Cryptography in Information Security - Revision history

Katmehm at 03:51, 12 April 2009

2009-04-12T03:51:58Z

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-	[[Image:public_key_cryptography_and_pgp.jpg\|Cryptography is Our Security\|right\|'''~~Description~~''']]	+	[[Image:public_key_cryptography_and_pgp.jpg\|thumb\|Cryptography is Our Security\|right\|'''Cryptography is Our Security''']]
	===Introduction===		===Introduction===
	The word [http://en.wikipedia.org/wiki/Cryptography cryptography] comes from two Greek words meaning "secret writing" and is the art and science of concealing meaning. [http://en.wikipedia.org/wiki/Cryptanalyst Cryptanalysis] is the breaking of codes. The basic component of cryptography is a		The word [http://en.wikipedia.org/wiki/Cryptography cryptography] comes from two Greek words meaning "secret writing" and is the art and science of concealing meaning. [http://en.wikipedia.org/wiki/Cryptanalyst Cryptanalysis] is the breaking of codes. The basic component of cryptography is a
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	==Cæsar cipher==		==Cæsar cipher==
	The action of a [http://en.wikipedia.org/wiki/Caesar Caesar] cipher is to replace each plaintext letter with one a fixed number of places down the alphabet. This example is with a shift of three, so that a B in the plaintext becomes E in the ciphertext		The action of a [http://en.wikipedia.org/wiki/Caesar Caesar] cipher is to replace each plaintext letter with one a fixed number of places down the alphabet. This example is with a shift of three, so that a B in the plaintext becomes E in the ciphertext
-	[[Image:Images.jpg\|right\|'''Caesar cipher machine''']]	+	[[Image:Images.jpg\|thumb\|Casear Cipher Machine\|right\|'''Caesar cipher machine''']]
	* '''EXAMPLE''':		* '''EXAMPLE''':
	The Caesar cipher is the widely known cipher in which letters are shifted. For example, if the key is 3, the letter A becomes D, B becomes E, and so forth, ending with Z becoming C. So the word "HELLO" is enciphered as "KHOOR." Informally, this cipher is a cryptosystem with:		The Caesar cipher is the widely known cipher in which letters are shifted. For example, if the key is 3, the letter A becomes D, B becomes E, and so forth, ending with Z becoming C. So the word "HELLO" is enciphered as "KHOOR." Informally, this cipher is a cryptosystem with:
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	==Vigènere cipher==		==Vigènere cipher==
	A longer key might obscure the statistics. The Vigenère cipher chooses a sequence of keys, represented by a string. The key letters are applied to successive plaintext characters, and when the end of the key is reached, the key starts over. The length of the key is called the [http://en.wikipedia.org/wiki/Period_of_the_cipher period of the cipher]. Because this requires several different key letters, this type of cipher is called [http://en.wikipedia.org/wiki/Polyalphabetic polyalphabetic].		A longer key might obscure the statistics. The Vigenère cipher chooses a sequence of keys, represented by a string. The key letters are applied to successive plaintext characters, and when the end of the key is reached, the key starts over. The length of the key is called the [http://en.wikipedia.org/wiki/Period_of_the_cipher period of the cipher]. Because this requires several different key letters, this type of cipher is called [http://en.wikipedia.org/wiki/Polyalphabetic polyalphabetic].
-	[[Image:Vigenere.png\|Vigenere Square Table\|180px\|right\|''' Vigenere square ''']]	+	[[Image:Vigenere.png\|thumb\|Vigenere Square Table\|180px\|right\|''' Vigenere square ''']]

	*'''EXAMPLE''': The first line of a limerick is enciphered using the key "BENCH," as follows:<br>		*'''EXAMPLE''': The first line of a limerick is enciphered using the key "BENCH," as follows:<br>

