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 IB MathematicsExtended Essay To what extent does an Arc Tan Encryption System ableto perform like Rivest-Shamir-Adleman?Number of Words: 2343  2017 – 2018Armin 12B (St. Francis)Stella Maris International SchoolGading Serpong, Indonesia     Table of Contents Introduction. 3 Theory of Arc Tan Crypt. 4 Practical Encoding of a Message using Arc TanCrypt. 10 Practical Decoding of a Message using Arc TanCrypt. 13 Data Collection for Encryption Test from Arc TanCrypt. 16 Introduction to RSA (Rivest, Shamir, Adleman).

17 Practical Encoding of a Message using RSA.. 18 Practical Decoding of a Message using RSA.. 19 Data Collection from RSA for Encryption Test.

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20 Conclusion. 20 Bibliography. 21   Introduction Withthe rapid development of network and multimedia technologies, privacy hasbecome a crucial part to the transmission of sensitive electronic data. Thus,ensuring the privacy for the transmission of sensitive electronic data plays astarring role in the eyes of many private users of several internet accountsand further private activity online.  Byensuring higher security in terms of cryptography for electronic data, we dohave to face algorithms and hours of mathematical studies and computerengineering.

By doing this Research, the author does have the intention to learnmore deeply about existing technologies and to propose a new improved method. Byproposing a new method later, the author intend to help people around the worldto feel safer whenever they must transfer private data through an untrustedconnection.Inthis essay, the author will first discuss about Tan Crypt which is the author’sown original idea. The author intends to find out to what extend does an ArcTan Encryption System perform like Rivest-Shamir-Adleman. To find out theanswer, the author will compare both encryption systems at the end of thisessay in the conclusion section.          Theory of Arc Tan Crypt all calculations arein DEG-ModeThis system does not represent a public key cryptosystem.It does make use of a single key that is used to encrypt and decrypt themessage. During the research, the author tried to find a way to encode datavery randomly using a simple key (or more than one key at the same time).

Sothe first step was to look around for something really unique in mathematicswhich could be used for encoding data. An example that was found by the authoris… Compared with elliptical graphs it does look morecompromising to be used for encryption when represented as graph as it can beseen below. Y – axis X – axis Figure 1:  When y = 789tan (2x+10)           When analyzing the graph, it can be seen that the lineis unregularly and seems very hard to find a general equation for this graphdirectly. The only important thing at this point is just to find a type ofgraph that does show a lot of random difference in y-values when the functionis altered by just a small number (e.g. 5,33,62). Y – axis X – axis Figure 2:  When y = 12tan (-3x+8)     Theequation for decrypting the message can be easily found by the following steps The first precaution in here is that when the Data isencrypted and decrypted using the two formulas stated above, it does affect theresult.

When the original data is compared with the encrypted and thendecrypted data, it is different. This problem is visualized below. Compared to be different In case thecalculation is done in the opposite way, which means that the encryptionformula and the decryption formula are exchanged with each other for thefollowing calculation process, the Original Data is the same with the DecryptedData. An graphical representation of this is shown below. Compared to be the same Practically, if the variable for ‘X’ has got a bigvalue of more than 6 digits (this value may vary due to different systems used,e.

g. GDC, Computer), it becomes hard to get the exactsame result after decryption as before doing the encryption.A simple example can be shown below using simple and freely-assigned values forthe three keys… Variable Value assigned A 2 B 3 X 010205000810 C 1 Z 12345678    = X Decryption…       Z = 12345677.999999999999999999999998The reason for this inaccuracy (.99999999999…

)is because the value for ? in ‘ ‘ is bigger than90 (if the angle is in degrees; e.g. radians -> 2?). The solution for thisproblems is to make sure that the value for ‘?’ needs to be inversed. This stepmakes sure that any value for ‘?’ is smaller than 1. Therefore, the formula forencryption will change.The objective for the following calculation is to meetthe objective or target stated above which is to decrease the size of thenumber that needs to be used as the value for ‘?’.   Thus, two new formulas can be summarized from theconclusion and calculation, as shown above, into the table below.

