 · 1. Free calculus calculator - calculate limits, integrals, derivatives and series step-by-step XOR and Symmetric Key Encryption. This encryption algorithm uses a sequence of non-negative integers x = [x [1],x [2]. A ROL (or ROR) encoded file has its bytes rotated by a certain number of bits (the key). By itself, the XOR encryption can be very robust if: It is based on a long key that will not repeat itself. Each character in the decoded string is xored sequentially against each character of the key we previously identified. Nov 11, 2023 · We have encryption function E (k,m)=k⊕m and decryption D (k,c)=k⊕c, which means that m1⊕m2=c1⊕c2. content message guessed rwise k n the k e revea primar is that imple t ent, an he May 8, 2024 · 0. ToString(); Pick colors from the image: Move your mouse over the image (or your drag your finger, on mobile), and click on the specific color that you want to select. Returns true when only one of the inputs is 1. If an Average Hamming Distance with a XOR key size of 9 is 19, when you divide by 9, you get 2. Note: A few things to remember while Sep 7, 2023 · The above operation does the following calculation internally. Dec 11, 2021 · Find the key for each block by performing single byte key XOR decryption. If you only see sensible results between 0x20 (space) and 0x7E then it is probably ASCII text and you could perform character-decoding instead. XOR stands for eXclusive OR, meaning it's the OR operation but without 1+1 being true. For example, x ^ y will return 0 if x and y are the same, and a non-zero value if they are different. Fingers crossed they fix some of this in next week’s patch. The problem statement, defined above, is based on Cryptopals Set 1 Challenge 6. If you write out the binary representation, you can see the pattern: encoded decoded. Divide the ciphertext by blocks, each with length equal to the one chosen at step 1. It also means that if one value is 1, the result will be the inverse of the other value. Guess key lengths. Below is a simple implementation in C++. Do Jun 22, 2020 · That way we know the key length and parts of the messages. Received Thanks: 890. We'll cover XORSearch, XORStrings, xorBruteForcer, brutexor and NoMoreXOR. Drop your audio file (s) in the song analyzer below and instantly get the Key in which a song was composed by magic. wikipedia. En. I'm trying to implement an extremely simple XOR encryption system in ComputerCraft for fun, here is the code: local function xorEncrypt(data, key) local klen = #key local. xor ("k"). bin for string 'message' Key length: 15 Partial Key: 3RyS3cR Plaintext: e XOR o is extr mmon as nent in mplex c By itse g a con peating simple er can y be br ng freq alysis. The possible key values range from 0 to 255 (in Dec 25, 2023 · This tells us how many times a single character has to go through XOR with Key, substitution, and permutation. Decrypting such messages manually can be time-consuming and complex, but our user-friendly The one-time pad is theoretically 100% secure. 333. The key can be exported in various formats including a CryptoAPI blob, CNG blob, PEM, XML, and ASN. Step 3: Click on the Export CSV button to download the data. First, let's recall the general outline of cracking repeating key XOR. for i = 1, #data do. Liveleakr. Because you can never have enough XOR-tools in your toolbox :-). For example, if we want to encrypt XOR with 134 as a key Jun 8, 2013 · Posts: 237. We would like to show you a description here but the site won’t allow us. XOR the bits at each position as below. Do you have any idea of the size of the key, because if it Dec 30, 2006 · XORSearch is a program to search for a given string in an XOR, ROL, ROT or SHIFT encoded binary file. the key makes some sense. But, we know that the flag starts with ```easyctf{```. exe. II. s = "". Xor key + new xor value = desire value. Feb 27, 2024 · For example, x ^ key will encrypt x using key, and x ^ key ^ key will decrypt x using key. Try decrypting the text with the key, and voilà, you get the original text. This truth table demonstrates the inclusive nature of the OR operation — ‘either this or that, or both’ lead to a true result. When data is XOR-encrypted with a repeating key and you known some of the plaintext, you can perform a simple Oct 4, 2017 · Xor key is typically 1 or 2 dword lines away. Yes encryption, unbreakable even. img1 ^ key = enc_img1. – Amit. In order for a cipher to be a perfect cipher, the encoding key should meet three conditions: must be a random string; Single bits are XORed (typical component of more complex ciphers) The round key will be XORed with the intermediate result and after that permuted and substituted. g. 4% 30: 14. We don't know the key length, but after few tries with consecutive -n 1, -n 2, -n 3 we finally try -n 6: Mar 25, 2018 · XOR is a digital logic gate that outputs true or 1 only when the two binary bit inputs to it are unequal i. So encrypting the cipher text with the plain text, we can find the key. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. XOR encryption is used in many symmetric ciphers, including AES, where the same key is used for both encryption and decryption. You can use X4 or X. C[i] = P[i] XOR K[i%K]; K[i%k] = ( K[i%k] +P[i] ) mod 64; Where P is a plain text, C is the encrypted text, K is a key and k denotes the length of the key K (note the caps is different for k and K). [Release]Xor Keys Finder for EP8 PS. This is the function used to encrypt/decrypt and the function that generates a random key with a determined length: var result = new StringBuilder(); for (int c = 0; c < text. For example, , , and . We expect a PE binary to contain a phrase This program cannot be run in DOS. join(cipher) privvyKey = raw_input("Enter the private key xortool A tool to do some xor analysis: - guess the key length (based on count of equal chars) - guess the key (base on knowledge of most frequent char) Usage: python xortool. 5% 18: 4. LC_ALL=C awk 'NF!=2{exit}; {printf "%c", xor($1, $2)}'. So the value shifts left one place and the most significant bit wraps around to the Mar 9, 2018 · For each column, find one instance of the value that is supposed to be an 'e' and XOR that data with the letter 'e' to find the Nth letter of the key. 111. The lowest overall distance is likely the key length. Sep 15, 2019 · 1 Introduction. Cannot retrieve latest commit at this time. World tier 3, Necro, and i burned my nightmare sigil. append(bin(ord(i))[2::])#add the conversion of the letters/characters #in your message from ascii to binary withoout the 0b in the front to your ciphered message list cipher = "". In a single byte XOR, the length of the key is one byte, so there can be only 255 possible keys (0x0 - 0xff) with the exception of 0 as the ke y because XORing any value with 0 will give the same value as result (that is, no encryption). Apr 25, 2017 · 31. This tool allows loading the number URL, which calculate Numeral System data with XOR operator. In the password field, enter the secret key. 4% 12: 4. in these case. Xorq has invented an encryption algorithm which uses bitwise XOR operations extensively. MAC and Corruption Detection. The ^ operator is overloaded for two types of operands: number and BigInt. The key stream is probably random binary data, so you can hex encode the key to take a better look at it. Please note that this IS a homework but it ain't mine, I'm just rather curious on how to solve it since I've always liked encryption Our XOR Decrypt online tool offers a convenient solution for unraveling XOR-encrypted texts effortlessly. Add this with “Fury against fate” side quest that i still can’t complete. message XOR key XOR key == message So your way to check that key is correct will indeed return all possible keys. data:0000000140023899 byte_140023899 db 0 . xor ("f"). bin message Searching XOR-encrypted xor_encrypted_text. XOR is a simple encryption method, but without the key, you can't know the value. XOR also gets used in the key shedule. XOR has (inter alia) these advantages when used for cryptography: Very fast computable, especially in hardware. If enough parameters are present, the rest of the key can be calculated. For BigInts, the operator returns a BigInt. zn vz ao my nk dy ut sz dd aj