Signatures detect changes to data objects. Numerous schemes are in use, especially the cryptographically secure standards SHA-1. We propose a novel signature scheme which we call algebraic signatures. The scheme uses the Galois Field calculations. Its major property is the sure detection of any changes up to a parameterized size. More precisely, we detect for sure any changes that do not exceed n-symbols for an n-symbol algebraic signature. This property is new for any known signature scheme. For larger changes, the collision probability is typically negligible, as for the other known schemes. We apply the algebraic signatures to the Scalable Distributed Data Structures (SDDS). We filter at the SDDS client node the updates that do not actually change the records. We also manage the concurrent updates to data stored in the SDDS RAM buckets at the server nodes. We further use the scheme for the fast disk backup of these buckets. We sign our objects with 4-byte signatures, instead of 20-byte standard SHA-1 signatures. Our algebraic calculus is then also about twice as fast.