Constructions of almost secure frameproof codes with applications to fingerprinting schemes

Q: what is the c-secure with -error family of binary codes?

the set of keys T , and π(j) = |T |−1. For q > c2, and any Rout, σ satisfyingRout < 1− σ c(c+ 1) , εin < σ < q − c2 q − c ,there exists a c-secure with ε-error family of binary codes C = {Cj}j∈T with length m = n l, rateR = RoutRin,error probability ε decreasing exponentially asε ≤ 2−m( 1−σ c Rin−(c+1)R+o(1)) + 2−qD(σ‖εin),and with a polynomial-time identification algorithm that runs in O(poly(m)).

Q: What is the family of fingerprinting codes proposed in [3]?

The family of fingerprinting codes proposed in [3] particularizes Construction 2 with Cout being a Reed-Solomon or an algebraic-geometric code, and Cin being a (c, c)-separating (i.e., c-secure frameproof) code.

Q: What is the convention for constructing -away from t-wise independence arrays?

(1)The arrays from the previous theorem can be used to construct ε-away from t-wise independence arrays with n = 2O(t+logM+log 1 ε ), which, in turn, lead to (M, t)-universal sets of size n = 2O(t+logM).

Q: What is the reason why the authors are interested in almost secure frameproof codes?

Since the authors are interested in almost secure frameproof codes, the authors do not require absolute separation, and the authors can tolerate a small bias on the elements of the probability vector p.

Q: what is the t-wise independence of a subset of a?

In other words, there is a subset S ⊆ {1, . . . ,M} of size t for which there is a vector a′ ∈ Ft2, with p = νS(a′;A)/n, such that|p− 2−t| > ε 2 .Substituting m = 1 and |p − 2−t| in Lemma 2 shows that A is not ε-away from t-wise independence, which contradicts the starting premise.

Question

1. What are the contributions mentioned in the paper "Constructions of almost secure frameproof codes with applications to fingerprinting schemes" ?

2. What are the future works in "Constructions of almost secure frameproof codes with applications to fingerprinting schemes" ?

3. What is the t-wise independence of a binary array?

4. What is the construction of the n,m-array?

Accepted Answer

This paper presents explicit constructions of fingerprinting codes.

Accepted Answer

Finally, the authors would like to note that even though a universal set is a separating code, the relationship between an almost universal set and an almost separating code is by no means evident and will they the subject of future research.

Accepted Answer

If M random variables X1, . . . , XM take uniformly at random the corresponding values of a row of a binary (n,M)-array A which is ε-away from t-wise independence, then any s ≤ t of such random variables behave like “almost independent” random variables, provided that ε is small.

Accepted Answer

If log2M > 3 log2 1 ε , then there is an explicit construction of a binary (n,M)-array that is ε-biased, withn ≤ 23/2 (log2M+log2 1 ε )+2.

Accepted Answer

Let c ≥ 2, t, M be integers such that M ≥ 2c, and let one of the following conditions be satisfiedCondition 1: c ≤ t < 2c, and 0 ≤ ε′ < 2−c+1, or Condition 2: t ≥ 2c, and 2−2c+1 + 2−t < ε′ < 2−c+1.Then, an ε′-away from t-wise independence (n,M)-array generates an ε-almost c-secure frameproof (n,M)-code, for anyε ≥Mc(1− 2−c + ε′/2)n. (5)Proof Let A be a binary (n,M)-array which is ε′-away from t-wise independence, and regard its columns as the codewords of a code C.First, observe that under the conditions given in the statement of the proposition, the array A need not yield a c-secure frameproof code.

Accepted Answer

the set of keys T , and π(j) = |T |−1. For q > c2, and any Rout, σ satisfyingRout < 1− σ c(c+ 1) , εin < σ < q − c2 q − c ,there exists a c-secure with ε-error family of binary codes C = {Cj}j∈T with length m = n l, rateR = RoutRin,error probability ε decreasing exponentially asε ≤ 2−m( 1−σ c Rin−(c+1)R+o(1)) + 2−qD(σ‖εin),and with a polynomial-time identification algorithm that runs in O(poly(m)).

Accepted Answer

The family of fingerprinting codes proposed in [3] particularizes Construction 2 with Cout being a Reed-Solomon or an algebraic-geometric code, and Cin being a (c, c)-separating (i.e., c-secure frameproof) code.

Accepted Answer

(1)The arrays from the previous theorem can be used to construct ε-away from t-wise independence arrays with n = 2O(t+logM+log 1 ε ), which, in turn, lead to (M, t)-universal sets of size n = 2O(t+logM).

Accepted Answer

Since the authors are interested in almost secure frameproof codes, the authors do not require absolute separation, and the authors can tolerate a small bias on the elements of the probability vector p.

Accepted Answer

In other words, there is a subset S ⊆ {1, . . . ,M} of size t for which there is a vector a′ ∈ Ft2, with p = νS(a′;A)/n, such that|p− 2−t| > ε 2 .Substituting m = 1 and |p − 2−t| in Lemma 2 shows that A is not ε-away from t-wise independence, which contradicts the starting premise.

Accepted Answer

On the other hand, an upper bound on the asymptotic rate is< 0.2835 for arbitrary (2, 2)-separating codes [22,16], and it was shown in [22] that it is < 0.108 for linear (2, 2)-separating codes.

Accepted Answer

the probability that z is generated by some other disjoint c-coalition, or equivalently, the ratio of descendants z ∈ descc(C) that are not c-uniquely decodable is at most Mc(1−2−c+ ε′/2)n.

Accepted Answer

The fact that the secure frameproof property is required with high probability, rather that in all the cases, is the main reason that allows us to obtain an improvement on the code rate, compared to ordinary secure frameproof (i.e., separating) codes.

Constructions of almost secure frameproof codes with applications to fingerprinting schemes

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Most frequently asked questions

1. What are the contributions mentioned in the paper "Constructions of almost secure frameproof codes with applications to fingerprinting schemes" ?

2. What are the future works in "Constructions of almost secure frameproof codes with applications to fingerprinting schemes" ?

3. What is the t-wise independence of a binary array?

4. What is the construction of the n,m-array?

5. What is the condition for a c-secure frameproof code?

6. what is the c-secure with -error family of binary codes?

7. What is the family of fingerprinting codes proposed in [3]?

8. What is the convention for constructing -away from t-wise independence arrays?

9. What is the reason why the authors are interested in almost secure frameproof codes?

10. what is the t-wise independence of a subset of a?

11. What is the upper bound on the asymptotic rate of a binary alphabet?

12. What is the probability that z is generated by a disjoint c-?

13. What is the main reason why the authors need to use secure frameproof codes?

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Citations

titleCombinatorial Constructions of Separating Codes

References

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Small-bias probability spaces: efficient constructions and applications

Simple Constructions of Almost k-wise Independent Random Variables

Towards 3-query locally decodable codes of subexponential length

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