Katmehm at 03:48, 12 April 2009

2009-04-12T03:48:43Z

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-	[[Image:public_key_cryptography_and_pgp.jpg\|right\|'''Description''']]	+	[[Image:public_key_cryptography_and_pgp.jpg\|Cryptography is Our Security\|right\|'''Description''']]
	===Introduction===		===Introduction===
	The word [http://en.wikipedia.org/wiki/Cryptography cryptography] comes from two Greek words meaning "secret writing" and is the art and science of concealing meaning. [http://en.wikipedia.org/wiki/Cryptanalyst Cryptanalysis] is the breaking of codes. The basic component of cryptography is a		The word [http://en.wikipedia.org/wiki/Cryptography cryptography] comes from two Greek words meaning "secret writing" and is the art and science of concealing meaning. [http://en.wikipedia.org/wiki/Cryptanalyst Cryptanalysis] is the breaking of codes. The basic component of cryptography is a
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	==Vigènere cipher==		==Vigènere cipher==
	A longer key might obscure the statistics. The Vigenère cipher chooses a sequence of keys, represented by a string. The key letters are applied to successive plaintext characters, and when the end of the key is reached, the key starts over. The length of the key is called the [http://en.wikipedia.org/wiki/Period_of_the_cipher period of the cipher]. Because this requires several different key letters, this type of cipher is called [http://en.wikipedia.org/wiki/Polyalphabetic polyalphabetic].		A longer key might obscure the statistics. The Vigenère cipher chooses a sequence of keys, represented by a string. The key letters are applied to successive plaintext characters, and when the end of the key is reached, the key starts over. The length of the key is called the [http://en.wikipedia.org/wiki/Period_of_the_cipher period of the cipher]. Because this requires several different key letters, this type of cipher is called [http://en.wikipedia.org/wiki/Polyalphabetic polyalphabetic].
-	[[Image:Vigenere.png\|180px\|right\|''' Vigenere square '''~~Vigenere Square Table~~]]	+	[[Image:Vigenere.png\|Vigenere Square Table\|180px\|right\|''' Vigenere square ''']]

	*'''EXAMPLE''': The first line of a limerick is enciphered using the key "BENCH," as follows:<br>		*'''EXAMPLE''': The first line of a limerick is enciphered using the key "BENCH," as follows:<br>

Katmehm at 03:45, 12 April 2009

2009-04-12T03:45:57Z

← Older revision		Revision as of 03:45, 12 April 2009
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	==Vigènere cipher==		==Vigènere cipher==
	A longer key might obscure the statistics. The Vigenère cipher chooses a sequence of keys, represented by a string. The key letters are applied to successive plaintext characters, and when the end of the key is reached, the key starts over. The length of the key is called the [http://en.wikipedia.org/wiki/Period_of_the_cipher period of the cipher]. Because this requires several different key letters, this type of cipher is called [http://en.wikipedia.org/wiki/Polyalphabetic polyalphabetic].		A longer key might obscure the statistics. The Vigenère cipher chooses a sequence of keys, represented by a string. The key letters are applied to successive plaintext characters, and when the end of the key is reached, the key starts over. The length of the key is called the [http://en.wikipedia.org/wiki/Period_of_the_cipher period of the cipher]. Because this requires several different key letters, this type of cipher is called [http://en.wikipedia.org/wiki/Polyalphabetic polyalphabetic].
-	[[Image:Vigenere.png\|180px\|right\|''' Vigenere square ''']]	+	[[Image:Vigenere.png\|180px\|right\|''' Vigenere square '''Vigenere Square Table]]

	*'''EXAMPLE''': The first line of a limerick is enciphered using the key "BENCH," as follows:<br>		*'''EXAMPLE''': The first line of a limerick is enciphered using the key "BENCH," as follows:<br>

