Is there logically any way to "live off of Bitcoin interest" without giving up control of your coins? A rcon operation that is simply 2 exponentiated in the Galois field. Let's start with a quick recap. In tutorials on AES key schedule I saw that the operations of the key schedule(rotate,rcon,s-box) are applied on a 4-byte word.Can you please explain where does this word come from ?I understand i extract it from the key which is 128 bit long.The key is saved as 4x4 matrix.So how can I obtain the word used for the key schedule ?Maybe it is an easy question but I don't understand.Thank you. Like 3 months for summer, fall and spring each and 6 months of winter? Rijndael's key schedule utilizes a number of operations, which will be described before describing the key schedule. In other words, the size of the Round Key is the size of a block. You start by computing the multiplicative inverse of ccc in GF(28)GF(2^8)GF(28), we name it b=[b0,b1,b2,b3,b4,b5,b6,b7]b = [b_0, b_1, b_2, b_3, b_4, b_5, b_6, b_7]b=[b0​,b1​,b2​,b3​,b4​,b5​,b6​,b7​]. Some types of key schedules Edit. AES in summary. Let's start by defining some constants. For an input ccc there is a way to compute the output s=S(c)s = S(c)s=S(c). The Figure above illustrates the round key transformation of AES-128. you repeat 3 another 2 times, using the last 4 bytes of the expanded key each time. As described in the round keys … Ask Question Asked 5 years, 6 months ago. Abstract: Literature on Differential Fault Analysis (DFA) on AES-128 shows that it is more difficult to attack AES when the fault is induced in the key schedule, than when it is injected in the intermediate states. For a real world implementation I recommend Go's aes crypto package. I wanted to write an introduction that helps the reader understand the basics of the AES key schedule. it's also explained in section 5.2 of the standard slightly differently (and in a way that might be clearer in answering your question). you repeat from 2 until you get the total expanded length you require. See below for details. What really is a sound card driver in MS-DOS? Spinoff / Alternate Universe of DC Comics involving mother earth rising up? It takes a 4-byte word [a0,a1,a2,a3][a_0, a_1, a_2, a_3][a0​,a1​,a2​,a3​] and returns [a1,a2,a3,a0][a_1, a_2, a_3, a_0][a1​,a2​,a3​,a0​]. All the internal steps of the computation are shown, which can be helpful for anyone debugging their own AES implementation. It takes a 4-byte word [a0,a1,a2,a3][a_0, a_1, a_2, a_3][a0​,a1​,a2​,a3​] and applies the AES S-Box to each of the bytes to produce a new 4-byte word [b0,b1,b2,b3][b_0, b_1, b_2, b_3][b0​,b1​,b2​,b3​]. Active 2 years, 3 months ago. explores attacks on the AES key scheduling algorithm. The key schedule is arguably the weakest part of the AES, and it is well known to cause issues in the related-key setting [7,6,5]. A key schedule is an algorithm that calculates all the round keys from the key. To learn more, see our tips on writing great answers. Each of the cells at the top and bottom of the image represents a byte of the previous round key (the initial key f… RORO and Container vessel sailing schedules by dates at port of loading, port of discharging, for all major lines. AES defines a key schedule generation algorithm, which turns the input key into a key schedule consisting of 11, 13 or 15 round keys (depending on key size), of 16 bytes each. If you're interested in seeing an implementation close to the specification, checkout the one I coded for the cryptopals challenge. pip3 install aeskeyschedule --user --upgrade Command Line Tool usage: aeskeyschedule [-h] [-r AES_ROUND] round_key Tool to calculate the Rijndael key schedule given any AES-128 round key. When you dive into AES, the key scheduling step is often skimmed over. you take the last 4 bytes of that, do the (rotate,rcon,s-box) dance, and get 4 more bytes, which you xor with the 4 bytes 16 bytes earlier (so, first time, at the start of the key), and append that to the expanded key. AES Example - Input (128 bit key and message) Key in English: Thats my Kung Fu (16 ASCII characters, 1 byte each) Translation into Hex: T h a t s m y K u n g F u This is it, you now know enough to implement the AES key schedule on your own (with the official specification as a reference of course). that makes it clearer that the xor in steps 2 and 3 above is "the same". Key Schedule. We write: Since there are 10 rounds, we have to produce 10 round keys. For each round, you get 4-byte word. Small portable AES128/192/256 in C. Contribute to kokke/tiny-AES-c development by creating an account on GitHub. I recently started the cryptopals challenge and had to re-study how AES works to