Implementation of RSA Algorithm with Chinese Remainder Theorem for Modulus N 1024 Bit and 4096 Bit

Desi Wulansari, Much Aziz Muslim & Endang Sugiharti
International Journal of Computer Science and Security (IJCSS), Volume (10) : Issue (5) : 2016 186
Implementation of RSA Algorithm with Chinese Remainder
Theorem for Modulus N 1024 Bit and 4096 Bit
Desi Wulansari desiwulans03@gmail.com
Department of Computer Science
Faculty of Mathematics and Natural Science
Semarang State University
Semarang, 50229, Indonesia
Much Aziz Muslim a212muslim@yahoo.com
Endang Sugiharti endanghs02@yahoo.com
Abstract
Cryptography has several important aspects in supporting the security of the data, which
guarantees confidentiality, integrity and the guarantee of validity (authenticity) data. One of the
public-key cryptography is the RSA cryptography. The greater the size of the modulus n, it will be
increasingly difficult to factor the value of n. But the flaws in the RSA algorithm is the time
required in the decryption process is very long. Theorem used in this research is the Chinese
Remainder Theorem (CRT). The goal is to find out how much time it takes RSA-CRT on the size
of modulus n 1024 bits and 4096 bits to perform encryption and decryption process and its
implementation in Java programming. This implementation is intended as a means of proof of
tests performed and generate a cryptographic system with the name "RSA and RSA-CRT Text
Security". The results of the testing algorithm is RSA-CRT 1024 bits has a speed of
approximately 3 times faster in performing the decryption. In testing the algorithm RSA-CRT 4096
bits, the conclusion that the decryption process is also effective undertaken more rapidly.
However, the flaws in the key generation process and the RSA 4096 bits RSA-CRT is that the
time needed is longer to generate the keys.
Keywords: Encryption and Decryption, RSA, RSA-CRT, 1024 bit and 4096 bit.
1. INTRODUCTION
When this information becomes essential for the people, the ease of access to information are
the main expectations. In addition to convenience, content information obtained is also influential
for most people. Security and confidentiality is an aspect that is important in order to prevent the
collapse of confidential information into the hands of those who are not responsible. A data can
be secured by way of encryption of the content of the data, it is commonly referred to
cryptography [1].
Cryptography is the study of mathematical techniques related to aspects of information security
such as data confidentiality, data integrity, data integrity, and authentication of data. One public-
key cryptography algorithm is the famous RSA cryptography. The RSA algorithm has a
cryptographic key pair that is a public key and a private key.

Mathematical operations on RSA consists of multiplying two prime numbers (p and q) are
selected randomly, the result of the multiplication is n. N The larger the size, the better the level of
security because it would deprive the attacker to break the factorization of the value of n. Then
proceed with the powers and operation of modular operations. Modular calculations on the
division process must leave the quotient (remainder value).
But despite the problems faced by RSA cryptography is secure so far it has been claimed [2], but
if the modulus n bits of its enormous size, the decryption process at the RSA algorithm requires a
long time. Additionally this process requires great complexity. So as to simplify the process of
powers between the two numbers modular required an efficient mathematical calculations.
Some of the popular mathematical calculations to solve this problem one of them is Chinese
Remainder Theorem (CRT), or more commonly referred to RSA-CRT [3]. This algorithm is a
mathematical calculation that his role in the RSA algorithm as modular exponential simplification
process. Modular powers RSA-CRT performed on the value of p and q [4]. In this study, there are
2 pieces the size of modulus n to be used is 1024 bits and 4096 bits. The resulting time of the
encryption and decryption process will be observed to obtain results and draw a conclusion.
2. THE ALGORITHM AND METHOD
In the key generation RSA-CRT decryption exponent (d) is not directly given to the private key but
can be calculated via the parameter dP, dQ and qInv which has a long half-size bits d. While the
private key RSA-CRT is set as Kprivate = (dP, dQ, qInv, p, q). While public key RSA-CRT is
equal to the system RSA algorithm, namely (n, e) so that the encryption algorithm is not change
is by using modular exponential function c = m
e
mod n.
2.1 RSA Algorithm
The RSA algorithm in the algorithm key generation, encryption and decryption are shown as
follows:
1) Choose two prime numbers p and q with p ≠ q.
2) Calculate the value of n = p q and ø(n) = (p - 1) (q - 1).
3) Choose a number e (1<e <ø(n)) with gcd (e, ø(n)) = 1.
4) Calculate the value of d = e
-1
on ℤø(n).
5) The key to the public is (n, e) and the private key is (n, d).
The encryption process using equation (1) as follows:
c = m
e
mod n (1)
Where c is the result of the encryption of the plaintext, plaintext is the original text [5] while e is
relatively prime numbers resulting from calculations ø(n) = (p - 1) (q - 1). m is the plaintext to be
encrypted, while n is the result of n = p q.
And the decryption algorithm of RSA to decrypt the chipertext, chipertext is the data that has
been encoded [6] using equation (2) as follows:
m = c
d
mod n (2)
Where m is the result of decryption, while c is the chipertext. d is the result of the inverse process
e modulus (p - 1) (q - 1).
2.2 RSA-CRT Algorithm
A RSA-CRT algorithm to generate the keys are as follows:
1) Choose two prime numbers p and q with p ≠ q provisions.
2) Calculate the value of n = p q and ø(n) = (p - 1) (q - 1).
3) Choose a number e (1<e <ø(n)) with gcd (e, ø(n)) = 1.
4) Calculate the value of d = e
-1
on ℤø(n).
5) Calculate the results dP = d mod (p - 1).
6) Calculate the results dQ = d mod (q - 1).

