# โณ๏ธAdversaries and message query theorems

Our innovative message query theorems redefine the boundaries of data integrity, ensuring that every piece of information exchanged within the Solareum network is not only secure but also transparently verifiable. We believe that in a world increasingly reliant on decentralized solutions, technical excellence in security is non-negotiable.

**Theorem 1** Let A be an adversary attacking S โฒ that makes no chosen message queries and at most one query to H1. Let วซ = SIGadv[A, S โฒ ; 0, QH0 , 1] be its advantage. Then there exists an adversary B for computing co-CDH, whose running time is about twice that of A, with advantage วซ โฒ = CDHadv[B โฒ (G0), G1] such that วซ โฒ โฅ วซ 2 โ วซ/N, where N = |R|, is the size of one coordinate in the image of H1, thus วซ โค (1/N) + โ วซ โฒ .

**Theorem 2** Let A be an adversary attacking S โฒ that makes no chosen message queries but potentially many queries to H1. Then there exists an adversary B attacking S โฒ , that makes only a single query to H1, and whose running time is about the same as A, such that

**Theorem 3** Let A be an adversary attacking S โฒ . Then there exists an adversary B attacking S โฒ , that makes no chosen message queries and whose running time is about the same as A, such that

**Corollary 1** For every adversary A attacking S there is a co-CDH algorithm B, whose running time is about twice that of A, such that

The proofs of which are left as an exercise to the reader.

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