Cryptography Tutorials - Herong's Tutorial Examples - v5.42, by Herong Yang
How to Calculate "M**e mod n"
This section discusses the difficulties of calculating 'M**e mod n'. The intermediate result of 'M**e' is too big for most programming languages.
If you are interested to apply the RSA encryption yourself manually, we need to learn how to calculate "M**e mod n" and "C**d mod n", which looks simple, but difficult to carry out.
First let's see how difficult is to calculate "C**d mod n" directly even with smaller numbers like "62**65 mod 133" as we saw in the previous example.
Here is a sample Perl script to calculate "62**65 mod 133":
# PowerModTest.pl # Copyright (c) HerongYang.com. All Rights Reserved. # print("\n"); print("Wrong answer:\n"); $c = 62**65 % 133; print("62**65 % 133 = ".$c."\n"); print("\n"); print("Correct answer:\n"); $c = 62*(((62**4%133)**4%133)**4%133) % 133; print("62*(((62**4%133)**4%133)**4%133) % 133 = ".$c."\n"); exit(0);
If you run it, you will get:
Wrong answer: 62**65 % 133 = 21 Correct answer: 62*(((62**4%133)**4%133)**4%133) % 133 = 6
So, why we are getting the wrong answer, if you use the expression "62**65 % 133" that matches the formula in encryption algorithm directly? It could be integer overflow on the intermediate result. I am not sure.
You can try this in PHP script too:
<?php # PowerModTest.php # Copyright (c) HerongYang.com. All Rights Reserved. # print("\n"); print("Wrong answer:\n"); $c = pow(62,65) % 133; print("pow(62,65) % 133 = ".$c."\n"); print("\n"); print("Correct answer:\n"); $c = 62*(pow(pow(pow(62,4)%133,4)%133,4)%133) % 133; print("62*(pow(pow(pow(62,4)%133,4)%133,4)%133) % 133 = ".$c."\n"); ?>
Here is the PHP script output. The direct expression also gives a wrong answer.
Wrong answer: pow(62,65) % 133 = 0 Correct answer: 62*(pow(pow(pow(62,4)%133,4)%133,4)%133) % 133 = 6
Conclusion, we can not carry out "M**e mod n" as two operations directly, because the intermediate result of "M**e" can be too big to be processed in exponentiation operations in most programming languages.
We need to find a different way to calculate "M**e mod n" correctly and efficiently.
Table of Contents
Introduction to AES (Advanced Encryption Standard)
DES Algorithm - Illustrated with Java Programs
DES Algorithm Java Implementation
DES Algorithm - Java Implementation in JDK JCE
DES Encryption Operation Modes
PHP Implementation of DES - mcrypt
Blowfish - 8-Byte Block Cipher
Secret Key Generation and Management
Cipher - Secret Key Encryption and Decryption
►Introduction of RSA Algorithm
What Is Public Key Encryption?
RSA Public Key Encryption Algorithm
Illustration of RSA Algorithm: p,q=5,7
Illustration of RSA Algorithm: p,q=7,19
Proof of RSA Public Key Encryption
►How to Calculate "M**e mod n"
Efficient RSA Encryption and Decryption Operations
Proof of RSA Encryption Operation Algorithm
RSA Implementation using java.math.BigInteger Class
Introduction of DSA (Digital Signature Algorithm)
Java Default Implementation of DSA
Private key and Public Key Pair Generation
PKCS#8/X.509 Private/Public Encoding Standards
Cipher - Public Key Encryption and Decryption
OpenSSL Introduction and Installation
OpenSSL Generating and Managing RSA Keys
OpenSSL Generating and Signing CSR
OpenSSL Validating Certificate Path
"keytool" and "keystore" from JDK
"OpenSSL" Signing CSR Generated by "keytool"
Migrating Keys from "keystore" to "OpenSSL" Key Files
Certificate X.509 Standard and DER/PEM Formats
Migrating Keys from "OpenSSL" Key Files to "keystore"