7) Calculate the results qInv = q
-1
mod p.
8) Results Kpublic = (e, n) and Kprivate = (dP, dQ, qInv, p, q).
The formula to determine the private key d by using equation (3) as follows:
d e mod ø(n) = 1 (3)
equation (3) is equivalent to the following equation (4):
d ≡ e
-1
(mod ℤø(n)) (4)
Where d is the result of the inverse process e modulus (p - 1) (q - 1), and e is relatively prime
numbers resulting from calculations ø(n) = (p - 1) (q - 1).
Encryption algorithm RSA-CRT serves to convert plaintext into chiperteks. Encryption algorithm
RSA-CRT is equal to the RSA encryption algorithm. The encryption process using the following
algorithm:
1) Input plaintext and Kpublic = (n, e).
2) Calculate the result of the formula encryption c = m e mod n.
Where c is the result of the encryption of the plaintext, while e is relatively prime numbers
resulting from calculations ø(n) = (p - 1) (q - 1). m is the plaintext to be encrypted, while n is the
result of n = p q.
To decrypt chiperteks, using the private key (dP, dQ, qInv, p, q) The RSA-CRT decryption
algorithm to restore chiperteks into plaintext is as follows:
1) Input chiperteks and Kprivate = (dP, dQ, qInv, p, q).
2) Calculate the value x1 = C
dP
mod p.
3) Calculate the value x2 = C
dQ
mod q.
4) Calculate h = qInv (x1 - x2) mod p.
5) The results of M = x2 + h q.
Where M is the result of decrypting encrypted plaintext message after (chiperteks). To find the
result x2, first calculating the results of x1. After that, the search continues in the process of
calculating x2 + h q to find the value of h.
2.3 Testing
Testing was conducted using the data in the form of text with each having a different size. Here is
a table of text data that is used both the character and its size in bytes is shown in Table 4.1.
No. Text Size (byte)
1 Hai 3 byte
2 SEGERA BERKUMPUL NANTI SORE AKAN ADA
RAPAT PENTING
50 byte
3 anGKasAdiRganTara666679 23 byte
4 Rp 95.345.000,- 15 byte
5 Segera Lakukan Re-shuffle Kabinet Kerja! 40 byte
TABLE 1: Text and Data List Size in Bytes.
In accordance with Table 1, the first text data of plaintext is used lower case letters. The second
text data using upper case letters. The third text data using a combination of upper case, lower
case and numbers. The fourth data meggunakan combination of upper case, lower case,
numbers and symbols. While in the fifth the data using a combination of upper case, lower case,
and symbols.