Katmehm at 03:43, 12 April 2009

2009-04-12T03:43:36Z

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	*[http://msdn2.microsoft.com/en-us/library/ms998382.aspx patterns & practices Security Engineering Explained]		*[http://msdn2.microsoft.com/en-us/library/ms998382.aspx patterns & practices Security Engineering Explained]
	*[http://www.opensecurityarchitecture.org Open Security Architecture- Controls and patterns to secure IT systems]		*[http://www.opensecurityarchitecture.org Open Security Architecture- Controls and patterns to secure IT systems]
		+
		+	--[[User:Katmehm\|Katmehm]] 23:43, 11 April 2009 (EDT)

Katmehm at 03:41, 12 April 2009

2009-04-12T03:41:57Z

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	=References=		=References=

-	1. ~~''Onion Routing for Anonymous Communications.''~~ [http://en.~~wikipedia~~.~~org~~/~~wiki~~/~~Cryptography~~]	+	<div class="references-small">
-	<br>	+	* [[Ross Anderson\|Ross J. Anderson]]: <cite>[http://www.cl.cam.ac.uk/~rja14/book.html Security Engineering: A Guide to Building Dependable Distributed Systems]</cite>, ISBN 0-471-38922-6
-	[http://en.~~wikipedia~~.~~org~~/~~wiki~~/~~Caesar_cipher~~]	+	* [[Morrie Gasser]]: [http://cs.unomaha.edu/~stanw/gasserbook.pdf <cite>Building a secure computer system</cite>] ISBN 0-442-23022-2 1988
-	[http://en.~~wikipedia~~.~~org~~/~~wiki~~/~~Encrypt~~]	+	* [[Stephen Haag]], [[Maeve Cummings]], [[Donald McCubbrey]], [[Alain Pinsonneault]], [[Richard Donovan]]: <cite>Management Information Systems for the information age</cite>, ISBN 0-07-091120-7
-	[http://en.~~wikipedia~~.~~org~~/~~wiki~~/~~Plaintext~~]	+	* [[E. Stewart Lee]]: [http://www.cl.cam.ac.uk/~mgk25/lee-essays.pdf <cite>Essays about Computer Security</cite>] Cambridge, 1999
-	[http://en.~~wikipedia~~.org/~~wiki~~/~~Cipher~~]	+	* [[Peter G. Neumann]]: [http://www.csl.sri.com/neumann/chats4.pdf <cite>Principled Assuredly Trustworthy Composable Architectures</cite>] 2004
-	[~~http~~:/~~/en~~.~~wikipedia~~.~~org~~/~~wiki~~/~~Julius_Caesar]~~	+	* [[Paul A. Karger]], [[Roger R. Schell]]: [http://www.acsac.org/2002/papers/classic-multics.pdf<cite>Thirty Years Later: Lessons from the Multics Security Evaluation</cite>], IBM white paper.
		+	* [[Robert C. Seacord]]: <cite>Secure Coding in C and C++</cite>. Addison Wesley, September, 2005. ISBN 0-321-33572-4
		+	* [[Clifford Stoll]]: <cite>Cuckoo's Egg: Tracking a Spy Through the Maze of Computer Espionage</cite>, Pocket Books, ISBN 0-7434-1146-3
		+	</div>
		+

	=See Also=		=See Also=
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	* [[Conventional_Encryption_Algorithms]]		* [[Conventional_Encryption_Algorithms]]
	* [[Payment_Card_Industry_Data_Security_Standard]]		* [[Payment_Card_Industry_Data_Security_Standard]]
-	* [http://en.wikipedia.org/wiki/Information_security Information Security]
-	* [[Piggybacking]]
	* [[Information_Security_References]]		* [[Information_Security_References]]
	* [[Security_and_Storage_Mediums]]		* [[Security_and_Storage_Mediums]]

