**Cryptography Tutorials - Herong's Tutorial Examples** - v5.42, by Herong Yang

Proof of RSA Public Key Encryption

This section describes steps to prove RSA Public Key Encryption algorithm. Fermat's little theorem is the key part of the proof.

To proof the RSA public key encryption algorithm, we need to proof the following:

Given that: p and q are 2 distinct prime numbers n = p*q m = (p-1)*(q-1) e satisfies 1 > e > n and e and m are coprime numbers d satisfies d*e mod m = 1 M satisfies 0 => M > n C = M**e mod n the following is true: M == C**d mod n

One way to prove the above is to use steps presented by Avi Kak at https://engineering.purdue.edu/kak/compsec/NewLectures/Lecture12.pdf

(a) Prove that "M**(e*d) mod p == M mod p" is true:

M**(e*d) mod p = M**(k1*m+1) mod p # because "d*e mod m = 1" = (M**(k1*m))*M mop p # factoring 1 M out = (M**(k1*(q-1)*(p-1)))*M mod p # because "m = (p-1)*(q-1)" = (M**(k2*(p-1)))*M mod p # set "k2 = k1*(q-1)" = ((M**(p-1) mod p)**k2)*M mod p # rearranging "mod p" If M and p are coprime numbers: M**(e*d) mod p = (1**k2)*M mod p # Fermat's little theorem: # M**(p-1) mod p = 1 # when M and p are coprimes = M mod p # done Else M == k*p must be true # because p is a prime number M mod p == 0 M**(e*d) mod p == 0 M**(e*d) mod p == M mod p # done

(b) Prove that "M**(e*d) mod q == M mod q" is true in the same process as above.

(c) Prove that "M == C**d mod n" is true:

M**(e*d) mod p == M mod p (1) # see proof (a) M**(e*d) - M == k3*p (2) # because of modulus operation M**(e*d) mod q == M mod q (3) # see proof (b) M**(e*d) - M == k4*q (4) # because of modulus operation M**(e*d) - M == k5*p*q (5) # because of (2) and (4) M**(e*d) - M == k5*n (6) # because n = p*q M mod n == M**(e*d) mod n (7) # because of modulus operation M == M**(e*d) mod n (8) # because 0=< M < n M == (M**e mod n)**d mod n (9) # moving "mod n" M == C**d mod n (10) # because C = M**e mod n

See you can see, the key part of the proof process is the "Fermat's little theorem", which says that if p is a prime number, then for any integer a, the number "a**p - a" is an integer multiple of p. See http://en.wikipedia.org/wiki/Fermat%27s_little_theorem for more details.

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

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"