In the first test, the key will be raised by 1024 bits. The results of the private key and public key
obtained at random. The value of e generated by the system randomly in accordance with the
provisions e (1 <e <ø(n)) with gcd (e, ø(n)) = 1 is e = 3. Since 1024 bit key size for the number of
numbers generated long enough on the value of p, q and n. Here below is the result of the RSA
key generation and a 1024-bit RSA-CRT:
p = 10277651387301176987572317325479097395047746036448
09259710908734340250231174881775912743279205016377
33979655095360153800010911836260705198915653142336
10391
Values of p key generated as many as 155 characters in decimal. While following the generation
of the keys on the value of q is:
q = 11477990785403537825829714057028719300154941210754
98780912268468497672034368407332370186424197056338
84617243251249168877722536665242249074891552040518
84923
The value of q raised key has a number no different from p key values as many as 154
characters. The system is processing the steps to find the value of the multiplication of two prime
numbers p and q are prime numbers by multiplying the result of both the n = p q. As a result of
the value of n are as follows:
n = 11796678791903279657847637919373441536852260211807
21756891581313624284029594204420437699481533561502
05907103863786409262454101821911942847169605133844
12454088268637081919279132101575631548968303775379
00396870079715414202890886129151448171128723026655
67529874887655142902239658345808376618266144606443
149034893
While at RSA-CRT key dP, dQ and qInv raised key value shown below:
dP = 68517675915341179917148782169860649300318306909653
95064739391562268334874499211839418288528033442515
59864367302401025333406078908404701326104354282240
6927
dQ = 76519938569356918838864760380191462001032941405033
25206081789789984480229122715549134576161313708925
64114955008327792518150244434948327165943680270125
6615
qInv = 49376036358377184742338381585975903414025413474651
80142927287373201845362801213975002353013594980009
82781233309664942894019440739725040071111488679026
6219
The key value dP, dQ, and qInv raised as many as 155 characters in decimal. The key value is
generated through the key generation process has a number of characters as much as 308
decimal. This allows RSA and RSA-CRT can meet the security because the number of digits p
and q are more than 200 characters so difficult for an attacker to perform factorization of integers
to the variable n. Number of key characters n is the number of key digit number of digits p plus q
key. The total size of each key value p and q is equal to 512 bits.

The next stage is to convert the original message into chiperteks through an encryption process
using the public key (n, e). In the process of testing, there are 5 pieces of text that are used with
each having a different size. This test was conducted to determine the time complexity of the
algorithm RSA and RSA-CRT if the process of encryption and decryption of 1024 bits. The results
displayed time during the testing process are presented in Table 2 below.
TABLE 2: The Results of Comparison Time RSA encryption and RSA-CRT 1024 bits.
Table 2 is a comparison of the speed of RSA encryption and RSA-CRT 1024 bits. Comparison of
the time generated by the RSA encryption algorithm and RSA-CRT 1024 bits is not much
different. As for the results of a comparison between the decryption algorithm RSA and RSA-CRT
1024 bits is shown in Table 3.
TABLE 3: The Results of Comparison Time Decryption RSA and RSA-CRT 1024 bits.
According to Table 3, it can be observed that the RSA algorithm can refund the value of
chiperteks into plaintext (decryption) requires quite a long time. However, after the RSA
algorithm, written by the method of Chinese Remainder Theorem (CRT) produces a fairly short
period of approximately three times faster in performing decryption on chiperteks. The next the
process of testing 4096 bits of RSA cryptography. the tests performed same with previous testing,
which is at RSA 4096 bits of key generated first. As a result of generating RSA keys and RSA-
CRT 4096 bits in full accordance with the above image is presented as follows:
e = 7
p = 23004739437130670717767973787359593061506850243043
25607619686023618167468895402838699644725214215416
13517783225570705477157602704678080850641643354084
48788139925747826925716115259021998013764524545442
50132576754539162984708954775994111863949367749188
98596798892762703721897660754269396787575347317861
72005032900309977955865353849663122559450766116200
19900697594319430653517455044561213715149587144358
12808105760534953251122857223443037295142422728392
77452518994866804326273059852039122733640793942419
51467961267922123771714470172589855471403925099003
99600933703299618940494790244562326436331743244891
88104721410869189
q = 21155458138467361615630020886747232977290579067606
44093817952201383907421669404140461298040861018512
94356397719420120661533112364966103895975470611284
File Encryption of RSA (nano second) Encryption of RSA-CRT (nano second)
1 330,130 371,148
2 316,803 348,592
3 330,131 358,845
4 344,485 375,250
5 351,661 382,426
File Decryption of RSA (nano second) Decryption of RSA-CRT (nano second)
1 103,534,963 29,710,330
2 102,998,756 29,400,697
3 103,437,564 29,477,594
4 103,300,180 29,538,084
5 103,515,483 29,730,836