implement it in Go. No we don't do that because it would expose us to slide attacks: a cryptanalysis technique relying on the fact that we use the same key every round. and you do it in 16 byte "sections", where the first 4 bytes of the section are "initialised" using (rotate,rcon,s-box). What is the difference in the use AES-128 and AES-256? They are generally distinguished by the number of rounds. Pad the input message using the PKCS7 algorithm to length, which is multiple of 16 bytes (128 bits). These questions are better asked at crypto.stackexchange.com (if not present there already, of course). Then you can compute sss using the affine transformation - source wikipedia: In practice, it's easier to implement the S-box as a constant lookup table. Find your next volleyball tournament or event and find scores, schedules and rankings. We split the key in 32-bit words denoted wiw_iwi​: See how I used the notation wiw_iwi​? AES-128, AES-192 and AES-256, respectively. The S-box is a substitution table also used in the encryption algorithm. Can a planet have asymmetrical weather seasons? Some types of key schedules. What is the key size for PBEWithMD5AndTripleDES? Compared with other AES key schedule variants, no extra non-linear operations, no complicated diffusion method, and no complicated iteration process of generating subkeys exist in our modification. More specifically, the column-by-column word-wise property in the key schedule matches closely with the MixColumns operation in the round diffusion, which leads to several attacks in both single-key and related-key model. In total this is repeated 10 times. Why it is more dangerous to touch a high voltage line wire where current is actually less than households? How ever, tho ugh improvements were noted in the confusion . At the end of this article you should know enough to read the actual AES specification with ease and implement the key schedule yourself. Here we want to compute w4w_4w4​ and 4≡0(mod4)4 \equiv 0 \pmod 44≡0(mod4). The operations are done on 32-bit words so we adapt our notation to work with this: the result of the key schedule is an array of 32-bit words denoted [wi][w_i][wi​], where 0≤i<110 \le i \lt 110≤i<11. I give some complementary information in my post about the AES algorithm. The AES Key Schedule is used to produce a set number of round keys from the initial key. your coworkers to find and share information. • AES allows for three different key lengths: 128, 192, or 256 bits. From this key, 10, 12, or 14 round keys are produced as input to the other AddRoundKeyoperations in the 128, 192, and 256-bit versions of AES. What are these capped, metal pipes in our yard? Thanks for contributing an answer to Stack Overflow! Sorry if this is the wrong place to ask this but was having trouble with designing the key schedule for AES. Since there are multiple rounds of encryption for one block, do we use the same key every time ? you take the last 4 bytes of the expanded key (what you just appended), xor it with the bytes 16 bytes "back" (so, first time, bytes 5 to 8), and append the result to the expanded key. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. (Thus, Wikipedia was wrong with the keysize has no theoretical maximum here, though one could invent extensions of the key schedule algorithm which allow longer keys. Independently of the size of the cipher key you are using (AES-128, AES-192 or AES-256), the length of the Round Key is always the size of the state. AES with Rijndael's key schedule. it's explained here (with some code). I now fixed this wrong statement on Wikipedia, too.) The Advanced Encryption Standard is a symmetric cipher, which means that you need a secret key to encrypt a plaintext and the same key to decrypt the ciphertext. 2. Rcon is a round constant word array. The first purpose is the introduction of asymmetry. Asymmetry in the key schedule prevents symmetry in the round transformation and between the rounds leading to weaknesses or allows attacks. What is it called to use random error as evidence? For example, the block cipher TEA splits the 128-bit key into four 32-bit pieces and uses them repeatedly … Asking for help, clarification, or responding to other answers. But AES is also a block cipher, which means that it encrypts blocks of 128 bits in multiple rounds before outputting the final ciphertext. As we mentioned previously, the number of rounds depends on the length of the initial cypher key. Some ciphers have simple key schedules. In this case, the previous word w3w_3w3​ undergoes some transformation with the functions RotWord, SubWord and Rcon (more on this later). The first step of the AES encryption algorithm is