Pairing-Based Broadcast Encryption with Compact Collusion-Secure Parameters
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Solution Overview
Problem
Existing broadcast encryption schemes face challenges in achieving short parameters, particularly in ciphertexts, public keys, and secret keys, while maintaining adaptive security against unbounded collusions, which is essential for ensuring that adversaries with arbitrary attributes do not learn the message content.
Innovation Solution
A pairing-based broadcast encryption scheme with O(N 1/3)-sized parameters is developed, utilizing a novel approach that encodes set membership using degree 3 polynomials and employs new techniques for quadratic reconstruction, achieving adaptive security under the bilateral k-Lin assumption in prime-order bilinear groups.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If pairing-based broadcast encryption schemes use traditional construction methods, then security against unbounded collusions is achieved, but parameter sizes (ciphertext, public key, secret key) become linear in N
Solution Approach 1:
The patent segments the attribute space by representing sets as degree-3 polynomials over finite fields, where each element in the universe can be represented as a point in a 3-dimensional space. This segmentation allows the system to work with polynomial evaluations rather than direct set representations, reducing parameter sizes from linear to cubic root scaling.
Solution Approach 2:
The patent changes the fundamental parameter representation from direct set membership indicators to polynomial coefficients and evaluation points. By transforming the problem into polynomial arithmetic over finite fields, the system achieves O(N^(1/3)) parameter sizes while maintaining security, representing a fundamental parameter change from linear to sublinear scaling.
2Quantity of substance
If the scheme uses degree-3 polynomial encoding for set membership, then parameter sizes are reduced to O(N^(1/3)), but the construction complexity increases
Solution Approach 1:
The patent introduces polynomial arithmetic as an intermediary layer between set membership and encryption operations. Instead of directly working with set representations, the system uses polynomial evaluations at specific points as intermediaries, which simplifies the overall construction and enables efficient computation despite the mathematical sophistication required.
Solution Approach 2:
The patent replaces traditional cryptographic construction mechanisms with polynomial-based algebraic structures. By substituting conventional encryption building blocks with polynomial arithmetic in bilinear groups, the system achieves more efficient parameter sizes while relying on well-established mathematical assumptions like the Decisional Linear assumption.
3Productivity
If the scheme achieves adaptive security with O(N^(1/3)) parameters, then efficiency is improved, but the requirement for bilinear groups and specific cryptographic assumptions increases
Solution Approach 1:
The patent moves the cryptographic construction into a different mathematical dimension by utilizing bilinear groups and pairing-based cryptography. This dimensional shift from standard symmetric or asymmetric cryptography to pairing-based systems in bilinear groups enables the O(N^(1/3)) efficiency while providing strong security guarantees under the Decisional Linear assumption.
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AI summary
A broadcast network can use a pairing-based broadcast encryption scheme for N users with O(N 1/3) -sized parameters. A pairing-based ciphertext-policy attribute-based encryption (CP-ABE) scheme for the class of degree 3 polynomials can be implemented with compact parameters. The public key, ciphertext and secret keys comprise O(n) group elements, where n is input length for the function. The constructions achieve adaptive security against unbounded collusions, and rely on the (bilateral) k-Lin assumption in prime-order bilinear groups.