Proof of DSA Digital Signature Algorithm

This section describes steps to prove DSA digital signature algorithm. Fermat's little theorem is the key part of the proof.

To proof the DSA digital signature algorithm, we need to proof the following:

Given: 
   p is a prime number                     (1)
   q is a prime number                     (2)
   (p-1) mod q = 0                         (3)
   1 < g < p                               (4)
   g = h**((p–1)/q) mod p                (5.1)
   g**q mod p = 1                        (5.2)
   0 < x < q                               (6)
   y = g**x mod p                          (7)
   h                                       (8)
   0 < k < q                               (9)
   r = (g**k mod p) mod q                 (10)
   r > 0                                  (11)
   k*i mod q = 1                          (12)
   s = i*(h+r*x) mod q                    (13)
   s > 0                                  (14)
   s*w mod q = 1                          (15)
   u1 = h*w mod q                         (16)
   u2 = r*w mod q                         (17)
   v = (((g**u1)*(y**u2)) mod p) mod q    (18)
   
prove: 
   v = r

One way to prove this is to use steps presented by William Stallings at http://mercury.webster.edu/aleshunas/COSC%205130/K-DSA.pdf

Fermat's little theorem: 
   h**(p-1) mod p = 1                                       # as (21)

Rewrite expression:
   g**(f*q) mod p 
      = (h**((p–1)/q) mod p)**(f*q) mod p                   # by (5.1)
      = h**(f*q*(p-1)/q) mod p
      = h**(f*(p-1)) mod p
      = (h**(p-1) mod p)**f mod p     
      = 1**f mod p                                          # by (21)
      = 1                                                   # as (22)

Rewrite expression:
   y**u2 mod p 
      = y**(r*w mod q) mod p                                # by (17)
      = (g**x mod p)**(r*w mod q) mod p                     # by (7)
      = g**(x*(r*w mod q)) mod p
      = g**(x*(r*w-f1*q) mod p
      = g**(x*r*w-x*f1*q) mod p
      = g**((x*r*w mod q)+f2*q-x*f1*q) mod p
      = (g**(x*r*w mod q)*(g**(f*q)) mod p
      = (g**(x*r*w mod q)*1) mod p                          # by (22)
      = g**(x*r*w mod q) mod p                              # as (24)


Rewrite expression:
   v = (((g**u1)*(y**u2)) mod p) mod q                      # by (18)
      = (((g**(h*w mod q))*(y**u2)) mod p) mod q            # by (16)
      = (((g**(h*w mod q))*(g**(x*r*w mod q))) mod p) mod q # by (24)
      = ((g**((h*w mod q)+(x*r*w mod q))) mod p) mod q  
      = ((g**(((h*w+x*r*w) mod q)+f*q)) mod p) mod q  
      = (((g**((h*w+x*r*w) mod q))*(g**(f*q))) mod p) mod q 
      = (((g**((h*w+x*r*w) mod q))*1)) mod p) mod q         # by (22)
      = ((g**((h*w+x*r*w) mod q)) mod p) mod q            
      = ((g**(((h+x*r)*w) mod q)) mod p) mod q              # as (25)                

Rewrite expression:
   k*s mod q 
      = k*(i*(h+r*x) mod q) mod q                           # by (13)
      = (k*i mod q)*((h+r*x) mod q) mod q
      = 1*((h+r*x) mod q) mod q                             # by (12)
      = (h+r*x) mod q                                       # as (26)

Apply (26) to (25):
   v = ((g**((k*s*w) mod q)) mod p) mod q         
      = (g**((k mod q)*(s*w mod q) mod q) mod p) mod q 
      = (g**((k mod q)*(1) mod q) mod p) mod q              # by (15)
      = (g**(k mod q) mod p) mod q     
      = (g**(k + f*q) mod p) mod q     
      = (g**k mod p)*(g**(f*q) mod p) mod q     
      = (g**k mod p)*1 mod q                                # by (22)
      = (g**k mod p) mod q
      = r                                                   # by (10)

Done.

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.

It is interesting to see that both RSA and DSA are based on "Fermat's little theorem".

Last update: 2013.

Table of Contents

 About This Book

 Cryptography Terminology

 Cryptography Basic Concepts

 Introduction to AES (Advanced Encryption Standard)

 Introduction to DES Algorithm

 DES Algorithm - Illustrated with Java Programs

 DES Algorithm Java Implementation

 DES Algorithm - Java Implementation in JDK JCE

 DES Encryption Operation Modes

 DES in Stream Cipher 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

 RSA Implementation using java.math.BigInteger Class

Introduction of DSA (Digital Signature Algorithm)

 What Is a Digital Signature?

 What Is DSA (Digital Signature Algorithm)?

 Illustration of DSA Algorithm: p,q=7,3

 Illustration of DSA Algorithm: p,q=23,11

 Illustration of DSA Algorithm with Different k and h

Proof 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

 MD5 Mesasge Digest Algorithm

 SHA1 Mesasge Digest Algorithm

 OpenSSL Introduction and Installation

 OpenSSL Generating and Managing RSA Keys

 OpenSSL Managing Certificates

 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"

 Using Certificates in IE (Internet Explorer)

 Using Certificates in Firefox

 Using Certificates in Google Chrome

 Outdated Tutorials

 References

 PDF Printing Version