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By Alt R., Vignes J.

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5σ 2 2n−τ ±prp Advauth (t , 2σ) + n + n OCB1[E,τ ] (t, σ) ≤ AdvE 2 2 −1 ✷ where t = t + cnσ for some absolute constant c. Propositions 1 and 2 establish that n = 128 and n = 64 satisfy the requirement for unique representations. They also guarantee that there is no representative of 1 within [1 .. 2n/2 ] × {0, 1}. To see this, note that the propositions imply that (0, 0) is the only representative for 1 within a space I1 × I2 that includes [1 .. 2n/2 ] × {0, 1}, and so there can be no representative of 1 within a subspace of I1 × I2 that excludes (0, 0).

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K ∈ G of bases and a set I1 × · · · × Ik ⊆ Zk of allowed indices. We say that the choice of parameters provides unique representations if for every (i1 , . . , ik ), (j1 , . . , jk ) ∈ I1 × · · · × Ik we have that ✷ α1i1 · · · αkik = α1j1 · · · αkjk implies (i1 , . . , ik ) = (j1 , . . , jk ). In other words, representable points are uniquely representable: any group element α1i1 · · · αkik that can be represented using allowed indices can be represented in only one way (using allowed indices).

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10th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics by Alt R., Vignes J.

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