For any finite field F, every (F, 1)-linear secret sharing scheme Σ is equivalent to a monotone span program of size σT(Σ) (see [121] for more details). Since the bounds studied in this section are bounds on the total information ratio of (F, 1)-linear secret sharing schemes, we have the same results for the size of monotone span programs. Next, we present a formulation of the Razborov rank measure [147] that is adapted to the context of secret sharing and access structures.
5.8.1 Razborov Rank Measure
Let Γ be an access structure on P , and let U ⊆ Γ and V ⊆ Γcbe two families of subsets.
For any U0 ⊆ U and V0 ⊆ V , we say that the Cartesian product U0× V0 is a (U, V )-
rectangle. For each i ∈ P , define the (U, V )-rectangle Ri = (U × V ) ∩ (T{i}× F{i}).
Let F be a field and let A be any |U | × |V | matrix over F with rows indexed by elements of U and columns indexed by elements of V . The restriction of A to the rectangle R = U0× V0 is the submatrix A R obtained by setting to 0 all entries not indexed by
elements of R.
Definition 5.27 ([147]). Let Γ ⊆ P(P ) be an access structure, U ⊆ Γ, V ⊆ Γc. Let F be a field and let A be a |U | × |V | matrix over F. If rank(A) > 0, the rank measure of Γ with respect to A is given by
µA(Γ) =
rank(A)
maxR∈RΓ(U,V )rank(A R)
.
If rank(A) = 0, we set µA(Γ) = 0. We accordingly define the rank measure of Γ as
µ(Γ) = max
A µA(Γ),
where the maximum is taken over all families of subsets U ⊆ Γ, V ⊆ Γcand all matrices
A of the form stated above.
Razborov [147] showed that the rank measure of a monotone Boolean function is a lower bound on the size of the shortest formula for this function (see Section 5.9). Later, G´al [162] proved that the rank measure is also a lower bound on the size of monotone span programs. Taking into account the connection between monotone span programs and linear secret sharing schemes mentioned above, we obtain that the rank measure is a lower bound on the optimal information ratio for linear secret sharing schemes. More concretely, we have the following result.
Theorem 5.28. Let Γ ⊆ P(P ) an access structure, U ⊆ Γ, V ⊆ Γc. Let F
q be a field
and let A be a |U | × |V | matrix over Fq. Then,
µA(Γ) ≤ λTq,1(Γ).
In the following theorem, we study the behavior of the rank measure when we add or delete subsets from an access structure.
Proposition 5.29. Let Γ, Γ0 ⊆ P(P ) be access structures, U ⊆ Γ, V ⊆ Γc. Let F be a
field and let A be a |U | × |V | matrix over F. Then, there exist U0 ⊆ Γ0, V0 ⊆ Γ0c and a |U0| × |V0| matrix A0 such that
µA(Γ) ≤ µA0(Γ0) + dist(Γ, Γ0).
Proof. Set U0 = U ∩ Γ0 and V0= V ∩ Γ0c, and let A0 be the restriction of A to U0× V0. Then, observe that |U \U0| ≤ |Γ\Γ0|, since U \U0 = U \Γ0 and U ⊆ Γ. Similarly, we see that |V \V0| ≤ |Γ0\Γ| by using Γc\Γ0c = Γ0\Γ. Since A0 is the submatrix obtained by
setting to 0 all rows of A indexed by U \U0 and all columns indexed by V \V0, we have rank(A) ≤ rank(A0) + |U \U0| + |V \V0| ≤ rank(A0) + dist(Γ, Γ0).
Let RΓ(U, V ) = {R1, . . . , Rn} and RΓ0(U0, V0) = {R0
1, . . . , Rn0}. Since R0i= Ri∩(U0×V0),
we have that A0R0iis a submatrix of A Ri, and thus rank(A Ri) ≥ rank(A
0
max R∈RΓ(U,V ) rank(A R) ≥ max R0∈R Γ0(U0,V0) rank(A0 R0).