59555691938605475679143738342023160406595656207125
28688746932060932901087146005724913383679676462914
25111818061111355369782150031417084873696513986408
44192179218027352730330751601408746261807993884875
24967320631383650474726049321139689267017927472741
10537452174438073647055888088907175371475637066399
84238358606270733851231054506236027168917410258644
27791811620232357156852830968051904448345249071423
20636813950865049249173520526338375849746056641291
62456646596400321
The number of characters on the keys p which successfully raised at RSA 2048 bit is 309
decimal, while key of q has a amount key characters as much as 617 decimal numbers too. Each
bit size of the prime numbers p and q primes of 1024 bits. While the key to a successful n
generated by the system is shown below:
n = 48667580214856711939608782982782654039474506145845
74729970275369021184937552979829698643289613989846
83810552175168649301276459103458598106550569350829
48311009766352440629735219005499596542366070318534
80636971426314636748719898530766702838739895624734
48715744017165496189610977062483011525926678939822
72304610282641009699295207757304332305369455979079
39400905726040909340376155455675230054966686745866
92395189350670883360136306662666024712982620043885
38817140835873762553894885582232091236765630370531
09409367448536939942326971709204992241443326968470
24842004195083050919973103956139746085705747721397
44220740017250377516579114375393032367960575734662
40985818086971124687153459889224102603129735966422
11090526509550618506660960501059059488768062423285
12293404637063857288861320321710486620724762865102
49155387283656915866544132912949177207955024183987
90808335434916939030718293314405213286026542653019
99274031454436177644289766421305117169389794695480
59405996132629013419458712153312629847526858358158
24272174866343408814248059867483504208140459395031
78208260352887452739471407601502459892739849796801
55601913500035969339798326846347431188891315946829
06095369125180465493978443533193024155728462316573
003954935742262620738051208609669
While on the RSA-CRT key dP, dQ and qInv are raised key value shown below:
dP = 19718348088969146329515406103451079767005871636894
21949388302305958429259053202433171124050183613213
83015242764774890408992230889724069300549980017786
70389834222069565936328098793447426868941021038950
71542208646747853986893389807994953026242315213590
55940113336653746047340852075088054389350297701024
33147171057408552533598874728282676479529228099600
17057740795130940560157818609623897470128217552306
96692662080458531358105306191522603395836362338622
37816444852742975136805479873176390914549251950645
29829681086790391804326688719362733261203364370574
85372228888542530520424105923910565516855779924193

04089761209316447
dQ = 12088833221981349494645726220998418844166045181489
39482181686972219375669525373794549313166206296293
11060798696811497520876064208552059369128840349305
48317538250631700388082136195441805946626089261214
44964998246891961657764083431842807647816957978808
14349610320635060211304085732238334213540865135090
53824102410301344417331858057947855006747425077071
57124183217933514556986313897794108152581672841566
34592829813964613512603364622232671640843221180799
90993347775011847914989174003563444096524234433511
01595320925847061232487331981743945399054428040813
26078179400494313856670583157907643342712032366452
35689512340800183
qInv = 17051326736583810883461788367598009671919646056446
43952558276557513946796179792294274207504509178165
47966000508916302886307980077711929444707031141893
22283717684812224892576002579728239682870705699779
48012900511259205040902736384145544011139640514949
60945017559503486335527275786234519161944495122863
34422942279408855885048789639567717310566182085612
56070805354303949559081130908069676256821267928508
95815215089426511367951153482283654043014516843543
89848650490862896117503781687512694020741855664799
94390754123303566174870359306737794612248588426900
86378489771115286793762730265134351318346311832925
03935686836335671
The key value of dP, dQ, and qInv is raised as much as 617 characters in decimal. N number of
key characters obtained through the multiplication of prime numbers p and q prime numbers as
many as 1,213 decimal. This key is meets criteria of security because the number of characters
that raised more than 200 decimal. But the key generation process with a size of 4096 bits takes
longer than the key generation process with 1024 bits. The last stage traversed by comparing the
speed of encryption and decryption of the RSA algorithm with RSA-CRT 4096 bits. The results of
the comparison of the encryption process is presented in Table 4.
TABLE 4: The Results of Comparison Time RSA encryption and RSA-CRT 4096 bits
According to the Table 4 can be drawn a response that the encryption process on the RSA
algorithm and RSA-CRT 4096 bits have no significant difference. For the comparison of the
decryption process at RSA and RSA-CRT 4096 bits is shown in Table 5.
File Encryption of RSA (nano second) Encryption of RSA-CRT (nano second)
1 6,623,256 6,643,762
2 6,634,535 6,573,018
3 6,626,332 6,591,474
4 6,660,166 6,659,142
5 6,706,303 6,714,506

