Cisco - New World Operations

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3 The Concentration of the ith Bit for Certain N In the previous section we found an expression of the asymptotic behavior of the coefficients of P which is hard to interpret. It is clear that the heavy coefficients of P will be around the points that annihilate the denominator, but otherwise it is not trivial to show that there exists a set Γ of size poly(n/ ) such that ||P − P|Γ ||22 ≤ . In this section we prove that if N = r2i+1 ± m and either r ∈ poly(n) or m ∈ poly(n) then P is concentrated. The result is a consequence of the two following lemmas.

Lemma 9. The following map is injective π : [− N2−1 , N2−1 ] −→ α −→ [0, 2i−1 − 1] ×[0, r] × {±1} × {±1} μα , sα , sδ ) ( λα , where sα = 1 if α ≥ 0 and −1 otherwise and sδ = 1 if δα ≥ 0 and −1 otherwise. Proof. Reducing equations 3 and 4 modulo 2i it is clear that λα and sδ determine δα modulo 2i and therefore δα . Then α can be computed from π(α) using equations 5 or 6. 2 The Concentration of the ith Bit for All N As a result of lemma 9 of the above section, we can describe the elements of ZN as regions in [0, 2i−1 − 1] × [0, r] × {±1} × {±1}.

A problem of computing d from e, N and some partial information on d. This approach has been further extended by Ernst et al. [7] for several partial key exposure situations. [7], thereby solving an open problem raised in their paper. In order to state our improvement we need some notations; see the next section for the precise definition. Let (e, N ) be an RSA public key pair and let d be its corresponding secret key. Here as usual we use N = (the bit-length of N ) as S. Jarecki and G. ): PKC 2009, LNCS 5443, pp.

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