RSA Encryption Algorithm - Revision history

Holtzmt: /* See Also */

2009-04-12T18:53:34Z

Holtzmt: /* Example */

2009-04-12T18:48:22Z

Example

← Older revision		Revision as of 18:48, 12 April 2009
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	6. Public Key is (k,e), Private Key is (k,d), Public Key = (51, 3), Private Key = (51, 11)		6. Public Key is (k,e), Private Key is (k,d), Public Key = (51, 3), Private Key = (51, 11)

-	7. Encrypt message using Private Key with ''c = (m^e(mod k)'' . Assume a message m=22 cipher text ''c = ((m^e)mod k) = 22^3 mod 51 = 10648 mod 51 = 40 ''	+	7. Encrypt message using Private Key with ''c = ((m^e)mod k)'' . Assume a message m=22 cipher text ''c = ((m^e)mod k) = 22^3 mod 51 = 10648 mod 51 = 40 ''

-	8. Decrypt cipher text using Private Key with ''m' = ((c^d) mod k)''. Decrypted message m’ = 40^11 mod 51 = 22	+	8. Decrypt cipher text using Private Key with ''m' = ((c^d) mod k)''. Decrypted message m’ = ((40^11) mod 51) = 22

	==Digital Signing==		==Digital Signing==

Holtzmt: /* Example */

2009-04-12T18:47:38Z

Example

← Older revision		Revision as of 18:47, 12 April 2009
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	Choose ''e'' = 3		Choose ''e'' = 3

-	5. Compute d such that 5d = 1 mod 32, d = 11 such that 311 = 1 mod 32	+	5. Compute d such that 3d = 1 mod 32, d = 11 such that 311 = 1 mod 32

-	6. Public Key is (T(k),e), Private Key is (T(k),d), Public Key = (51, 3), Private Key = (51, 11)	+	6. Public Key is (k,e), Private Key is (k,d), Public Key = (51, 3), Private Key = (51, 11)

-	7. Encrypt message using Private Key with ''c = (m^e(mod T(k)))'' . Assume a message m=22 cipher text ''c = ((m^e)mod T(k)) = 22^3 mod 51 = 10648 mod 51 = 40 ''	+	7. Encrypt message using Private Key with ''c = (m^e(mod k)'' . Assume a message m=22 cipher text ''c = ((m^e)mod k) = 22^3 mod 51 = 10648 mod 51 = 40 ''

-	8. Decrypt cipher text using Private Key with ''m' = ((c^d) mod T(k))''. Decrypted message m’ = 40^11 mod 51 = 22	+	8. Decrypt cipher text using Private Key with ''m' = ((c^d) mod k)''. Decrypted message m’ = 40^11 mod 51 = 22

	==Digital Signing==		==Digital Signing==

Holtzmt: /* See Also */

2009-04-12T18:46:06Z

Holtzmt: /* Example */

2009-04-12T18:40:09Z

Example

← Older revision		Revision as of 18:40, 12 April 2009
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	Choose ''e'' = 3		Choose ''e'' = 3

-	5. ~~compute~~ d such that 5d = 1 mod 32, d = 11 such that 311 = 1 mod 32	+	5. Compute d such that 5d = 1 mod 32, d = 11 such that 311 = 1 mod 32

-	6. ~~public key~~ is (T(k),e), ~~private key~~ is (T(k),d), Public Key = (51, 3), Private Key = (51, 11)	+	6. Public Key is (T(k),e), Private Key is (T(k),d), Public Key = (51, 3), Private Key = (51, 11)

-	7. Encrypt message using Private Key with ''c = (m^e(mod k))'' . Assume a message m=22 cipher text ''c = ((m^e)mod k) = 22^3 mod 51 = 10648 mod 51 = 40 ''	+	7. Encrypt message using Private Key with ''c = (m^e(mod T(k)))'' . Assume a message m=22 cipher text ''c = ((m^e)mod T(k)) = 22^3 mod 51 = 10648 mod 51 = 40 ''

-	8. Decrypt cipher text using Private Key with ''m' = ((c^d) mod k)''. Decrypted message m’ = 40^11 mod 51 = 22	+	8. Decrypt cipher text using Private Key with ''m' = ((c^d) mod T(k))''. Decrypted message m’ = 40^11 mod 51 = 22

	==Digital Signing==		==Digital Signing==

Holtzmt: /* Example */

2009-04-12T18:38:51Z

Example

← Older revision		Revision as of 18:38, 12 April 2009
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	6. public key is (T(k),e), private key is (T(k),d), Public Key = (51, 3), Private Key = (51, 11)		6. public key is (T(k),e), private key is (T(k),d), Public Key = (51, 3), Private Key = (51, 11)

-	7. Encrypt message using Private Key with ''c = (m^e)(mod k)'' . Assume a message m=22 cipher text ''c = ((m^e)mod k) = 22^3 mod 51 = 10648 mod 51 = 40 ''	+	7. Encrypt message using Private Key with ''c = (m^e(mod k))'' . Assume a message m=22 cipher text ''c = ((m^e)mod k) = 22^3 mod 51 = 10648 mod 51 = 40 ''