Katmehm at 03:39, 12 April 2009

2009-04-12T03:39:27Z

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	=See Also=		=See Also=
		+
	* [[Conventional_Encryption_Algorithms]]		* [[Conventional_Encryption_Algorithms]]
		+	* [[Payment_Card_Industry_Data_Security_Standard]]
		+	* [http://en.wikipedia.org/wiki/Information_security Information Security]
	* [[Piggybacking]]		* [[Piggybacking]]
		+	* [[Information_Security_References]]
		+	* [[Security_and_Storage_Mediums]]
		+
	=External Links=		=External Links=
		+
		+	*[http://security.practitioner.com/introduction/ An Introduction to Information Security]
		+	*[http://www.logicalsecurity.com/resources/resources_articles.html Introduction to Security Governance]
		+	*[http://www.coesecurity.com/services/resources.asp COE Security - Information Security Articles]
		+	*[http://www.davidstclair.co.uk/example-security-templates/example-internet-e-mail-usage-policy-2.html Example Security Policy]
		+	*[http://www.iwar.org.uk/comsec/ IWS - Information Security Chapter]
		+	*[http://www.issa.org/ Information Systems Security Association]
		+	*[http://msdn2.microsoft.com/en-us/library/ms998382.aspx patterns & practices Security Engineering Explained]
		+	*[http://www.opensecurityarchitecture.org Open Security Architecture- Controls and patterns to secure IT systems]

Katmehm at 03:33, 12 April 2009

2009-04-12T03:33:52Z

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	[[Image:Vigenere.png\|180px\|right\|''' Vigenere square ''']]		[[Image:Vigenere.png\|180px\|right\|''' Vigenere square ''']]

-	*'''EXAMPLE''': The first line of a limerick is enciphered using the key "BENCH," as follows:	+	*'''EXAMPLE''': The first line of a limerick is enciphered using the key "BENCH," as follows:<br>
-	~~Key~~ <br>B ENCHBENC HBENC HBENCH BENCHBENCH<br>	+	Key B ENCHBENC HBENC HBENCH BENCHBENCH<br>
	Plaintext A LIMERICK PACKS LAUGHS ANATOMICAL<br>		Plaintext A LIMERICK PACKS LAUGHS ANATOMICAL<br>
	Ciphertext B PVOLSMPM WBGXU SBYTJZ BRNVVNMPCS<br>		Ciphertext B PVOLSMPM WBGXU SBYTJZ BRNVVNMPCS<br>
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	=See Also=		=See Also=
-		+	* [[Conventional_Encryption_Algorithms]]
		+	* [[Piggybacking]]
	=External Links=		=External Links=

Katmehm at 03:30, 12 April 2009

2009-04-12T03:30:36Z

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	[[Image:Vigenere.png\|180px\|right\|''' Vigenere square ''']]		[[Image:Vigenere.png\|180px\|right\|''' Vigenere square ''']]

-	*'''EXAMPLE''': The first line of a limerick is enciphered using the key "BENCH," as follows.<br>	+	*'''EXAMPLE''': The first line of a limerick is enciphered using the key "BENCH," as follows:
-	~~Key~~ B ENCHBENC HBENC HBENCH BENCHBENCH<br>	+	Key <br>B ENCHBENC HBENC HBENCH BENCHBENCH<br>
	Plaintext A LIMERICK PACKS LAUGHS ANATOMICAL<br>		Plaintext A LIMERICK PACKS LAUGHS ANATOMICAL<br>
	Ciphertext B PVOLSMPM WBGXU SBYTJZ BRNVVNMPCS<br>		Ciphertext B PVOLSMPM WBGXU SBYTJZ BRNVVNMPCS<br>

Katmehm at 03:28, 12 April 2009

2009-04-12T03:28:43Z

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	[http://en.wikipedia.org/wiki/Cipher]		[http://en.wikipedia.org/wiki/Cipher]
	[http://en.wikipedia.org/wiki/Julius_Caesar]		[http://en.wikipedia.org/wiki/Julius_Caesar]
		+
		+	=See Also=
		+
		+	=External Links=