Given a rectangle R ∈ RΓ(U, V ), let R0 = R ∩ (U0× V0). Note that A0 R0 is a submatrix
of A R, and so rank(A R) ≥ rank(A0 R0). Since the map RΓ(U, V ) → RΓ0(U0, V0)
given by R 7→ R ∩ (U0× V0) is exhaustive, we get the inequality max R∈RΓ(U,V ) rank(A R) ≥ max R0∈R Γ0(U0,V0) rank(A0 R0).
By using the previous inequalities, we see that µA(Γ) =
rank(A)
maxR∈RΓ(U,V )rank(A R)
≤ rank(A 0) + dist(Γ, Γ0) maxR0∈R0 Γ0(U0,V0)rank(A 0 R0) ≤ µA0(Γ0) + dist(Γ, Γ0).
Theorem 5.30. Let Γ, Γ0 ⊆ P(P ) be access structures. Then |µ(Γ) − µ(Γ0)| ≤ dist(Γ, Γ0).
Proof. Let A be the |U |×|V | matrix such that µ(Γ) = µA(Γ), and let A0be the restriction
of A to U0 × V0, where U0 = U ∩ Γ0 and V0 = V ∩ Γ0c. By Proposition 5.29 we
have µ(Γ) ≤ µA0(Γ0) + dist(Γ, Γ0). Now, by definition µA0(Γ0) ≤ µ(Γ0), so µ(Γ) ≤
µ(Γ0) + dist(Γ, Γ0).
Note that the behavior of the rank measure bound is different from that of λTq,1. If we extend the bound on Corollary 5.10 to λTwe have that |λTq,`(Γ)−λTq,`(Γ0)| ≤ n·dist(Γ, Γ0) for every two access structures Γ and Γ0.
Recently, in [163], the rank measure bound has been used to prove that for every prime p there exist access structures Γp for which λTq,1(Γp) = 2Ω(n) for every finite field Fq
of characteristic different from p. Let P = P2∪ P3, where P2 = {1, . . . , n} and P3 =
{n + 1, . . . , 2n}. Let Γ be the access structure P with Γ|P2 = Γ
2 and Γ| P3 = Γ
3satisfying
that for every A ∈ min Γ either A ⊆ P2 or A ⊆ P3. This access structure satisfies
λTq,1(Γ) = 2Ω(n) for every finite field Fq.
5.8.2 Critical Subfamilies
The next technique provides lower bounds on the size of the shares for linear secret sharing schemes. It was first introduced in [148].
Definition 5.31. Let Γ be an access structure and let H ⊆ min Γ. We say that H is a critical subfamily for Γ, if every H ∈ H contains a set TH ⊆ H, |TH| ≥ 2, such that the
following two conditions are satisfied
• The set TH uniquely determines H in the subfamily H: No other set in H contains
• For any subset Y ⊆ TH, the set SY = ∪A∈H, A∩Y 6=∅A \ Y does not contain any
member of min Γ.
Theorem 5.32 ([148]). Let Fq be a finite field. Let Γ be an access structure and let H
be a critical subfamily for Γ. Then λTq,1(Γ) ≥ |H|.
Given a critical subfamily for an access structure Γ, it is easy to construct a critical subfamily for an access structure Γ0 obtained by deleting some subsets from Γ or from min Γ. However, it is not easy to find a critical subfamily for access structures that are obtained by adding subsets to Γ or to min Γ.
Lemma 5.33. Let H be the critical subfamily for an access structure Γ. Let Γ0 be an access structure with min Γ0 ⊆ min Γ and | min Γ \ min Γ0| = `, and let Γ00 be an access
structure with Γ00 ⊆ Γ and |Γ \ Γ00| = `. Then there exist two critical subfamilies H0 and H00 for Γ0 and Γ00, respectively, with |H0| ≥ |H| − ` and |H00| ≥ |H| − `.
Proof. The families of subsets H0= H ∩ min Γ0 and H00= H ∩ Γ00are critical subfamilies for Γ0 and Γ00, respectively.