en>David Eppstein: /* References */ authorlink Matthias Aschenbrenner

2014-12-21T04:21:29Z

References: authorlink Matthias Aschenbrenner

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	~~[[File:Lai Massey scheme diagram en.svg\|right]]~~

	~~The '''Lai-Massey scheme''' is a cryptographic structure used in the design~~ of ~~[[block cipher]]s.<ref>Aaram Yun~~, ~~Je Hong Park, Jooyoung Lee:~~ [http://~~eprint~~.~~iacr~~.~~org~~/~~2007~~/~~347.pdf Lai~~-~~Massey Scheme and Quasi~~-~~Feistel Networks]. ''IACR Cryptology''<~~/~~ref><ref>Serge Vaudenay: [http://lasec~~.~~epfl.ch/pub/lasec/doc/Vau99c~~.~~ps On the Lai-Massey Scheme~~]. ~~''ASIACRYPT'99''</ref> It is used~~ in ~~[[International Data Encryption Algorithm\|IDEA]] and [[IDEA NXT]]~~.

	~~== Construction details ==~~
	~~Let~~ <~~math~~>~~\mathrm F~~<~~/math~~> be the ~~round function and <math>\mathrm H</math> a half-round function and let <math>K_0~~,~~K_1,\ldots,K_n</math> be~~ the ~~sub-keys~~ for the ~~rounds <math>0,1,\ldots,n</math> respectively~~.

	~~Then~~ the ~~basic operation is as follows:~~

	~~Split~~ the ~~plaintext block into two equal pieces~~, (~~<math>L_0</math>~~, <~~math~~>~~R_0~~<~~/math~~>)

	For ~~each round <math>i =0,1,\dots,n</math>, compute~~

	~~:<math>(L_{i+1}~~',~~R_{i+1}~~'~~) = \mathrm H~~(~~L_i' + T_i,R_i' + T_i~~)~~</math>~~

	~~where <math>T_i = \mathrm F(L_i' - R_i'~~, ~~K_i)</math> and <math>(L_0'~~,~~R_0') = \mathrm H(L_0,R_0)</math>~~

	~~Then~~ the ~~ciphertext is~~ <~~math~~>~~(L_{n+1}, R_{n+1}) = (L_{n+1}',R_{n+1}')~~<~~/math~~>.

	~~Decryption of a ciphertext <math>~~(~~L_{n+1}, R_{n+1}~~)~~</math> is accomplished by computing for <math>i=n~~,n-1,~~\ldots~~,0<~~/math~~>

	:<~~math~~>(~~L_i',R_i'~~) ~~= \mathrm H^{-1}~~(~~L_{i+1}' - T_i, R_{i+1}' - T_i~~)~~</math>~~

	~~where <math>T_i = \mathrm F(L_{i+1}' - R_{i+1}'~~,~~K_i)</math>~~ and ~~<math>(L_{n+1}',R_{n+1}')=\mathrm H^{~~-1}(~~L_{n+1},R_{n+1}~~)<~~/math~~>

	~~Then~~ <~~math~~>~~(L_0~~,~~R_0) = (L_0~~',~~R_0~~'~~)</math>~~ is ~~the plaintext again~~.

	~~The Lai-Massey scheme offers security properties similar~~ to ~~those of~~ the ~~[[Feistel structure]]~~. ~~It also shares its advantage over a [[substitution~~-~~permutation network]] that the round function~~ <~~math~~>~~\mathrm F~~<~~/math~~> ~~does not have~~ to ~~be invertible~~.

	~~The half-round function is required~~ to ~~prevent a trivial distinguishing attack (<math>L_0-R_0 = L_{n+1}-R_{n+1}</math>)~~. ~~It commonly applies an orthomorphism <math>\sigma</math> on~~ the ~~left hand side~~, ~~that is~~,
	~~:<math>\mathrm H~~(L, R) = (~~\sigma(L)~~,R)~~</math>~~
	~~where both <math>\sigma</math> and <math>x\mapsto \sigma(x)-x</math> are [[permutation]]~~s ~~(in the mathematical sense, that is~~, a ~~bijection – not a [[permutation box]]). Since there are no orthomorphisms for bit blocks (groups of size <math>2^n</math>),~~ "~~almost orthomorphisms~~" ~~are used instead.~~

	~~<math>\mathrm H</math> may depend on~~ the ~~key~~. ~~If it doesn't~~, the ~~last application can be omitted, since its inverse is known anyway. The last application is commonly called "round <math>n~~.5<~~/math~~>~~" for a cipher that otherwise has~~ <~~math~~>~~n</math> rounds.~~

