**Cryptography Tutorials - Herong's Tutorial Examples** - v5.40, by Dr. Herong Yang

Illustration of DSA Algorithm with Different k and h

This section provides a tutorial example to illustrate how DSA digital signature algorithm works with same DSA key paramters but different k and h values.

In the third example, I want to use the same DSA key parameters, the same hash value, and a different k value of 7.

The same DSA key parameters are used:

{23,11,4,8} # the public key: {p,q,g,y} {23,11,4,7} # the private key: {p,q,g,x}

The process of generating a digital signature with the same private key {p,q,g,x}={23,11,4,7} and a different k value of 7 can be illustrated as:

h = 3 # the hash value as the message digest k = 7 # selected: 0 < k < q r = 8 # computed: r = (g**k mod p) mod q = (4**7 mod 23) mod 11 i = 8 # computed: k*i mod q = 1: 7*i mod 11 = 1 s = 10 # computed: s = i*(h+r*x) mod q = 8*(3+8*7) mod 11 {8,10} # the digital signature: {r,s}

The process of verifying the digital signature {r,s}={8,10} with the same public key {p,q,g,y}={23,11,4,8} can be illustrated as:

h = 3 # the hash value as the message digest w = 10 # computed: s*w mod q = 1: 10*w mod 11 = 1 u1 = 8 # computed: u1 = h*w mod q = 3*10 mod 11 = 8 u2 = 3 # computed: u2 = r*w mod q = 8*10 mod 11 = 3 v = 8 # computed: v = (((g**u1)*(y**u2)) mod p) mod q # = (((4**8)*(8**3)) mod 23) mod 11 = 8 v == r # verification passed

There is no problem. The digital signature is still good.

Now let's try on a different hash value of h=9 and generate the digital signature with same private key {p,q,g,x}={23,11,4,7}.

h = 9 # the hash value as the message digest k = 7 # selected: 0 < k < q r = 8 # computed: r = (g**k mod p) mod q = (4**7 mod 23) mod 11 i = 8 # computed: k*i mod q = 1: 7*i mod 11 = 1 s = 3 # computed: s = i*(h+r*x) mod q = 8*(9+8*7) mod 11 {8,3} # the digital signature: {r,s}

Here is the verification process with the same public key {p,q,g,y}={23,11,4,8}:

h = 9 # the hash value as the message digest w = 4 # computed: s*w mod q = 1: 3*w mod 11 = 1 u1 = 3 # computed: u1 = h*w mod q = 9*4 mod 11 = 3 u2 = 10 # computed: u2 = r*w mod q = 8*4 mod 11 = 10 v = 8 # computed: v = (((g**u1)*(y**u2)) mod p) mod q # = (((4**3)*(8**10)) mod 23) mod 11 = 8 v == r # verification passed

Looks very good.

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

RSA Implementation using java.math.BigInteger Class

►Introduction of DSA (Digital Signature Algorithm)

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

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"