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K let Ni ∈ N be pairwise relatively prime numbers, and let fi (x) ∈ Z[x] be polynomials of degree δ. k Then there exists a unique polynomial f (x) modulo M := i=1 Ni such that f (x) ≡ fi (x) (mod Ni ) (2) The polynomial f (x) can be determined in time O(δ log2 M ). k M Proof. Let M := i=1 Ni , Mi := N and Mi be the inverse of Mi modulo Ni for i i = 1, . . , k. The existence of such an inverse is guaranteed by gcd(Mi , Ni ) = 1. Then k Mi Mi fi (x) f (x) = i=1 Solving Systems of Modular Equations in One Variable 41 is the desired solution.

2567, pp. 62–70. Springer, Heidelberg (2002) 18. ps 19. : Analysis and Improvements of NTRU Encryption Paddings.. In: Yung, M. ) CRYPTO 2002. LNCS, vol. 2442, pp. 210–225. Springer, Heidelberg (2002) 20. : Trading One-Wayness Against Chosen-Ciphertext Security in Factoring-Based Encryption.. , Chen, K. ) ASIACRYPT 2006. LNCS, vol. 4284, pp. 252–266. Springer, Heidelberg (2006) 21. : Digital Signatures and Public-Key Functions as Intractable as Factorization. 1 Proof. For each pair (ri , mi ) ∈ preimg(e), we define ai = p ∗ g ∗ ri + f ∗ mi where, as usual, f, g are the secret and auxiliary key respectively.

In table 2 we summarize the main instantiations of NTRU1 (for further details the reader is referred to [5, Section 2]). Sometimes, for efficiency reasons, a combination of the above sets might be used. For example in NTRU-2001 q might be a prime or in NTRU-2005 Lr and F might belong in X (d) which denotes the set of (binary) polynomials of the from b1 + b2 ∗ b3 where bi are very sparse binary polynomials with d 1s. 1 Recently, in order to secure against attacks presented in [11], the NTRU parameters have been revised in [7].

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Block Ciphers And Cryptanalysis by FauzanMirza

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