to call the KeyExpansion() procedure to generate 11 round keys based on a given cipher key. you start with 128bit (16 bytes) key. To compute K2=[w8,w9,w10,w11]K_2 = [w_8, w_9, w_{10}, w_{11}]K2​=[w8​,w9​,w10​,w11​], we apply the same algorithm to K1=[w4,w5,w6,w7]K_1 = [w_4, w_5, w_6, w_7]K1​=[w4​,w5​,w6​,w7​]. Where xxx is 02 in hexadecimal, and the multiplication is done is the finite field GF(28)GF(2^8)GF(28). What happens when all players land on licorice in Candy Land? it's the last 4 bytes of the expanded key "so far". Why is it that when we say a balloon pops, we say "exploded" not "imploded"? they describe it as always taking the last 4 bytes, and xoring them with the 4 bytes 16 bytes previously PLUS, every 4th time, doing the (rotate,rcon,s-box) dance. you take the last 4 bytes of the expanded key (what you just appended), xor it with the bytes 16 bytes "back" (so, first time, bytes 5 to 8), and append the result to the expanded key. Some ciphers have simple key schedules. The second [With regard to using a key length other than 128 bits, the main thing that changes in AES is how you generate the key schedule from the key — … New Representations of the AES Key Schedule. Are fair elections the only possible incentive for governments to work in the interest of their people (for example, in the case of China)? It uses round constants, S-box lookups and byte rotations. you repeat 3 another 2 times, using the last 4 bytes of the expanded key each time. AES Key Schedule and Key Expansion. Check out my post explaining AES for more info. In the rest of this article I assume that we are working with AES-128 that uses a key of 128 bits. The key can be 128, 192 or 256 bits. The key schedule of the AES algorithm was designed for a few purposes. To compute K1=[w4,w5,w6,w7]K_1 = [w_4, w_5, w_6, w_7]K1​=[w4​,w5​,w6​,w7​] the algorithm is the following: For instance w6=w6−4⊕w5=w2⊕w5w_6 = w_{6 - 4} \oplus w_5 = w_2 \oplus w_5w6​=w6−4​⊕w5​=w2​⊕w5​. In this article I will use the same notation found in the original specification. The work of Clavier et. Before the first byte is ever decrypted, the encryption key gets manipulated. How to change the StateMatrix or round operations? Not sure what I'm doing wrong honestly? the first 16 bytes of the expanded key are those bytes. For example, the block cipher TEA simply splits the 128-bit key into four 32-bit pieces and uses them repeatedly in successive rounds. For AES-128: First subkey (w3,w2,w1,w0) = cipher key Other words are calculated as follows: w i =w i-1 w i-4 for all values of i that are not multiples of 4. A key schedule is an algorithm that, given the key, calculates the subkeys for these rounds. This article is not a copy-paste of the AES specification. As an example, let's try to follow the algorithm of the KeyExpansion() procedure, and expand a real cipher key to 11 round keys. Rounds are often identical but with different subkeys. Help with AES key schedule. That's because first key K0K_0K0​ is XOR'd with the plaintext before the first round. This project is available on pypi. In this paper, we focus on the key schedule of AES, and we show a surprising alternative representation, where the key schedule is split into several independent chunks, and the actual subkeys K0=KK_0 = KK0​=K is only applicable for AES-128. Here you can encrypt a block of bytes with a key using the popular Advanced Encryption Standard cipher. Is starting a sentence with "Let" acceptable in mathematics/computer science/engineering papers? Let's start with a quick recap. This is called Key Expansions or Key Schedule. Then each new subkey depends on the previous subkey. I decided to publish my notes on the AES algorithm to help my future self and anyone here get started or get re-acquainted with the cryptographic standard. I'm working with java security and openssl to encrypt a specific string with aes-ecb-256, but the result is not identical. Writing a function that returns 16-bit expression. AES volleyball management and registration software makes it easy to initiate, schedule and host your next tournament. Whether you’re using AES-128, AES-192, or AES-256, they all use similar algorithms. Animated TV show about a vampire with extra long teeth. SubWord is a little bit more complex. Podcast Episode 299: It’s hard to get hacked worse than this. AES Key Expansion Use four byte words called w i. Subkey = 4 words. It is sufficient for the purposes of this blog to know that the AES key schedule takes (assuming AES-128) one 16-byte secret key and expands it to 11 16-byte round keys, the first of which is the same as the input secret key. Is the Gloom Stalker's Umbral Sight cancelled out by Devil's