Katmehm at 03:27, 12 April 2009

2009-04-12T03:27:08Z

← Older revision		Revision as of 03:27, 12 April 2009
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	==Data Encryption Standard==		==Data Encryption Standard==
	The Data Encryption Standard (DES) was designed to encipher sensitive but nonclassified data.		The Data Encryption Standard (DES) was designed to encipher sensitive but nonclassified data.
-	It is [http://en.wikipedia.org/wiki/Bit-oriented bit-oriented], unlike the other ciphers we have seen. It uses both [ http://en.wikipedia.org/wiki/Transposition_cipher transposition] and [http://en.wikipedia.org/wiki/Substitution_cipher substitution]	+	It is [http://en.wikipedia.org/wiki/Bit-oriented bit-oriented], unlike the other ciphers we have seen. It uses both [http://en.wikipedia.org/wiki/Transposition_cipher transposition] and [http://en.wikipedia.org/wiki/Substitution_cipher substitution]
	and for that reason is sometimes referred to as a [http://en.wikipedia.org/wiki/Product_cipher product cipher]. Its input, output, and key are each 64 bits long.		and for that reason is sometimes referred to as a [http://en.wikipedia.org/wiki/Product_cipher product cipher]. Its input, output, and key are each 64 bits long.
	The sets of 64 bits are referred to as blocks.		The sets of 64 bits are referred to as blocks.
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	One of the keys would be publicly known; the other would be		One of the keys would be publicly known; the other would be
	kept private by its owner. Classical cryptography requires the sender and recipient to share a common key.		kept private by its owner. Classical cryptography requires the sender and recipient to share a common key.
-	Public key cryptography does not. If the encipherment key is public, to send a secret message simply	+	Public key cryptography does not. If the encipherment [http://en.wikipedia.org/wiki/Key_(cryptography) key] is public, to send a secret message simply
	encipher the message with the recipient's public key. Then send it. The recipient can decipher it using his		encipher the message with the recipient's public key. Then send it. The recipient can decipher it using his
	private key.		private key.
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	-kAlice = 5 and		-kAlice = 5 and
	-kBob = 7.		-kBob = 7.
-	Their public keys are	+	Their [http://en.wikipedia.org/wiki/Public_key public keys] are
	*KAlice = 175 mod 53 = 40 and		*KAlice = 175 mod 53 = 40 and
	*KBob = 177 mod 53 = 6		*KBob = 177 mod 53 = 6
	Suppose Bob wishes to send Alice a message. He computes a shared secret key by enciphering Alice's public key using his		Suppose Bob wishes to send Alice a message. He computes a shared secret key by enciphering Alice's public key using his
-	private key:	+	[http://en.wikipedia.org/wiki/Private_key private key]:
	-SBob,Alice = KAlice<br>		-SBob,Alice = KAlice<br>
	*kBob mod p = 407 mod 53 = 38<br>		*kBob mod p = 407 mod 53 = 38<br>
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	==RSA==		==RSA==
-	is an exponentiation cipher. Choose two large prime numbers p and q, and let n = pq. The totient(n) of n is the number of numbers less than n with no factors in common with n.	+	is an [http://en.wikipedia.org/wiki/Exponentiation exponentiation cipher]. Choose two large [http://en.wikipedia.org/wiki/Prime_numbers prime numbers] p and q, and let n = pq. The totient(n) of n is the number of numbers less than n with no factors in common with n.
-	The security of [http://www.cas.mcmaster.ca/wiki/index.php/Public_Key_Encryption_Algorithms RSA] is based on the difficulty of integer factorization: Finding large primes and multiplying them together is easy. However, from the product to find the factors is hard. There is no known efficient general technique to solve this problem.	+	The [http://en.wikipedia.org/wiki/Security security] of [http://www.cas.mcmaster.ca/wiki/index.php/Public_Key_Encryption_Algorithms RSA] is based on the difficulty of [http://en.wikipedia.org/wiki/Integer_factorization integer factorization]: Finding large primes and multiplying them together is easy. However, from the product to find the factors is hard. There is no known efficient general technique to solve this problem.
	*'''EXAMPLE''': Let n = 10. The numbers that are less than 10 and are relatively prime to (have no factors in common with) n are 1, 3, 7, and 9. Hence, [http://en.wikipedia.org/wiki/Totient totient](10) = 4. Similarly, if n = 21, the numbers that are relatively prime to n are 1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, and 20. So totient(21) = 12		*'''EXAMPLE''': Let n = 10. The numbers that are less than 10 and are relatively prime to (have no factors in common with) n are 1, 3, 7, and 9. Hence, [http://en.wikipedia.org/wiki/Totient totient](10) = 4. Similarly, if n = 21, the numbers that are relatively prime to n are 1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, and 20. So totient(21) = 12

