On Negation Complexity of Injections, Surjections and Collision-Resistance in Cryptography

  title={On Negation Complexity of Injections, Surjections and Collision-Resistance in Cryptography},
  author={Douglas Miller and Adam Scrivener and Jesse Stern and Muthuramakrishnan Venkitasubramaniam},
  journal={IACR Cryptol. ePrint Arch.},
Goldreich and Izsak (Theory of Computing, 2012) initiated the research on understanding the role of negations in circuits implementing cryptographic primitives, notably, considering one-way functions and pseudo-random generators. More recently, Guo, Malkin, Oliveira and Rosen (TCC, 2015) determined tight bounds on the minimum number of negations gates (i.e., negation complexity) of a wide variety of cryptographic primitives including pseudo-random functions, error-correcting codes, hardcore… 



The Power of Negations in Cryptography

The study of monotonicity and negation complexity for Bool-ean functions has been prevalent in complexity theory as well as in computational learning theory, but little attention has been given to it

Limiting negations in non-deterministic circuits

Learning circuits with few negations

This paper studies the structure of Boolean functions in terms of the minimum number of negations in any circuit computing them, a complexity measure that interpolates between monotone functions and the class of all functions.

More on the complexity of negation-limited circuits

A main technical lemma on functions computed at NEGATION gates in negationlimited circuits computing symmetric functions is proved, and a number of lower bounds on the size and depth of negation-limited circuits are shown.

On the complexity of negation-limited Boolean networks

The theorem of Markov precisely determines the number r of NEGATION gates necessary and sucient to compute a system of boolean functions F, and some lower bound techniques for negation-limited circuits are introduced.

The gap between monotone and non-monotone circuit complexity is exponential

The proof is immediate by combining the Alon—Boppana version of another argument of Razborov with results of Grötschel—Lovász—Schrijver on the Lovász — capacity of a graph.

Limiting Negations in Constant Depth Circuits

It is proven that when polynomial size circuit families of constant depth are considered: l negations are no longer sufficient and there is a matching upper bound: for any $\epsilon > 0$, everything computable by constant depth threshold circuits can be computed by constant Depth threshold circuits using negations asymptotically.

Universal one-way hash functions and their cryptographic applications

A Universal One-Way Hash Function family is defined, a new primitive which enables the compression of elements in the function domain and it is proved constructively that universal one- way hash functions exist if any 1-1 one-way functions exist.

A Superpolynomial Lower Bound for a Circuit Computing the Clique Function with At Most (1/6) log log n Negation Gates

A new approach is developed by combining the three approaches: the restriction applied for constant depth circuits[Has], the approximation method applied for monotone circuits[Raz2] and boundary covering developed in the present paper, to derive strong lower bounds on the size of a Boolean circuit that computes the clique function with a limited number of negation gates.