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Then Alice s public key is the polynomial h ( z )= fq(z) *g(z) (mod q ) . The ploynomial h ( z ) ,along with the parameters N , p , and 4 , are made public. Alice s private key consists of t.he polynomials f(x) and f,(z). To summarize; we have Public key: h(z) Private key: ( f ( z )f,(z)) , where h(z)= fq(z) g(z) (mod q ) and f ( z )* f,(z) = 1 (mod q ) . * Bob sends Alice an encrypted as follows. Bob first selects a polynomial M ( z ) E L , that represents the plaintext message. Recall that the coefficients of the message ploynomial M ( z ) are in the range - ( p - 1)/2 and ( p - l ) / 2 and that q is much larger than p . Consequently, the message M ( z ) can be viewed as a small polynomial modulo q , in the sense that the vector of coefficients has small Euclidean length. Bob then chooses a random blinding polynomial r ( z ) E L, and uses Alice s public key to compute the ciphertext message C(z) (also a polynomial) as
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which he sends to Alice. To decrypt Bob s message, Alice computes
The coefficients of a(.) are chosen to be in the interval -4/2 to q / 2 (it is crucial that the coefficients be taken in this interval before the next step in the decryption). Then Alice computes b(z)= u ( z ) (mod p ) . Although it is not obvious, Alice recovers the message M ( z ) by computing
Below we give an intuitive explanation why NTRU decryption works, but first we give an example. To illustrate the NTRU algorithm, we use the example found at [108]. Suppose that we select NTRU parameters N = 11, q = 32, p = 3, and the sets of polynomials L f = L(4,3), L, = L ( 3 , 3 ) , and L, = L( 3, 3) . Then to generate her private key, Alice selects a polynomial f ( z ) E L f , that is, a polynomial of degree ten with four +1 coefficients and three -1 coefficients, and all remaining coefficients set to 0. She also chooses a polynomial g(z), where g(z) E L,. Suppose that the selected polynomial are f ( z ) = -1
g ( z ) = -1
+ z + x2 z4 + z6 + 2 9 z10 E L f + x2 + z3+ z5 - 2 8 5 1 0 E L,.
PUBLIC K E Y SYSTEMS
Next, Alice computes f,(z) and fq(. :), the inverses of f ( x ) modulo p and q. respectively. Using the algorithm in Table 6.3, she finds
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Alice's private key consists of the pair of polynomials ( f ( z )f,, ( z ) ) . To generate her public key h ( z ) ,Alice computes
h(x) = P f & )