ECDSA Signature Compression via Public Key Inversion
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Solution Overview
Problem
Existing ECDSA signature verification methods incur significant additional costs due to the need for extra elliptic curve group operations when compressing signatures, which is undesirable in bandwidth-limited applications like two-dimensional barcodes and RFID tags.
Innovation Solution
A method is introduced to compress ECDSA signatures by substituting the integer s with a smaller value c, where s ≡ c^d mod n, using the extended Euclidean algorithm to find suitable values for c and d, allowing for verification without the private key, and reducing the number of elliptic curve group operations required.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Loss of substance
If ECDSA signatures are compressed by truncating s by omitting 2b bits, then bandwidth is conserved, but verification cost increases by approximately 2^2b elliptic curve group operations
Solution Approach 1:
Instead of truncating the signature component s and paying the verification cost, the patent inverts the approach by compressing the public key Q to q by omitting bits. This shifts the compression burden to the public key while maintaining verification efficiency, as the verifier only needs to perform standard ECDSA verification operations without the exponential overhead associated with truncated signatures.
Solution Approach 2:
The patent extracts and removes redundant bits from the public key Q to create a compressed version q. By identifying and eliminating the most significant bits that can be derived or reconstructed, the public key size is reduced without compromising the security or functionality of the ECDSA verification process.
2Loss of substance
If signature compression is implemented to conserve bandwidth in applications like two-dimensional barcodes and RFID tags, then transmission efficiency improves, but verification complexity increases
Solution Approach 1:
The patent applies compression to the public key rather than the signature, inverting the traditional approach. This allows bandwidth conservation in bandwidth-constrained environments while avoiding the exponential verification complexity that would result from signature truncation, making it suitable for applications like RFID tags and two-dimensional barcodes.
3Quantity of substance
If the signature component s is truncated to reduce size, then bandwidth consumption decreases, but the number of elliptic curve group operations required for verification increases exponentially
Solution Approach 1:
Rather than truncating the signature component s which causes exponential verification overhead, the patent compresses the public key Q to q. This inversion of the compression target maintains constant verification complexity while achieving the desired reduction in transmitted data size.
Data Source
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AI summary
An improved compression scheme for compressing an ECDSA signature is provided. The scheme substitutes the integer s in a signature (r, s) by a smaller value c. The value c is derived from s and another value d, d being small enough such that c is smaller than s. The compressed signature (r, c) is verified by computing a value using r and e, e being a hash of a message m, and using this value with a value R recovered from r to derive the value d. The value s can then be recovered and the full signature then recovered and verified.