TABLE 5: The Results of Comparison Time Decryption RSA and RSA-CRT 4096 bits
The comparison is shown in Table 5 states that after the RSA algorithm is combined with the
Chinese Remainder Theorem (CRT) is more effective to accelerate the process of decryption of a
message, more or less able to speed up the decryption process as much as three times faster
than the use of the RSA algorithm usual, though the size of the bits of used 4 times larger than
1024 are bits.
After the entire the process of testing is done, the next will produce a statement that can be
drawn as a conclusion. The results of the testing of all five text data after the encryption and
decryption process in modulus n 1024 bits and 4096 bits of will be loaded in graphical form. The
following is a graph of the time complexity of the encryption process on text data to the one
shown in Figure 1.
FIGURE 1: The Results of Comparison Time Encryption RSA and RSA-CRT.
Results graph of the time complexity of the encryption process in Figure 1 shows that the
difference between the encryption process on RSA and RSA-CRT did not experience much
change although the modulus n which is used differently. So also with the results of the
encryption process on text data of 2 to 5 text data equally showed no significant differences in
time. Hereafter the time complexity is the result of the decryption process text data to the one
shown in Figure 2.
0
1000000
2000000
3000000
4000000
5000000
6000000
7000000
1024 4096
time(nanosecond)
Modulus (n)
The Result of Encryption RSA vs RSA-CRT on Text 1
RSA
RSA-CRT
File Decryption of RSA (nano second) Decryption of RSA-CRT (nano second)
1 5,853,890,548 1,304,281,228
2 5,842,122,477 1,149,450,813
3 5,906,614,132 1,171,997,568
4 5,944,532,789 1,311,686,767
5 5,945,090,537 1,168,936,106

FIGURE 2: The Results of Comparison Time Decryption RSA and RSA-CRT
Based on the graph in Figure 2, Theorem Chinese Remainder Theorem (CRT) in the RSA is very
effective to accelerate the process of decryption of the message even though the value of the
modulus n which is used differently. Decryption speeds have increased by about 3-fold compared
with the use of RSA. Increased speed is also generated at the data into text data to the 2 to 5.
3. CONCLUSION
The results of the test declared that the correct algorithm RSA-CRT has a rate of approximately 3
times faster in performing the decryption process although using the modulus n of different sizes.
However, shortcomings exist in key generation process of RSA and RSA-CRT 4096 bits because
of the time it takes much longer.
4. REFERENCES
[1] Muslim, M. A., & Prasetiyo, B. “Implementation Twofish Algorithm for Data Security in a
Communication Network using Library Chilkat Encryption Activex.” Journal of Theoretical and
Applied Information Technology, 84 (3), 370. 2016.
[2] Chen, C., Wang, T., & Tian, J. “Improving Timing Attack on RSA-CRT via Error Detection and
Correction Strategy.” Information Sciences, 232, 464-474. 2013.
[3] Lu, Y., Zhang, R., & Lin, D. “New Partial Key Exposure Attacks on CRT-RSA with Large
Public Exponents.” In International Conference on Applied Cryptography and Network
Security (pp. 151-162), Springer International Publishing. June, 2014.
[4] Quisquater, J. J., & Couvreur, C. Fast Decipherment Algorithm for RSA Public-Key
Cryptosystem. Electronics Letters, 18(21), 905-907. 1982.
[5] Abutaha, Mohammed, et al. “Survey Paper: Cryptography is the Science of Information
Security.” International Journal of Computer Science and Security (IJCSS), 5.3: 298. 2011.
[6] Alawadhi, R., & Nair, S. “A Crypto-System with Embedded Error Control for Secure and
Reliable Communication.” International Journal of Computer Science and Security (IJCSS),
7(2), 48. 2013.
0
1E+09
2E+09
3E+09
4E+09
5E+09
6E+09
7E+09
1024 4096
time(nanosecond)
Modulus (n)
The Result of Decryption RSA vs RSA-CRT on Text 1
RSA
RSA-CRT

Implementation of RSA Algorithm with Chinese Remainder Theorem for Modulus N 1024 Bit and 4096 Bit

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