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Chosen Ciphertext Attack
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In 2000, Jaulrnes and Joux [73]developed a clever chosen-ciphertext attack on NTRU. Their discovery resulted in changes to the recommended parameter sets by NTRU Cryptosystems, Inc. Since the attack also is effective on OAEPlikc padding, which was originally proposed for use with NTRU, other padding methods are now used with NTRU. For our overview of the attack, we will reference the NTRU parameter sets given in [138] and reproduced here in Table 6.4. Table 6.4: Previous Recommended NTRU Parameters
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Recall that the NTRU decryption process consists of first computing
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f(.) * M ( T ) = f ( ~ r) r )* h(z) f ( x ) * M ( x ) (mod 4 ) * ( = f ( z ) * p r ( ~*)f,(x) * g(z) f ( x ) * M ( z )(mod q )
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followed by f p ( x ) * u ( x ) (mod p ) which usually yields the plaintext message M ( z ) .For appropriate parameter choices. the coefficients of the polynomial p r ( x )*y(x) + f ( z ) * M ( x )lie iii the range -q/2 and q/2. Consequently, the polynomial pr*g(cr) + f(x)* M ( z )(mod q ) is the same as the truc (riorimodular) polynomial, t h a t is; the mod q has no effect. The idea of Jaulmes and Joux's chosen-ciphertext attack is to construct ciphertexts, which result in intermediate polynomials whose modular values differ from the true values. For cxample, suppose Trudy cliooses a ciphcrtext polynomial which is of tlic form C ( x ) = yh(5) y, where y is an integer and h(z)is Alice's public key. The NTRU decryption algorithm yields
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* C(n:)= f ( x )* y h ( z ) + yf(z) (mod = yyf(.z.) * h ( z )+ y f ( ~ ) (mod = /dJ..) ) (mod 411 +yf(x
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where f ( x ) and g(z) both have coefficients in (0, 1, -I}. It follows that the polynomial jy(z) y f ( z ) has coefficicnts in (0, y, -y, 2y, -2y}. If Trudy has chosen y so that>y < q / 2 arid 2y > q / 2 . then the decryption process reduces
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only the coefficients equal to f 2 y modulo q and these coefficients are selected so as to lie between - q / 2 and q/2. Now suppose that a single coefficient of u ( z )is f 2 y , say, ai = +2y. Then u ( z ) (mod q ) = yg(z) y f ( z ) - qzz, and the final decrypted output is
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Furthermore, if Trudy chose y to be a multiple of p , then the final decrypted output collapses to
z ( z ) = - f p ( x ) * 9x2
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Trudy then recovers Alice s private key f ( x ) by computing
f ( z ) = -422
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In general, the polynomial y f ( z ) yyg(z) may have none or several coefficients equal to f 2 y . In these cases, the above attack would not work. However, this chosen-ciphertext attack can be generalized and it can be practical, even for stronger security parameters [138]. The intersection polynomial w(x) of polynomials u(x) and u(x) is defined to be 2 W(.) = wo W 1 2 2 0 2 2 ... wN-IxN-l,
We say that polynomials u ( x ) and v ( x )have a collision when they have the same non-zero coefficient in a corresponding term. In the attack discussed above, the intersection polynomial of f ( x )and g ( 2 ) was the polynomial w(x) = xi,that is, f ( x )and g(z) had one collision. Using the security parameters corresponding to N = 107 in Table 6.4, Jaulmes and Joux found that the probability of one collision occurring between f ( x ) and g ( 2 ) was 0.13. Therefore, for this particular choice of parameters, the chosen ciphertext attack using C ( x )= yh(z) y is successful approximately thirteen percent of the time and, in these instances, it easily recovers f ( z ) . For higher security parameters, the number of expected collisions between f ( x )and g(%) is too high and Alice s private key f ( x )cannot be recovered in this manner when using the chosen ciphertext C(z) = yh(z) y. In these cases, chosen ciphertext messages of the form