Linear equation over a ring

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The Lai-Massey scheme is a cryptographic structure used in the design of block ciphers. It is used in IDEA and IDEA NXT.

Construction details

Let $\mathrm {F}$ be the round function and $\mathrm {H}$ a half-round function and let $K_{0},K_{1},\ldots ,K_{n}$ be the sub-keys for the rounds $0,1,\ldots ,n$ respectively.

Then the basic operation is as follows:

Split the plaintext block into two equal pieces, ($L_{0}$, $R_{0}$)

For each round $i=0,1,\dots ,n$, compute

$(L_{i+1}',R_{i+1}')=\mathrm {H} (L_{i}'+T_{i},R_{i}'+T_{i})$

where $T_{i}=\mathrm {F} (L_{i}'-R_{i}',K_{i})$ and $(L_{0}',R_{0}')=\mathrm {H} (L_{0},R_{0})$

Then the ciphertext is $(L_{n+1},R_{n+1})=(L_{n+1}',R_{n+1}')$.

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

$(L_{i}',R_{i}')=\mathrm {H} ^{-1}(L_{i+1}'-T_{i},R_{i+1}'-T_{i})$

where $T_{i}=\mathrm {F} (L_{i+1}'-R_{i+1}',K_{i})$ and $(L_{n+1}',R_{n+1}')=\mathrm {H} ^{-1}(L_{n+1},R_{n+1})$

Then $(L_{0},R_{0})=(L_{0}',R_{0}')$ 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 $\mathrm {F}$ does not have to be invertible.

The half-round function is required to prevent a trivial distinguishing attack ($L_{0}-R_{0}=L_{n+1}-R_{n+1}$). It commonly applies an orthomorphism $\sigma$ on the left hand side, that is,

$\mathrm {H} (L,R)=(\sigma (L),R)$

where both $\sigma$ and $x\mapsto \sigma (x)-x$ are permutations (in the mathematical sense, that is, a bijection – not a permutation box). Since there are no orthomorphisms for bit blocks (groups of size $2^{n}$), "almost orthomorphisms" are used instead.

$\mathrm {H}$ 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 $n.5$" for a cipher that otherwise has $n$ rounds.

Literature

• X. Lai. On the design and security of block ciphers. ETH Series in Information Processing, vol. 1, Hartung-Gorre, Konstanz, 1992
• X. Lai, J. L. Massey. 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, p. 33