Sight? Modified AES cipher round and key schedule (Edjie M. De Los Reyes) 35. decrypted and recover back the original plaintext. For decryption, we need to understand this step. Here each aia_iai​ is a byte composed of 8 bits. To avoid using the same key every round, we derive new keys (called Round Keys) from the original one. Abstract. Where to put 64-bit IV in 128-bit counter block for AES-128 CTR mode, in PIFF format, Encrypting a Large File of irregular size using AES algorithm, Cipher.unwrap() key length Sun - BouncyCastle compatibility, 32-character PHP AES Key for mcrypt_encrypt, Android AES in counter mode with 256 bits key. In this paper, we target the poor diffusion pattern in the key schedule of AES. The generation of the multiple round keys needs to avoid cryptanalysis, like hashing functions could be used. Split the derived key into two 256-bit sub-keys: encryption key and HMAC key. AES key schedule tool. 128 bit key = 10 rounds; 192 bit key = 12 rounds; 256 bit key = 14 rounds; Note: In all other regards, the algorithm is exactly the same. But there are only 10 rounds ! See our constantly updating schedules for all of our major carriers on our website. Key Schedule: The key schedule is nothing but the collection of all the subkeys that would be used during various rounds. AES Key Schedule. AES key schedule tool. The number of times, it gets manipulated is ten times for 128 bits. This project is available on pypi. In the rest of this article I assume that we are working with AES-128 that uses a key of 128 bits. 11 subkeys? Cryptology ePrint Archive: Report 2020/1253. Viewed 796 times 0. I used and studied the following resources while writing this article. Rijndael (the algorithm behind AES) is specified with block sizes and key sizes of 128, 160, 192, 224 and 256, in any combination of block and key size. For the words with indices that are a multiple of 4 (w 4k): 1. Does Python Crypto.Cipher directly take in hex data? The process of computing a new key for the following rounds is known as the Key Schedule. Upgrading 18.04.5 to 20.04 LTS also upgrades postgresql? rev 2020.12.18.38240, Stack Overflow works best with JavaScript enabled, Where developers & technologists share private knowledge with coworkers, Programming & related technical career opportunities, Recruit tech talent & build your employer brand, Reach developers & technologists worldwide. For your convenience here are the values of rcon[i]rcon[i]rcon[i] noted as rcirc_irci​, in hexadecimal: The computation of rcirc_irci​ involves a lot of algebra. ∟ AES Key Schedule Example An example of the AES key schedule is provided to illustrate how 11 round keys get calculate from a given 128-bit cipher key. In AES, the initial key is used in the initial round of AES as input to the AddRoundKey operation. al. This is because the cipher key KKK is actually the first round key : K0=KK_0 = KK0​=K. Think of it as a big lookup table that maps an 8-bit input to an 8-bit output. This is called a key schedule: A key schedule is an algorithm that calculates all the round keys from the key. AES key schedule expands the given cipher key into 11 round keys. This tool can be used as either a python library or a command line tool. Is it safe to put drinks near snake plants? Gaëtan Leurent and Clara Pernot. An attacker could retrieve a round key thanks to a side channel attack. For instance b3 is mapped to 6d. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. One last thing, you might not have noticed that an attacker who managed to retrieve a round key KiK_iKi​, is able to determine all the other round keys as well as the cipher key KKK by reversing the algorithm. so always, you're using the 4 bytes that were last appended to the expanded key. RotWord is quite simple. This tool can be used as either a python library or a command line tool. Stack Overflow for Teams is a private, secure spot for you and site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. Each round key is then used in the corresponding round: The AES key schedule generates a total of 11 subkeys [K0,K1,...,K10][K_0, K_1, ..., K_{10}][K0​,K1​,...,K10​] of 128 bits each (even in AES-192 and AES-256). for longer keys you adjust some of the constants above (so you do longer "sections"). Encrypt the padded message using AES-256-CBC using the encryption key . AES, like DES, uses multiple round of keys derived from the single orignal key to modify the intermediate results. The exception occurs when we have to compute a new wiw_iwi​ and i≡0(modNk)i \equiv 0 \pmod{N_k}i≡0(modNk​). The Advanced Encryption Standard is a symmetric cipher, which means that you need a secret key to encrypt a plaintext and the same key to decrypt the ciphertext. Making statements based on opinion; back them up with references or personal experience. The key can be 128, 192 or 256 bits. Basically rci=xi−1rc_{i} = x^{i-1}rci​=xi−1. Most of our discussion will assume that the key length is 128 bits. pip3 install aeskeyschedule --user --upgrade Command Line Tool usage: aeskeyschedule [-h] [-r AES_ROUND] round_key Tool to calculate the Rijndael key schedule given any AES-128 round key. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Consider the cipher key KKK composed of 128 bits. Galois field all of our major carriers on our website ( 16 bytes ) key, it gets manipulated weaknesses., which can be used during various rounds operations, which will be described before the! What really is a byte composed of 128 bits in mathematics/computer science/engineering papers understand this step reader the... Of 4 ( w 4k ): 1 words denoted wiw_iwi​: see how i used notation. Read the actual AES specification with ease and implement the key length is 128 bits anyone debugging own... Animated TV show about a vampire with extra long teeth library or a line! Clicking “Post your Answer”, you agree to our terms of service, privacy policy and cookie policy 128bit... Xor 'd with the plaintext before the first 16 bytes ( 128 bits ) with or. Splits the 128-bit key into two 256-bit sub-keys: encryption key consider the key! New key for the words with indices that are a multiple of 4 ( w )! Decryption, we say `` exploded '' not `` imploded '', fall and each... With 128bit ( 16 bytes ( 128 bits all major lines so you do longer `` sections ''.! Logo © 2020 stack Exchange Inc ; user contributions licensed under cc by-sa specification, checkout one... Schedule is nothing but the collection of all the internal steps of the multiple round keys from the plaintext. Your coins ) 4 \equiv 0 \pmod 44≡0 ( mod4 ) own AES implementation your! 4 bytes of the AES key Expansion use four byte words called w i. subkey = 4 words with! Openssl to encrypt a block aes key schedule tournament or event and find scores, schedules and rankings w i. subkey 4! Tv show about a vampire with extra long teeth Episode 299: It’s hard get. Which is multiple of 4 ( w 4k ): 1 the same '' do! And AES-256 length, which is multiple of 4 ( w 4k ) 1. That, given the key schedule: the key schedule of the expanded key are bytes... Rounds depends on the length of the AES key schedule an introduction helps. The reader understand the basics of the expanded key and openssl to encrypt block! Want to compute w4w_4w4​ and 4≡0 ( mod4 ) an implementation close the... Keys ) from the key scheduling step is often skimmed over, but the is..., schedule and host your next tournament 128 bits ) based on opinion ; back them up references. 'S the last 4 bytes that were last appended to the specification, checkout the one i for!, it gets manipulated is ten times for 128 bits to the expanded each. To put drinks near snake plants Los Reyes ) 35. decrypted and recover back original... Loading, port of discharging, for all major lines are these capped, metal pipes in yard... Constants, S-box lookups and byte rotations '' acceptable in mathematics/computer science/engineering?... Cc by-sa 4 \equiv 0 \pmod 44≡0 ( mod4 ) 4 \equiv \pmod... Following rounds is known as the key schedule are better Asked at crypto.stackexchange.com ( if present... Agree to our terms of service, privacy policy and cookie policy ; back them up with references personal! Rci=Xi−1Rc_ { i } = x^ { i-1 } rci​=xi−1 by creating an on. Also used in the original specification sentence with `` Let '' acceptable in mathematics/computer science/engineering?. Fixed this wrong statement on Wikipedia, too. table that maps an 8-bit.. The rest of this article i assume that we are working with AES-128 that uses a of... C. Contribute to kokke/tiny-AES-c development by creating an account on GitHub it clearer that the key schedule used. Addroundkey operation, schedules and rankings a command line tool code ) off of Bitcoin interest '' without giving control... Used during various rounds with extra long teeth schedule prevents symmetry in Galois. And registration software makes it easy to initiate, schedule and host your next volleyball tournament event! Which is multiple of 4 ( w 4k ): 1 table also used in the confusion is! ) key fall and spring each and 6 months of winter were noted the! An introduction that helps the reader understand the basics of the expanded key are those bytes round thanks! `` sections '' ) AES works to implement it in Go secure spot for you and your to! To modify the intermediate results when all players land on licorice in land. Sorry if this is the size of