Encryption Formula Decryption Formula  Thus, the validity for this formula is tested as shownbelow using the same values as used in the example above. Decryption… X= -0.32962962962962962962962962962959     Practical Encoding of a Message using Arc TanCryptFor the first time, let a simple message to be encodedusing this cryptography system.    ABE HJ1.     Now, it is neededto convert this text into cipher which is using numbers. In this case, we willuse our own Encoding System but you can use any other Cipher – Codec (e.

g.ASCII) as long as the Decoder of the text (e.g.

your friend) also use the sameCipher – Codec A B C D E F G H I J K L M 01 02 03 04 05 06 07 08 09 10 11 12 13                           N O P G R S T U V W X Y Z 14 15 16 17 18 19 20 21 22 23 24 25 26  Note:A space will be replaced by a zero in this Encoding SystemUsing this Table, we will convert the letters and allthe spaces into numbers; this is a single number, so it is named ‘PrimaryData’. All the numbers are combined into one single number            A-> 01             |          B-> 02             |           (space) -> 00  |          …    ABE HJ 010205000810Note:This number is called Primary Data aspreferred by the author2.      Nowwe will have to think of three different encryption keys, it has to be asunique as possible by using different numbers (non-zeroreal number, ) ofrandom difference. It doesn’t matter if it’s even or odd, it just has to beunique and as long as possible Key 1= -0.

16329           Key 2= -1347.34323Key3= 34343133.     Now the authorwill make use of the actual encryption Formula which is… Themeaning or definition of those variables mentioned above like ‘A’ and ‘B’ arewritten below in the table. Variable Definition A The value of the first key B The value of the second key Z The primary data C The value of the third key X The secondary data  The values are assigned to the corresponding variablesas shown in the table below Variable Value assigned A -0.16329 B -1347.

34323 Z 010205000810 C 3434313 X Encrypted Data (Secondary Data)     4.     So now all thevalues are put into the formula and calculated                   A      B                                    C              X    5.      Theresult represents the final encrypted data or ‘Secondary Data’ which can now beused to be transferred or shared to the receiver of the data2548.9518443723587130981426922727Note:This number is called Secondary Data aspreferred by the author Practical Decoding of a Message using Arc TanCrypt Note: Same Message is used for decoding as has been encoded in          previous section above      Asthe receiver of the data, the author will use the following steps to decryptthe Secondary Data back to the Primary Data.1.

     Todecrypt the data, the receiver must make use of a decryption formula. Thedecryption formula is as follows… In this formula, there are some variables seen buttheir meanings are still the same as before during the encryption process.Below are the definitions of the variables. Variable Definition Z The secondary data A The value of the first key C The value of the third key B The value of the second key X The primary data  2.     Practically, thereceiver assigns all the known values, including the encryption key itselfwhich are three keys, to its corresponding values as defined above.

Variable Value X 2548.9518443723587130981426922727 A -0.16329 C 3434313 B -1347.34323 Z Decrypted Data (Primary Data)  3.     Furthermore, allthe variables are put into the formula and the decrypted data can be retrieved. 4.

      Thisstep is only necessary if the length of the number is odd. For example, Number Length Even or odd? Action 1234 4 Even no action 4567890 7 Odd (Add zero before its number) 04567890 02340003243 11 Odd 002340003243  So in this case, the number ‘10205000810’ is an oddnumber and therefore needs to have an additional ‘0’ (zero) on its left end.  0102050008105.     Now the code canbe simply reconverted back into the original message. 01 -> A             |          02-> B             |           00 -> (space)  |          …010205000810 ABE HJData Collection for Encryption Test from ArcTanCrypt Before the encryption test is being done later on,there need to be more examples of different data to obtain more reliable test results.     Data ID Content of Data Purpose 1 ABE HJ Simple example 2 1234567890 Identic Cumulative difference 3 AAAAAAAAAAAA Repetitive data                                                                                                                                                                            All steps as they havebeen done in the sections above to encrypt the data using Arc TanCrypt (Step1-5) are repeated and the corresponding results put in the table below; thesame corresponding keys are being used (Key 1= -0.16329; Key 2= -1347.34323;Key 3= 3434313)    Data ID Content of Data Encrypted Data 1 ABE HJ 2548.