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	==HMAC==		==HMAC==
	Alice wants to send Bob a message of n bits. She wants Bob to be able to verify that the message he		Alice wants to send Bob a message of n bits. She wants Bob to be able to verify that the message he
-	receives is the same one that was sent. So she applies a mathematical function, called a checksum	+	receives is the same one that was sent. So she applies a mathematical function, called a [http://en.wikipedia.org/wiki/Checksum checksum]
	function, to generate a smaller set of k bits from the original n bits. This smaller set is called the checksum		function, to generate a smaller set of k bits from the original n bits. This smaller set is called the checksum
	or message digest. Alice then sends Bob both the message and the checksum. When Bob gets the		or message digest. Alice then sends Bob both the message and the checksum. When Bob gets the
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	that the message has not been changed.		that the message has not been changed.

-	*'''EXAMPLE''': The parity bit in the ASCII representation is often used as a single-bit checksum. If	+	*'''EXAMPLE''': The [http://en.wikipedia.org/wiki/Parity_bit parity bit] in the [http://en.wikipedia.org/wiki/ASCII ASCII] representation is often used as a [http://en.wikipedia.org/wiki/Single-bit single-bit checksum]. If
	odd parity is used, the sum of the 1-bits in the ASCII representation of the character, and the		odd parity is used, the sum of the 1-bits in the ASCII representation of the character, and the
	parity bit, is odd. Assume that Alice sends Bob the letter "A."		parity bit, is odd. Assume that Alice sends Bob the letter "A."
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	number is odd, Bob knows that the message has arrived unchanged.		number is odd, Bob knows that the message has arrived unchanged.

-	HMAC is a generic term for an algorithm that uses a keyless hash function and a cryptographic key to	+	HMAC is a [http://en.wikipedia.org/wiki/Generic generic] term for an algorithm that uses a keyless [http://en.wikipedia.org/wiki/Hash_function hash function] and a cryptographic key to
	produce a keyed hash function [594]. This mechanism enables Alice to validate that data Bob sent to her is		produce a keyed hash function [594]. This mechanism enables Alice to validate that data Bob sent to her is
	unchanged in transit. Without the key, anyone could change the data and recompute the message		unchanged in transit. Without the key, anyone could change the data and recompute the message
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	The need for HMAC arose because keyed hash functions are derived from cryptographic algorithms. Many		The need for HMAC arose because keyed hash functions are derived from cryptographic algorithms. Many
	countries restrict the import and export of software that implements such algorithms. They do not restrict		countries restrict the import and export of software that implements such algorithms. They do not restrict
-	software implementing keyless hash functions, because such functions cannot be used to conceal	+	software implementing keyless hash functions, because such functions cannot be used to [http://en.wikipedia.org/wiki/Conceal conceal information]. Hence, HMAC builds on a keyless hash function using a cryptographic key to create a keyed
-	information. Hence, HMAC builds on a keyless hash function using a cryptographic key to create a keyed	+
	hash function.		hash function.