	~~== Literature ==~~
	* X. Lai. ''[http://e-collection.library.ethz.ch/eserv/eth:38650/eth-38650-02.pdf On the design and ~~security~~ of ~~block ciphers]''. ETH Series in Information Processing~~, ~~vol. 1~~, ~~Hartung-Gorre~~, ~~Konstanz~~, ~~1992~~
	* X. Lai, J. ~~L. Massey. [http://www.isiweb.ee.ethz.ch/archive/massey_pub/pdf/BI319~~.~~pdf A proposal for~~ a ~~new block encryption standard]. ''Advances in Cryptology EUROCRYPT'90''~~, ~~Aarhus~~, ~~Denemark, LNCS 473, p. 389-404, Springer, 1991~~
	* Serge Vaudenay: ''A Classical Introduction to Cryptography'', [http://books.google.de/books?id=DVLEql738mwC&pg=PA33 p. ~~33]~~

	~~== References ==~~
	~~<references/>~~
	~~{{Cryptography block}}~~

	~~[[Category:Block ciphers]]~~

en>D.Lazard: /* Properties of effective rings */ clarifiction

2014-01-05T21:34:17Z

Properties of effective rings: clarifiction

New page

{{technical|date=October 2013}}
[[File:Lai Massey scheme diagram en.svg|right]]

The '''Lai-Massey scheme''' is a cryptographic structure used in the design of [[block cipher]]s.<ref>Aaram Yun, Je Hong Park, Jooyoung Lee: [http://eprint.iacr.org/2007/347.pdf Lai-Massey Scheme and Quasi-Feistel Networks]. ''IACR Cryptology''</ref><ref>Serge Vaudenay: [http://lasec.epfl.ch/pub/lasec/doc/Vau99c.ps On the Lai-Massey Scheme]. ''ASIACRYPT'99''</ref> It is used in [[International Data Encryption Algorithm|IDEA]] and [[IDEA NXT]].

== Construction details ==
Let <math>\mathrm F</math> be the round function and <math>\mathrm H</math> a half-round function and let <math>K_0,K_1,\ldots,K_n</math> be the sub-keys for the rounds <math>0,1,\ldots,n</math> respectively.

Then the basic operation is as follows:

Split the plaintext block into two equal pieces, (<math>L_0</math>, <math>R_0</math>)

For each round <math>i =0,1,\dots,n</math>, compute

:<math>(L_{i+1}',R_{i+1}') = \mathrm H(L_i' + T_i,R_i' + T_i)</math>

where <math>T_i = \mathrm F(L_i' - R_i', K_i)</math> and <math>(L_0',R_0') = \mathrm H(L_0,R_0)</math>

Then the ciphertext is <math>(L_{n+1}, R_{n+1}) = (L_{n+1}',R_{n+1}')</math>.

Decryption of a ciphertext <math>(L_{n+1}, R_{n+1})</math> is accomplished by computing for <math>i=n,n-1,\ldots,0</math>

:<math>(L_i',R_i') = \mathrm H^{-1}(L_{i+1}' - T_i, R_{i+1}' - T_i)</math>

where <math>T_i = \mathrm F(L_{i+1}' - R_{i+1}',K_i)</math> and <math>(L_{n+1}',R_{n+1}')=\mathrm H^{-1}(L_{n+1},R_{n+1})</math>

Then <math>(L_0,R_0) = (L_0',R_0')</math> is the plaintext again.

The Lai-Massey scheme offers security properties similar to those of the [[Feistel structure]]. It also shares its advantage over a [[substitution-permutation network]] that the round function <math>\mathrm F</math> does not have to be invertible.

The half-round function is required to prevent a trivial distinguishing attack (<math>L_0-R_0 = L_{n+1}-R_{n+1}</math>). It commonly applies an orthomorphism <math>\sigma</math> on the left hand side, that is,
:<math>\mathrm H(L, R) = (\sigma(L),R)</math>
where both <math>\sigma</math> and <math>x\mapsto \sigma(x)-x</math> are [[permutation]]s (in the mathematical sense, that is, a bijection – not a [[permutation box]]). Since there are no orthomorphisms for bit blocks (groups of size <math>2^n</math>), "almost orthomorphisms" are used instead.

<math>\mathrm H</math> may depend on the key. If it doesn't, the last application can be omitted, since its inverse is known anyway. The last application is commonly called "round <math>n.5</math>" for a cipher that otherwise has <math>n</math> rounds.

== Literature ==
* X. Lai. ''[http://e-collection.library.ethz.ch/eserv/eth:38650/eth-38650-02.pdf On the design and security of block ciphers]''. ETH Series in Information Processing, vol. 1, Hartung-Gorre, Konstanz, 1992
* X. Lai, J. L. Massey. [http://www.isiweb.ee.ethz.ch/archive/massey_pub/pdf/BI319.pdf A proposal for a new block encryption standard]. ''Advances in Cryptology EUROCRYPT'90'', Aarhus, Denemark, LNCS 473, p. 389-404, Springer, 1991
* Serge Vaudenay: ''A Classical Introduction to Cryptography'', [http://books.google.de/books?id=DVLEql738mwC&pg=PA33 p. 33]

== References ==
<references/>
{{Cryptography block}}

[[Category:Block ciphers]]

Linear equation over a ring - Revision history

en>David Eppstein: /* References */ authorlink Matthias Aschenbrenner

en>D.Lazard: /* Properties of effective rings */ clarifiction