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One issue that may arise when attempting to evaluate the security of a round function for a block cipher is that the analysis of the round function does not treat the round key space and the message space as merely sets but as more sophisticated structures. For example, if the message space is $\{0,1\}^{n}$, then the message space has extra mathematical structure since $\{0,1\}^{n}$ is always a Boolean algebra. Furthermore, the round function may not be treated as simply a function but as a circuit used to compute a function. This may be a problem since a block cipher may appear to be more or less secure than it really is simply because such a block cipher has been evaluated in the context of the additional structure on the round key space and the message space.

In what ways can one evaluate the round function of a block cipher while treating the round key space and message space as simply sets without any additional mathematical structure except for the round function itself?

Let $K,X$ be finite sets. Let $F:K\times X\rightarrow X$ be a function, and if $k\in K$, then let $F_{k}:X\rightarrow X$ be the function defined by $F_{k}(x)=F(k,x)$ whenever $x\in X$. Suppose that each $F_{k}$ is a bijection. Then we shall say that $F$ is a block cipher round function.

Suppose that $F_{i}:K_{i}\times X_{i}\rightarrow X_{i}$ is a block cipher round function for $i\in\{1,2\}$. Suppose that $\phi:K_{1}\rightarrow K_{2},\psi:X_{1}\rightarrow X_{2}$ are bijections. Then we say that $(\phi,\psi)$ is an isomorphism from $F_{1}$ to $F_{2}$ if $\psi(F_{1}(k,x))=F_{2}(\phi(k),\psi(x))$ whenever $k\in K_{1},x\in X_{1}$, and we say that $F_{1}$ and $F_{2}$ are isomorphic if there is some isomorphism from $F_{1}$ to $F_{2}$. We shall say that a function $M$ is round function invariant parameter if $M(F)=M(G)$ whenever $F,G$ are isomorphic block cipher round functions.

A round function invariant parameter $M$ is said to be an objective measure of security if $M(F)\in[-\infty,\infty]$ for each round function $F$ and where a higher value of $M(F)$ suggests that the round function $F$ is more secure.

What objective measures of security have been evaluated or estimated for modern round functions? Have any of these objective measures of security been taken into consideration when evaluating the cryptographic security of block ciphers, cryptographic hash functions, or other cryptographic objects (such as in the NIST cryptographic competitions)?

For this post, I am interested not only in block ciphers that are used for symmetric encryption but also in block ciphers that are used for other purposes such as constructing cryptographic hash functions using the Davies-Meyer construction.

Example

In some cases, one can recover some of the mathematical structure on $K,X$ from the round function $F$ and use this additional structure to obtain round function invariant parameters.

Suppose that $K,X$ are vector spaces over the field $F_{p}$ with $p$ elements for some prime $p$. Suppose that $\iota:K\rightarrow X$ is a vector space isomorphism. Let $P:X\rightarrow X$ be a permutation, and suppose that $F:K\times X\rightarrow X$ is the block cipher round function defined by letting $F(k,x)=\iota(k)+P(x)$. Define ternary operations $L,M$ on the sets $K,X$ such that $L(j,k,l)=j-k+l$ whenever $j,k,l\in K$ and $M(x,y,z)=x-y+z$ whenever $x,y,z\in M$. Then the operations $L,M$ are first order definable in the two sorted structure $(K,X,F)$. The operations $L,M$ should be thought of as operations that are similar to vector space operations but where the spaces $(K,L),(X,M)$ do not have a defined basepoint that one can set to be the origin.

Now, suppose that $s$ is a positive integer. To make our construction more convenient, suppose that $\dim(K)=\dim(X)=p^{N}-1$. Now, let $V$ be a uniform randomly selected $N$-dimensional affine subspace of $X$, and let $k_{1},\dots,k_{u}$ be uniform randomly selected elements of $K$. Then define $$\mathcal{R}_{s}=\dim(\{F_{k_{1}}\circ\dots\circ F_{k_{u}}(v)\mid v\in V\}).$$

One can therefore define a round function invariant parameter $M_{s}$ so that $M_{s}(F)=E(\mathcal{R}_{s})$ (and one can set $M_{s}(F)=0$ whenever the random variable $\mathcal{R}_{s}$ does not make sense for the block cipher round function $F$). Here a higher value of $M_{s}(F)$ suggests a higher level of non-linearity and cryptographic security for the block cipher round function $F$. One can define many other similar round function invariant parameters $N$ so that $N(F)$ is a measure of the non-linearity of the block cipher round function $F$.

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Quite a bit of structure is actually definable in terms of the block cipher round function, and from this structure, one can produce block cipher round function invariants that measure cryptographic security.

A portion of this answer is essentially the same as my previous answer here, so in that case, we shall omit the proofs of the results.