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	For our purposes, three aspects of cryptography require study. Classical cryptography uses a single key		For our purposes, three aspects of cryptography require study. Classical cryptography uses a single key
	shared by all involved. Public key cryptography uses two keys, one shared and the other private. Both		shared by all involved. Public key cryptography uses two keys, one shared and the other private. Both
-	types of cryptosystems can provide secrecy and origin authentication (although classical cryptography	+	types of cryptosystems can provide secrecy and [http://en.wikipedia.org/w/index.php?title=Special%3ASearch&search=origin+authentication origin authentication] (although classical cryptography
	requires a trusted third party to provide both). Cryptographic hash functions may or may not use a secret		requires a trusted third party to provide both). Cryptographic hash functions may or may not use a secret
	key and provide data authentication.		key and provide data authentication.
-	All cryptosystems are based on substitution (of some quantity for another) and permutation (scrambling of	+	All cryptosystems are based on substitution (of some quantity for another) and [http://en.wikipedia.org/wiki/Permutation permutation] (scrambling of
	some quantity). Cryptanalysis, the breaking of ciphers, uses statistical approaches (such as the Kasiski		some quantity). Cryptanalysis, the breaking of ciphers, uses statistical approaches (such as the Kasiski
	method and differential cryptanalysis) and mathematical approaches (such as attacks on the RSA method).		method and differential cryptanalysis) and mathematical approaches (such as attacks on the RSA method).
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	=Research Issues=		=Research Issues=
-	Cryptography is an exciting area of research, and all aspects of it are being studied. New secret key ciphers incorporate techniques for defeating differential and linear cryptanalysis. New public key ciphers use simple	+	Cryptography is an exciting area of research, and all aspects of it are being studied. New secret key ciphers incorporate techniques for defeating differential and [http://en.wikipedia.org/wiki/Linear_cryptanalysis linear cryptanalysis]. New public key ciphers use simple instances of [http://en.wikipedia.org/wiki/NP-hard NP-hard] problems as their bases, and they cast those instances into the more general framework of the NP-hard problem. Other public key ciphers revisit well-studied, difficult classical problems
-	instances of NP-hard problems as their bases, and they cast those instances into the more general framework of the NP-hard problem. Other public key ciphers revisit well-studied, difficult classical problems	+
	(such as factoring) and use them so that mathematically breaking the cipher is equivalent to solving the hard problem. Still others are built on the notion of randomness (in the sense of unpredictability). Cryptanalytic techniques are also improving. From the development of differential cryptanalysis came linear cryptanalysis. The use of NP-hard problems leads to an analysis of the problem underlying the cipher to reduce it to the simpler, solvable case. The use of classical mathematical problems leads to the application of advanced technology to make the specific problem computable; for example, advances in technology		(such as factoring) and use them so that mathematically breaking the cipher is equivalent to solving the hard problem. Still others are built on the notion of randomness (in the sense of unpredictability). Cryptanalytic techniques are also improving. From the development of differential cryptanalysis came linear cryptanalysis. The use of NP-hard problems leads to an analysis of the problem underlying the cipher to reduce it to the simpler, solvable case. The use of classical mathematical problems leads to the application of advanced technology to make the specific problem computable; for example, advances in technology
	have increased the sizes of numbers that can be factored, which in turn lead to the use of larger primes as the basis for ciphers such as RSA.		have increased the sizes of numbers that can be factored, which in turn lead to the use of larger primes as the basis for ciphers such as RSA.
-	Advances in both cryptography and cryptanalysis lead to a notion of "provable security." The issue is to prove under what conditions a cipher is unbreakable. Then, if the conditions are met, perfect secrecy is obtained. Similar issues arise with cryptographic ~~protocols~~ (~~some of which the next chapters will explore~~). This leads to the area of assurance and serves as an excellent test base for many assurance techniques	+	Advances in both cryptography and cryptanalysis lead to a notion of "[http://en.wikipedia.org/wiki/Provable_security provable security]." The issue is to prove under what conditions a cipher is unbreakable. Then, if the conditions are met, perfect secrecy is obtained. Similar issues arise with cryptographic [http://en.wikipedia.org/wiki/Protocol_(computing) protocols]. This leads to the area of assurance and serves as an excellent test base for many assurance techniques

	=References=		=References=