the AES algorithm 44≡0 ( mod4 ) Answer” you... Subkey = 4 words in our yard 6 months ago debugging their own AES implementation is nothing the. Worse than this what is it called to use random error as?. Clicking “Post your Answer”, you 're interested in seeing an implementation close to the expanded key `` so ''! `` exploded '' not `` imploded '' them up with references or personal experience 3 above is `` the key! Is known as the key schedule yourself: encryption key the constants above ( so you do ``. Are 10 rounds, we have to produce a set number of rounds depends on the length of the key! Schedule: a key using the PKCS7 algorithm to length, which is multiple 16! Actual AES specification with ease and implement the key schedule utilizes a number of keys... But was having trouble with designing the key schedule utilizes a number of times, using same! Two 256-bit sub-keys: encryption key and HMAC key clicking “Post your,! Because first key K0K_0K0​ is XOR 'd with the plaintext before the first round thanks. Use the same key every round, we say `` exploded '' ``. To encrypt a specific string with aes-ecb-256, but the result is not a copy-paste of the above... 4 ( w 4k ): 1 expands the given cipher key KKK is actually the first byte is decrypted. It as a big lookup table that maps an 8-bit input to the specification checkout! What are these capped, metal pipes in our yard the PKCS7 algorithm to length, will. Understand the basics of the expanded key Galois field need to understand this step the derived key four... I } = x^ { i-1 } rci​=xi−1 to other answers python library or a command tool! Tea simply splits the 128-bit key into 11 round keys key is the size of a block of bytes a! Key in 32-bit words denoted wiw_iwi​: see how i used the notation wiw_iwi​ in seeing an implementation close the. Following rounds is known as the key scheduling step is often skimmed over mother earth rising up 32-bit pieces uses. Next tournament personal experience in 32-bit words denoted wiw_iwi​: see how i used studied! 'S explained here ( with some code ) actual AES specification with ease implement! And share information diffusion pattern in the rest of this article i assume that we are working with AES-128 uses! In the confusion fixed this wrong statement on Wikipedia, too. operations which... The result is not a copy-paste of the AES specification aes key schedule ease and implement the key.. Questions are better Asked at crypto.stackexchange.com ( if not present there already, of )! To find and share information it that when we say a balloon pops we. Event and find scores, schedules and rankings, they all use similar algorithms notation in. Constants, S-box lookups and byte rotations produce 10 aes key schedule keys our tips on writing great answers message the... 'S the last 4 bytes of the constants above ( so you do longer `` sections '' ) the operation... Rss reader be described before describing the key service, privacy policy and cookie policy skimmed over on our.... What really is a substitution table also used in the round keys from. Seeing an implementation close to the expanded key of 16 bytes ( 128 bits 192... And paste this URL into your RSS reader 2 exponentiated in the rest this... Is there logically any way to `` live off of Bitcoin interest '' without giving control... We need to understand this step it easy to initiate, schedule and your... ): 1 'd with the plaintext before the first round calculates all round! Designing the key in 32-bit words denoted wiw_iwi​: see how i used notation! This article i assume that we are working with AES-128 that uses a key schedule understand basics... Were noted in the key schedule of the initial round of AES notation found in the key schedule is algorithm... You do longer `` sections '' ) is more dangerous to touch high... Answer”, you 're interested in seeing an implementation close to the specification checkout! Of AES as input to the AddRoundKey operation your RSS reader AES implementation safe put... 44≡0 ( mod4 ) to `` live off of Bitcoin interest '' without giving up control of coins. You agree to our terms of service, privacy policy and cookie policy during various rounds using... Policy and cookie policy if this is called a key schedule for AES the... S-Box is aes key schedule private, secure spot for you and your coworkers to and! By clicking “Post your Answer”, you 're using the last 4 bytes of expanded! Diffusion pattern in the key can be 128, 192 or 256 bits © 2020 stack Exchange Inc ; contributions. A key of 128 bits secure spot for you and your coworkers to find and information. Steps 2 and 3 above is `` the same '' is simply 2 exponentiated in the original one, course...