FIT5034: RSA and ECC design and implementation
A Simple Implementation of RSA and ECC
Cryptosystem
Xiaoyu Qin (21637881)
xyqin1@student.monash.edu
1. Introduction
RSA is a widely used encryption method and also a powerful digital signature algorithm and
however, ECC can more or less replace it in some places. Especially, shorter key size is used in ECC
comparing to RSA with a same level of security. To learn the algorithm of RSA and ECC and also to
compare them, they are implemented in a simple way and the implementation is discussed in
this report.
2. Thinking of cryptosystems
Both RSA and ECC are asymmetric cryptosystems, which support encryption, decryption, digital
signature and also Diffie-Hellman key exchange [1,2]. They support Diffie-Hellman key exchange
in a similar way, but do other tasks differently [1,2]. In ECC, the public key or the public key with
the pre-shared information is much more complex than the private key [1,2] and the signature
and verification algorithms are also totally different [1,3], while RSA always does the two tasks in
the same couple with a same algorithm and just opponent parameters. Plus, to do the encryption,
RSA just need the public key but ECC need the message receiver’s public key and also a private
key form the message sender [1,2]. Actually, what is done with ECC is actually to add the
symmetric session key in ECC Diffie-Hellman key exchange on the message [2], which can actually
also be done with RSA. However, the way to do encryption and decryption in RSA is easier than
those in ECC, which just need the public key or the private key of the message receiver [1,2]. In
one sentence, RSA and ECC do the other tasks rather than the Diffie-Hellman key exchange in
really different way.
3. Software design and engineering
Java is the programming language chosen to implement RSA and ECC. As the reason discussed
before, it is hard to really implement a unified interface of the two methods. However, there are
some algorithms and basic terms which can be reused. There are the UML class diagrams which
describe the design of the implementation appended on this report. Because some details are
not quite important, there might be some parts not shown in the diagrams.
In ECC, the algorithm can be described in some objects and thus they should be repackaged into
messages which can be used to describe the messages and to be stored or transferred. In RSA,
there are just numbers, so the design is much simpler than ECC.
There are some algorithms are implemented into math functions, such as prime number
generation and prime number generation, which are put into a stand along package. Similarly,
some basic functions are also put there because they are not just related to the cryptosystems
though they are not reused in this program, such as a message loader and the UI class.
Briefly, the programming process is from both the bottom and the top to the middle. Firstly, the
empty class and interfaces as a model is designed and also the math functions are realized. Then,
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FIT5034: RSA and ECC design and implementation
ECC and RSA are really implemented one by one and there were test after each of them are
finished. Especially, when implementing ECC, modelling the algorithm is done before filling the
algorithm into the empty classes.
4. Algorithms
Here are the algorithms used in the program. Most of them have actually more than one choice.
Function
Random
Number
Generation
Prime Number
Test
Greatest
Common
Divisor(GCD)
Position
Algorithm Reference
my.math.MTBigRandom
Mersenne
Twister
Wikipedia[4]
my.math.PrimeEngine
Miller
Rabin
Lecture
Note[1]
my.math.BigIntegerHelper
Euclid
Lecture
Algorithm Note[1]
Extend
Mod Inverse my.math.BigIntegerHelper
Euclid
Wikipedia[5]
Algorithm
ECC
The one
Imbedding
my.cryptosystem.asymmetric.ecc.ECCPlainText need q mod Lecture Note
message
4==3
Note: Mersenne Twister is widespread used to support security applications and also easy to be
implemented [4]. To generate e and d in RSA, Euclid Algorithm and Extend Euclid Algorithm can
be also replaced by Chinese Remainder Theorem, which is used in PGP. In this program, Extend
Euclid Algorithm is not used in ECC and the original modInverse method of BigInteger class is
used. There are also many ways to imbed messages into ECC points. Because the one suggested
by Dr. Phu Le is easy, it is used. And therefore, the q is generated to satisfy “q mod 4=3” [2].
According to some pagers about ECC, there are also some other types of equation for the curve
and the one chosen for this program, “y2=x3+ax+b” is a most simple one [6,7].
5. Test and result analysis
We tested the time length (in milliseconds) of different common tasks to compare the different
algorithms and the different key size.
Task
RSA Keygen
RSA Keygen
ECC Keygen
ECC Keygen
RSA
Encryption
RSA
Encryption
RSA
Decryption
Parameters
keysize=1024
keysize=512
Curve=P-521
Curve=P-192
time 1 time 2 time 3 time 4
7642
3861
12113
13708
34187
16533
4503
4122
316
267
275
272
47
45
50
51
keysize=1024,blocksize=64
224
210
223
219
keysize=512,blocksize=64
202
217
219
238
keysize=1024,blocksize=64
220
219
218
212
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FIT5034: RSA and ECC design and implementation
RSA
keysize=512,blocksize=64
197
197
198
214
Decryption
ECC
Curve=P-521,blocksize=22
3720
3715
3655
3645
Encryption
ECC
Curve=P-192,blocksize=22
329
359
360
358
Encryption
ECC
Curve=P-521,blocksize=22
3461
3488
3492
3520
Decryption
ECC
Curve=P-192,blocksize=22
310
313
344
343
Decryption
RSA
keysize=1024,Hash=SHA1
75
54
57
50
Signature
RSA
keysize=512,Hash=SHA1
244
67
71
71
Signature
RSA
keysize=1024,Hash=SHA1
61
53
58
57
Verification
RSA
keysize=512,Hash=SHA1
56
48
51
53
Verification
ECC
Curve=P-521,Hash=SHA1
269
228
219
223
Signature
ECC
Curve=P-192,Hash=SHA1
152
85
80
89
Signature
ECC
Curve=P-521,Hash=SHA1
409
404
414
412
Verification
ECC
Curve=P-192,Hash=SHA1
109
58
64
61
Verification
Obviously, in this program, RSA is mostly faster than ECC. But, generating RSA key is nightmare.
Also, we know that RSA uses longer key size than ECC with a same level of security, which means
that cost more on storage and transference. Thus, it is hard to say which is better than the other
one.
Actually, we also tested generating the curves in ECC. It takes too long to count the order. That is
why mostly the two pre-stored curve suggested by NIST is used in our program and using a
recommended curve is really necessary [7].
6. Conclusion
As a conclusion of this report, it is obvious that ECC is not bad but really a little bit more complex
than RSA and RSA can be still used because ECC is not always better than it. However, our work is
far from perfect. Some algorithms and also the software design are able to be optimized and also
more testing data can be collected, which should be the future work because of the limitation of
time.
7. References
[1]
Design and Implementation of RSA, Dr. Phu Le, .
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FIT5034: RSA and ECC design and implementation
[2]
Introduction to ECC and Implementation Consideration, Dr. Phu Le, .
[3]
ECC Digital Signature Algorithm and example, Dr. Phu Le, .
[4]
"Mersenne twister," WikiPedia.
[5]
"Extended Euclidean algorithm," WikiPedia.
[6]
N. Koblitz, "A course in number theory and cryptography," Discrete Applied Mathematics,
vol. 26, 1990.
[7]
NIST, "Recommended Elliptic Curves For Federal Government Use," 1999.
8. Appendix
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FIT5034: RSA and ECC design and implementation
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FIT5034: RSA and ECC design and implementation
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