Lai-Massey scheme

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