9518443723587130981426922727 2 ABCDEFGHIJK 2548.9520335559295211689030626686 3 AAAAAAAAAAAA 2548.9518443723587130141375417096            Introduction to RSA (Rivest, Shamir, Adleman)  (Page,Don. https://www.pagedon.com/wp-content/uploads/2010/03/rsa-encryption.jpg.28 March 2010.

Document. 22 January 2018)   Above is a simple representation ofhow RSA is applied as a public key cryptography. The first thing to do beforeencrypting and decrypting a message is to generate the public key itself. Thereare some rules and conditions that need to be considered before generating akey for RSA which are listed down below in the box. Choose two primes p and q and let n = pq.

Let e ? Z be positive such that gcd(e,?(n)) = 1. Compute a value for d ? Z such that de?1 (mod ?(n)). Our public key is the pair (n,e) and our private key is the triple (p,q,d). For any non-zero integer m

Decrypt c using m?cd (mod n).                                             (The Sage Development Team)To apply the RSA public key cryptography, the Authorwill use a Mathematics Program named SAGE to do all the calculation related toRSA. The following two sections (Practical Encoding of a Message using RSA,Practical Decoding of a Message using RSA) are adapted from Official SAGEMathematics Website (The Sage Development Team).Practical Encoding of a Message using RSAThe plain text that will be encrypted is “TEST”.

1.     Choose twoprimes p and q and let n = pq. 2.     If e representseuler’s number then let e ? Z be positive suchthat gcd(e,?(n)) = 1. 3.     Compute a valuefor d ? Z such that de?1 (mod ?(n)).

4.     Our public key isthe pair (n,e) and our private key is the triple (p,q,d).  5.

     For any non-zerointeger m

Document. 22 January 2018.)Data Collection from RSAfor Encryption Test Data ID Content of Data Encrypted Data 1 ABE HJ 1070917808190970274561765953 2 ABCDEFGHIJK 4294446786332150780921029517 3 AAAAAAAAAAAA 2618252287065532206260889871  ConclusionThe advantage of using the Tan Crypt is that whenproviding a valid encryption key is very easy which means there no specificrules that needs to be followed for generating the key. On the other hand, RSAneeds to have a special key for encryption and decryption which is complicatedto do. Tan Crypt does encrypt data into a decimal number which helps confusingcrackers because the cipher text is so far from the nature of the originalplaintext through using Tan Crypt. Improvement that could be done from Tan Crypt is thatit is not a public key cryptosystem.

Therefore, Tan Crypt cannot be as usefulas RSA for public key cryptography purposes. This is because the key forencryption and decryption is the same using Tan Crypt.The advantage of using RSA is that it has been alreadyproven to be reliable in the modern era of telecommunication. Even though RSAmay not be 100% secure but it still may have a better image in the minds ofexperts nowadays. In this case the process to generate prime numbersusing RSA method could be longer than the prediction because the first step is liketrial and error (see step 1 in RSA). Prime numbers need to be generated in RSA aredone by trial and error using the Mersenne Prime Number Rule.

Also, when theprimes are multiplicated with each other (p*q), the result becomes bigger thanthe data that is planned to be encrypted becoming a cipher text.      The Arc Tan Encryption System will able to evolvebecoming a public key cryptosystem if it applicable for telecommunicationpurposes like RSA. On the other hand, if the Arc Tan Encryption System is notapplicable in vast areas of Privacy, then it would not get the upper hand indomination versus other cryptosystems quantitatively in numbers of its usersand various applications for privacy purposes.   Bibliography Page, Don. https://www.pagedon.com/wp-content/uploads/2010/03/rsa-encryption.jpg.

28 March 2010. Document. 22 January 2018. The Sage Development Team.

Number Theory and the RSA Public Key Cryptosystem. 2017. Document. 22 January 2018.      

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