	8. Decrypt cipher text using Private Key with ''m' = ((c^d) mod k)''. Decrypted message m’ = 40^11 mod 51 = 22		8. Decrypt cipher text using Private Key with ''m' = ((c^d) mod k)''. Decrypted message m’ = 40^11 mod 51 = 22

Holtzmt: /* Example */

2009-04-12T18:38:35Z

Example

← Older revision		Revision as of 18:38, 12 April 2009
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	1. choose 2 prime numbers. For examples sake i=17, j=3		1. choose 2 prime numbers. For examples sake i=17, j=3

-	2. compute k=ij	+	2. compute k=ij, k = 17*3 = 51

-	~~k = 17*3 = 51~~	+	3. compute T(k)=(i-1)(j-1), T(k) = (17-1)(3-1) = 10*4 = 32
-		+
-	3. compute T(k)=(i-1)*(j-1)	+
-		+
-	T(k) = (17-1)(3-1) = 104 = 32	+

	4. Choose an integer ''e'' such that ''1 < e < T(k)'', and ''e'' and ''T(k)'' are [http://en.wikipedia.org/wiki/Coprime coprime].		4. Choose an integer ''e'' such that ''1 < e < T(k)'', and ''e'' and ''T(k)'' are [http://en.wikipedia.org/wiki/Coprime coprime].
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	Choose ''e'' = 3		Choose ''e'' = 3

-	5. compute d such that 5*d = 1 mod 32	+	5. compute d such that 5d = 1 mod 32, d = 11 such that 311 = 1 mod 32
-		+
-	d = 11 such that 3*11 = 1 mod 32	+
-		+
-	~~6. public key is (T(k),e), private key is (T(k),d)~~	+
-		+
-	~~Public Key = (51, 3), Private Key = (51, 11)~~	+
-		+
-	~~7. Encrypt message using Private Key with ''c = (m^e)(mod k)''~~	+

-	~~Assume a message m=22 cipher text ''c =~~ ((m^e)~~mod~~ k) = ~~22^3 mod~~ 51 = ~~10648 mod~~ 51 ~~= 40 ''~~	+	6. public key is (T(k),e), private key is (T(k),d), Public Key = (51, 3), Private Key = (51, 11)

-	8. ~~Decrypt cipher text~~ using Private Key with ''m' = ((c^d) mod k)''	+	7. Encrypt message using Private Key with ''c = (m^e)(mod k)'' . Assume a message m=22 cipher text ''c = ((m^e)mod k) = 22^3 mod 51 = 10648 mod 51 = 40 ''

-	Decrypted message m’ = 40^11 mod 51 = 22	+	8. Decrypt cipher text using Private Key with ''m' = ((c^d) mod k)''. Decrypted message m’ = 40^11 mod 51 = 22

	==Digital Signing==		==Digital Signing==

Holtzmt: /* Example */

2009-04-12T18:37:37Z

Example

← Older revision		Revision as of 18:37, 12 April 2009
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	==Example==		==Example==

-	1. choose 2 prime numbers. For examples sake i=11, j=5	+	1. choose 2 prime numbers. For examples sake i=17, j=3

	2. compute k=ij		2. compute k=ij

-	k = 11*5 = 55	+	k = 17*3 = 51

	3. compute T(k)=(i-1)*(j-1)		3. compute T(k)=(i-1)*(j-1)

-	T(k) = (11-1)(5-1) = 104 = 40	+	T(k) = (17-1)(3-1) = 104 = 32

	4. Choose an integer ''e'' such that ''1 < e < T(k)'', and ''e'' and ''T(k)'' are [http://en.wikipedia.org/wiki/Coprime coprime].		4. Choose an integer ''e'' such that ''1 < e < T(k)'', and ''e'' and ''T(k)'' are [http://en.wikipedia.org/wiki/Coprime coprime].
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	Choose ''e'' = 3		Choose ''e'' = 3

-	5. compute d such that 3*d = 1 mod 40	+	5. compute d such that 5*d = 1 mod 32

-	d = 13 such that 3*13 = 1 mod 40	+	d = 11 such that 3*11 = 1 mod 32

-	6. public key is (k,e), private key is (k,d)	+	6. public key is (T(k),e), private key is (T(k),d)

-	~~Private~~ Key = (50, 3), SK= (50, 13)	+	Public Key = (51, 3), Private Key = (51, 11)

	7. Encrypt message using Private Key with ''c = (m^e)(mod k)''		7. Encrypt message using Private Key with ''c = (m^e)(mod k)''

-	Assume a message M=10 cipher text ''''c = (m^e)(mod k)'' = 10^3 mod 50 = ~~100~~ mod 33 =13	+	Assume a message m=22 cipher text ''c = ((m^e)mod k) = 22^3 mod 51 = 10648 mod 51 = 40 ''
-		+
-	~~8. Decrypt cipher text using SK with M~~' ~~= C^D mod N:~~	+
-		+
-	~~Decrypted message M’ = C^D mod N = 13^7 mod 33 = 7~~	+

		+	8. Decrypt cipher text using Private Key with ''m' = ((c^d) mod k)''

		+	Decrypted message m’ = 40^11 mod 51 = 22

	==Digital Signing==		==Digital Signing==

Holtzmt at 18:23, 12 April 2009

2009-04-12T18:23:01Z

← Older revision		Revision as of 18:23, 12 April 2009
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	4. Choose an integer ''e'' such that ''1 < e < T(k)'', and ''e'' and ''T(k)'' are [http://en.wikipedia.org/wiki/Coprime coprime]. ''e'' is the Public Key Exponent.		4. Choose an integer ''e'' such that ''1 < e < T(k)'', and ''e'' and ''T(k)'' are [http://en.wikipedia.org/wiki/Coprime coprime]. ''e'' is the Public Key Exponent.

