Digital Signature different everytime - cryptography

Question

Digital signatures depend upon message and the secret key. Suppose receiver copies the message several times ( the reason for which we unique ids) , why will the signature be different (since the message and secret key are same)?

Answer 1

Bitcoin uses ECDSA, and the reason the signature is different for the same message and key is the random nonce, which is generated each time a signature is created.
An ECDSA signature is a pair (r,s) where r is the X coordinate of kG, and s = (m+r*x)/k (where k=nonce, m=message hash, x=private key, G=curve generator).
See also #3 in https://en.wikipedia.org/wiki/Elliptic_Curve_Digital_Signature_Algorithm#Signature_generation_algorithm

Answer 2

This answer is just a practical supplement to the theoretical answer provided above. This snippet digitally signs inside the for-loop the same message over and over again using the same private key and each time, it is printing out different digital signatures. Each of these signatures are valid and will verify using the signer's public key.
import java.security.Security;
import java.security.Signature;
import java.security.KeyFactory;
import java.security.KeyPairGenerator;
import java.security.SecureRandom;
import java.security.KeyPairGenerator;
import java.security.KeyPair;
import org.bouncycastle.util.encoders.Hex;
import org.bouncycastle.jce.ECNamedCurveTable;
import org.bouncycastle.jce.spec.ECParameterSpec;
import org.bouncycastle.jce.interfaces.ECPublicKey;
import org.bouncycastle.jce.spec.ECPublicKeySpec;
import org.bouncycastle.jce.provider.BouncyCastleProvider;
class liptic
{
static String the_message = "how are you, little tiger?";
public static void main(String args[]) throws Exception
{
Security.addProvider(new BouncyCastleProvider());
SecureRandom random = new SecureRandom();
ECParameterSpec ecSpec
=
ECNamedCurveTable.getParameterSpec("secp256k1");
KeyPairGenerator g =
KeyPairGenerator.getInstance("ECDH", "BC");
g.initialize(ecSpec, random);
KeyPair keyPair = g.generateKeyPair();
byte[] bob_pubkey = Hex.decode( "040581E4AEEEB1CEA57094D1AD97B8C721509B6E5D3690C70BBB8EB2C5FE8040"
+ "FB2C9B0A77EA2AD05C5E8DB499F647BC9A8BE829961950D6F5A45952C097CCB0BC");
Signature signer = Signature.getInstance("SHA256withECDSA", "BC");
ECPublicKeySpec pubkSpec = new ECPublicKeySpec(
ecSpec.getCurve().decodePoint(bob_pubkey), ecSpec);
ECPublicKey publicKey
= (ECPublicKey)KeyFactory.getInstance("ECDSA", "BC")
.generatePublic(pubkSpec);
byte[] signature = null;
byte[] input = the_message.getBytes();
signer.initSign(keyPair.getPrivate());
signer.update(input);
for (int i = 0; i < 5; i++) {
signature = signer.sign();
System.out.println(o(signature));
}
}
public static String o(byte[] bytes)
{
int n = 0;
StringBuilder sb = new StringBuilder();
for (byte b : bytes) {
sb.append(String.format("%02X", b));
n++;
if (n % 128 == 0) {
sb.append("\n");
}
}
return sb.toString();
}
}
Here are the digital signatures:
3046022100889CC42C4BAA07FF33AB34CADD8BCB0A44E77031D4A5F5A9849840DF3AB63FDA0221009CA5C49FC0EBE9F839A0CFAB18CEC91C9169FF439C1E2DFD724D06E2DB9FE258
30460221009D30465EFD3676982CBE12B998D41D012322C255594D5037F156143AEC7E7305022100A77FE7DEB580837A1A5A5D1B74334C56E9F26BA1834EE3AC93ECEB01349A6F1C
3045022100DF4AF153D808A9199C18C97E689F1214E7F59C621D6ECBAADFE206B83707CA2802203E592D0152E79E14084183206F4B6DBC95D68FBA2D23F65884A3B68FA79A4E04
304502202E9FA22709308D08106F76CBB6278B3F485A3C706EDA3FFAF5CE744D4B90E9510221009DD2370863D6C1CE36D828FF9B98347905F2D0856052C4A30B25DD00575B8921
3045022100AA46FEA1A80498E481D46B17EFD7FBE6656641CD719AF1F5DC0C77ADD334729D0220471472117374E0284074EBC81172E6271CA9D86F54AFCE6E6CF6863814EBF824
This is using the Elliptic Curve Digital Signature Algorithm.