If $G,H$ are groups, then we say that a function $\phi:G\rightarrow H$ is a heap homomorphism if there exists a group homomorphism $\psi:G\rightarrow H$ along with a $b\in H$ where $\phi(g)=b\psi(g)$ for all $b\in B.$ Equivalently, the mapping $\phi:G\rightarrow H$ is a heap homomorphism if and only if $\phi(xy^{-1}z)=\phi(x)\phi(y)^{-1}\phi(z)$ whenever $x,y,z\in G$. A bijective heap homomorphism is said to be a heap automorphism.

For this post, let $U_{0},\dots,U_{n-1},V_{0},\dots,V_{n-1}$ be groups. Suppose that $I_{i}:U_{i}\rightarrow V_{i}$ is a group isomorphism whenever $0\leq i<n$. Let $K=U_{0}\times\dots\times U_{n-1},X=V_{0}\times\dots\times V_{n-1}$. Then define a group isomorphism $\iota:K\rightarrow X$ by letting $\iota(u_{0},\dots,u_{n-1})=(I_{0}(u_{0}),\dots,I_{n-1}(u_{n-1}))$.

If $0\leq i<n$, then let $s_{i}:V_{i}\rightarrow V_{i}$ be an arbitrary bijection. Define a mapping $S:V_{0}\times\dots\times V_{n-1}\rightarrow V_{0}\times\dots\times V_{n-1}$ by letting $S(v_{0},\dots,v_{n-1})=(s_{0}(v_{0}),\dots,s_{n-1}(v_{n-1}))$. Let $\Gamma:X\rightarrow X$ be a heap automorphism. Let $P=\Gamma\circ S$, and define a mapping $F:K\times X\rightarrow X$ by letting $F(k,x)=\iota(k)P(x)$.

Proposition: The mappings $K^{2}\times X\rightarrow X,(j,k,x)\mapsto\iota(jk^{-1})x$ and $X^{3}\rightarrow X,(x,y,z)\mapsto xy^{-1}z$ are first order definable in $(K,X,F)$.

Let $\pi_{i}:X\rightarrow V_{i}$ be the projection group homomorphism. Let $\simeq_{i}$ be the equivalence relation on $X$ where we set $x\simeq_{i}y$ if and only if $\pi_{i}(x)=\pi_{i}(y)$. I claim that the set of equivalence relations $\{\simeq_{0},\dots,\simeq_{n-1}\}$ is higher order definable in $(K,X,F)$ under reasonable hypotheses about the block cipher $F$, but to prove this claim, we will need to go over a few lemmas. From the definability of $\{\simeq_{0},\dots,\simeq_{n-1}\}$, we can produce many block cipher round function invariants that measure non-linearity and also the avalanche effect.

For $0\leq i<n$, and $j\in U_{i}$, let $s_{i}^{j}$ be the permutation of $V_{i}$ defined by letting $s_{i}^{j}(v)=I_{i}(j)s_{i}(v)$.

If $0\leq i<n$, then let $H_{i}$ be the group of all permutations of $V_{i}$ generated by $s_{i}^{j}\circ(s_{i}^{k})^{-1},(s_{i}^{j})^{-1}\circ s_{i}^{k}$.

Let $H$ be the group of all permutations of $X$ generated by all permutations of the form $F_{j}^{-1}\circ F_{k},F_{j}\circ F_{k}^{-1}$. Observe that $F_{j}\circ F_{k}^{-1}(x)=\iota(jk^{-1})(x)$ and $F_{j}^{-1}\circ F_{k}(x)=S^{-1}[\Gamma^{-1}[\iota(j^{-1}k)]S(x)]$. In particular, $H$ is generated by the permutations of the form $x\mapsto\iota(m)(x)$ along with the permutations of the form $S^{-1}[\iota(m)S(x)]$. Said differently, $H$ is generated by the permutations of the form $$(x_{0},\dots,x_{n-1})\mapsto(I_{0}(m_{0})(x_{0}),\dots,I_{n-1}(m_{n-1})(x_{n-1}))$$ and $$(x_{0},\dots,x_{n-1})\mapsto(s_{0}^{-1}[I_{0}(m_{0})(s_{0}(x_{n-1}))],\dots,s_{n-1}^{-1}[I_{n-1}(m_{n-1})(s_{n-1}(x_{n-1}))]).$$ Thus, $H$ consists of all permutations of the form $h_{0}\times\dots\times h_{n-1}$ where $h_{i}\in H_{i}$ whenever $0\leq i<n$.

Therefore $H\simeq H_{0}\times\dots\times H_{n-1}$ as an external direct product. We may also write $H$ as an internal direct product $H_{0}^{\sharp},\dots,H_{n-1}^{\sharp}$ where $H_{i}^{\sharp}$ consists of all mappings $h\in H$ where if $h_{0},\dots,h_{n-1}$ are the mappings where $h(x_{0},\dots,x_{n-1})=(h_{0}(x_{0}),\dots,h_{n-1}(x_{n-1}))$ for all $x_{0},\dots,x_{n-1}$, then $h_{j}$ is the identity function whenever $j\neq i$. Then $H_{i}^{\sharp}\simeq H_{i}$ whenever $0\leq i<n.$

We say that a group $G$ is superindecomposible if whenever $A,B$ are subgroups of $G$ with $ab=ba$ whenever $a\in A,b\in B$ and $G=AB$, then $|A|=1$ or $|B|=1$ (let me know in the comments if you can think of a better word than 'superindecomposible').

