S. Pohlig and M. Hellman. An improved algorithm for computing discrete logarithms over gf(p) and its cryptographic significance. IEEE Transactions on Information Theory, 24:106--110, 1978.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....[x] in the curve. Next we explain how to find ff and fi as required in step 2. Recall that E a;b ZZ n Theta ZZ m where n = q fl 1 1 Delta Delta Delta q fl r r ; mjn and all the primes q i are less than d. To find ff; fi we use a simple generalization of the Pohlig Hellman algorithm [31] for discrete log in groups of smooth order. For simplicity we assume that all the fl i are equal to 1. The Pohlig Hellman algorithm generalizes to the case where fl i 1 as well. The method for finding ff; fi is to construct ff; fi (mod q i ) for all i = 1; r and then use chinese ....

....encryption scheme (D; E) over a plain text ring of size n can be broken in expected time O TBBF (n) exp( 1 o(1) 3 q log n log 2 log n) Proof To simplify the exposition we assume that n is square free. This restriction can be easily lifted using methods of Pohlig and Hellman [31] which will be discussed in section 4.3 (Lemma 4.3) Since one can factor integers in expected exp( 1 o(1) 3 q log n log 2 log n) time (see [20] it is possible to factor the plain text ring into a direct product of finite fields: ZZ n = Q s i=1 IF p i where the p i are distinct primes. ....

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S. Pohlig and M. Hellman. An improved algorithm for computing discrete logarithms over gf(p) and its cryptographic significance. IEEE Transactions on Information Theory, 24:106--110, 1978.

....that p Gamma 1 is smooth. This algorithm is analogous to the algorithm in Simon s paper [29] with the group Z k 2 replaced by Z p Gamma1 . The smooth case is not in itself an interesting accomplishment, since there are already polynomial time algorithms for classical computers in this case [25]; however, explaining this case is easier than explaining either the general case of discrete log or the factoring algorithm, and as the three algorithms are similar, this example will illuminate how the more complicated algorithms work. We will start our algorithm with x, g and p on the tape ....

S. Pohligand M. Hellman, "An improved algorithm for computing discrete logarithms over GF(p) and its cryptographic significance," IEEE Trans. Information Theory, Vol. 24, pp. 106--110 (1978).

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