-	5. Compute ''d'' such that ''ed = 1 mod(T(k))''. In other words ''ed mod (T(k)) = 0''. ''d'' is the Private Key Exponent.	+	5. Compute ''d'' such that ''ed = 1 mod(T(k))''. In other words ''ed mod (T(k)) = 1''. ''d'' is the Private Key Exponent.

	The two keys are a tuple. The public key consists of the Public Key Exponent, and the Modulous, (e,k). The Private Key consists of the Private Key Exponent, and again, the Modulous, (d,k).		The two keys are a tuple. The public key consists of the Public Key Exponent, and the Modulous, (e,k). The Private Key consists of the Private Key Exponent, and again, the Modulous, (d,k).
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	Person B wants to decrypt a message send from Person A. Person B uses their own private key (d,k) to decrypt the message in the following way. m = (c^d)(mod k)		Person B wants to decrypt a message send from Person A. Person B uses their own private key (d,k) to decrypt the message in the following way. m = (c^d)(mod k)
		+
		+	==Example==
		+
		+	1. choose 2 prime numbers. For examples sake i=11, j=5
		+
		+	2. compute k=ij
		+
		+	k = 11*5 = 55
		+
		+	3. compute T(k)=(i-1)*(j-1)
		+
		+	T(k) = (11-1)(5-1) = 104 = 40
		+
		+	4. Choose an integer ''e'' such that ''1 < e < T(k)'', and ''e'' and ''T(k)'' are [http://en.wikipedia.org/wiki/Coprime coprime].
		+
		+	Choose ''e'' = 3
		+
		+	5. compute d such that 3*d = 1 mod 40
		+
		+	d = 13 such that 3*13 = 1 mod 40
		+
		+	6. public key is (k,e), private key is (k,d)
		+
		+	Private Key = (50, 3), SK= (50, 13)
		+
		+	7. Encrypt message using Private Key with ''c = (m^e)(mod k)''
		+
		+	Assume a message M=10 cipher text ''''c = (m^e)(mod k)'' = 10^3 mod 50 = 100 mod 33 =13
		+
		+	8. Decrypt cipher text using SK with M' = C^D mod N:
		+
		+	Decrypted message M’ = C^D mod N = 13^7 mod 33 = 7
		+

Holtzmt at 17:16, 12 April 2009

2009-04-12T17:16:14Z

← Older revision		Revision as of 17:16, 12 April 2009
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	==See Also==		==See Also==
	*http://imps.mcmaster.ca/courses/SE-4C03-09/project/wiki-pages.html		*http://imps.mcmaster.ca/courses/SE-4C03-09/project/wiki-pages.html
		+	--[[User:Holtzmt\|Holtzmt]] 13:16, 12 April 2009 (EDT)

← Older revision		Revision as of 18:53, 12 April 2009
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	*http://www.cas.mcmaster.ca/wiki/index.php/Steganography_and_Digital_Watermarking		*http://www.cas.mcmaster.ca/wiki/index.php/Steganography_and_Digital_Watermarking
	*http://www.cas.mcmaster.ca/wiki/index.php/Data_Encryption_for_Storage_Devices		*http://www.cas.mcmaster.ca/wiki/index.php/Data_Encryption_for_Storage_Devices
-		+	--[[User:Holtzmt\|Holtzmt]] 14:53, 12 April 2009 (EDT)
-	--[[User:Holtzmt\|Holtzmt]] 14:46, 12 April 2009 (EDT)	+

← Older revision		Revision as of 18:46, 12 April 2009
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	==See Also==		==See Also==
	*http://imps.mcmaster.ca/courses/SE-4C03-09/project/wiki-pages.html		*http://imps.mcmaster.ca/courses/SE-4C03-09/project/wiki-pages.html
-	--[[User:Holtzmt\|Holtzmt]] 13:16, 12 April 2009 (EDT)	+	*http://www.cas.mcmaster.ca/wiki/index.php/Public_Key_Encryption_Algorithms
		+	*http://www.cas.mcmaster.ca/wiki/index.php/Public_Key_Authentication
		+	*http://www.cas.mcmaster.ca/wiki/index.php/Steganography_and_Digital_Watermarking
		+	*http://www.cas.mcmaster.ca/wiki/index.php/Data_Encryption_for_Storage_Devices
		+
		+	--[[User:Holtzmt\|Holtzmt]] 14:46, 12 April 2009 (EDT)