Theorem (Krull-Schmidt): Suppose that $G$ is a group that satisfies the ascending chain condition and descending chain condition on normal subgroups (any finite group satisfies this property). Furthermore, suppose that $G$ is written as an internal direct product of non-trivial directly indecomposible subgroups in two different ways, namely $G=G_{0}\times\dots\times G_{n-1}$ and $G=H_{0}\times\dots H_{n-1}$. Then there exists a permutation $\rho$ of $\{0,\dots,n-1\}$ such that $G_{i}\simeq H_{\rho(i)}$ for $0\leq i<n$ and where $$G=G_{0}\times\dots\times G_{r}\times H_{\rho(r+1)}\times\dots H_{\rho(n-1)}$$ whenever $0\leq r\leq n-1$.

Lemma: Suppose that $G$ is a finite group. If $G$ is an internal direct product of superindecomposible subgroups, then whenever we factor $G$ as internal direct products $G=G_{0}\times\dots\times G_{m-1}$ and $G=H_{0}\times\dots\times H_{n-1}$, then $m=n$, and there is some permutation $\rho:\{0,\dots,n-1\}\rightarrow\{0,\dots,n-1\}$ where $H_{i}=G_{\rho(i)}$ for $0\leq i<n$.

Theorem: There exists a higher order formula $\phi$ such that if the groups $H_{0},\dots,H_{n-1}$ are indecomposible, then $(K,X,F)\models\phi(z)$ if and only if $z=\{\simeq_{0},\dots,\simeq_{n-1}\}$.

Proof: Observe that the group $H$ is higher order definable in the structure $(K,X,F)$. Furthermore, the group $H$ has a unique up-to-permutation factorization $H^{\sharp}_{0}\times\dots\times H^{\sharp}_{n-1}$ into its indecomposible factors. Therefore, the set of all indecomposible factors $\{H^{\sharp}_{0},\dots,H^{\sharp}_{n-1}\}$ is definable in $(K,X,F)$. Now, for each $i$, let $\hat{\simeq}_{i}$ be the equivalence relation where we set $x\hat{\simeq}_{i}y$ if and only if $h(x)=y$ for some $h\in H^{\sharp}_{i}$. Let $\simeq_{i}$ be the equivalence relation on $X$ generated by $\bigcup\{\hat{\simeq}_{j}\mid j\neq i\}$. Then $\{\simeq_{0},\dots,\simeq_{n-1}\}$ is definable from $\{H^{\sharp}_{0},\dots,H^{\sharp}_{n-1}\}$. Therefore, the set $\{\simeq_{0},\dots,\simeq_{n-1}\}$ is definable in $(K,X,F)$. $\square$

From the definability of $\{\simeq_{0},\dots,\simeq_{n-1}\}$, we can define quite a few block cipher round function invariants.

If $T_{1},T_{2}$ are groups, and $f_{i}:T_{i}\rightarrow T_{i}$ for $i\in\{1,2\}$, then we say that $f_{1},f_{2}$ are affinely equivalent if there exists heap automorphisms $L_{1}:T_{1}\rightarrow T_{2},L_{2}:T_{2}\rightarrow T_{1}$ where $f_{2}=L_{1}f_{1}L_{2}$. We say that a mapping $M$ is invariant under affine equivalence if $M(f)=M(g)$ whenever $f,g$ are affinely equivalent.

We say that a pair $(k,\Delta)$ is an S-boxification if $\Delta$ is a heap automorphism and whenever $x\simeq_{i}y$, then $\Delta\circ F_{k}(x)\simeq_{i}\Delta\circ F_{k}(y)$. The collection of all S-boxifications is higher order definable in $(K,X,F)$. Observe that there exists an S-boxification, namely the mapping $(e,\Gamma^{-1})$. Suppose now that $(k,\Delta)$ is an S-boxification. Then it is easy to show that each $\simeq_{i}$ is a congruence with respect to $\Delta\Gamma^{-1}$. Therefore $\Delta=E\Gamma^{-1}$ for some heap automorphism $E$ where each $\simeq_{i}$ is a congruence with respect to $E$. Therefore, $\Delta\circ F_{k}=E'\circ S$ for some heap automorphism $E'$. Therefore, the quotient algebra $(X,\Delta\circ F_{k})/\simeq_{i}$ is affinely equivalent to $(V_{i},s_{i})$ for $0\leq i<n$. From these observations, we obtain the following fact.

Fact: If the mapping $M$ is invariant under affine equivalence and definable in higher order logic, then the mapping $M^{+}$ defined by letting $M^{+}(F)=\{M(s_{0}),\dots,M(s_{n-1})\}$ is higher order definable in the structure $(K,X,F)$. Therefore, $M^{+}$ is a round function invariant parameter.

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