Michael O. Rabin and Je#ery O. Shallit. Randomized algorithms in number theory. Communications on Pure and Applied Mathematics, 39:239--256, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... that an integer is positive by proving knowledge of four values the squares of which sum up to the considered integer (in Z) again under the strong RSA assumption using additional parameters (n; g; h) Lagrange proved the an integer can always be represented as four squares and Rabin and Shallit [36] provide an ecient algorithm for it. We note that in our case the interval is symmetric and it therefore suces to prove that ( n 1) 2) 0 holds, which is more ecient. With these observations one can derive the following protocol for veri able decryption of a discrete logarithm from the ....

....1. If 62 or the ciphertext is malformed (e.g. if v 6= abs(v) the veri er outputs 1, and the protocol stops. In case (u; e; v) is a valid ciphertext w.r.t. label L, the prover decrypts it, thereby obtain m, and computes integers w 1 ; w 4 such that i=1 w i = n 1) 4 m (c.f. [36]) 25 2. If (u; e; v) indeed decrypts to log with label L, i.e. if = the decryptor sends 1 to the veri er, chooses t 1 ; t 5 2R [n=4] computes W 1 : g ; and M : g and sends W 1 , W 2 , W 3 , W 4 , and M to the veri er. The prover and the veri er ....

M. O. Rabin and J. O. Shallit. Randomized algorithms in number theory. Communications on Pure and Applied Mathematics, 39:239-256, 1986.

.... 0 b x 0. If C = g then let C a = C=g mod n, and C b = g =C mod n. Now we have to show that C a and C b are both commitment to positive integers. 2. It is a well known fact due to Lagrange that any non negative integer can be represented as a sum of four squares. Rabin and Shallit [RS86] gave an ecient algorithm for computing this representation. Thus the task is to show that a committed number is a sum of squares. 3. Showing that a committed number is a sum of several other committed values is straightforward. Let X and Y be two commitments. To show that a commitment Z is to ....

Michael O. Rabin and Jerey O. Shallit. Randomized algorithms in number theory. Communications in pure and applied mathematics, 39:239{ 256, 1986.

.... and proves in statistical zero knowledge that the committed number is equal to (mod jMj) 2) Prover finds a representation = 1 2 3 4 of . Such representation exists iff 0 as shown by Lagrange. An efficient algorithm for finding i was proposed by Rabin and Shallit [RS86]. Prover commits to ( 1 ; 2 ; 3 ; 4 ) and then proves in statistical zero knowledge that P 4 i=1 i = With suitable security parameters, a noninteractive version of this proof is 3366 8 dlog 2 He bytes long. For more details see [Lip01] 4 Auxiliary Proofs Range Proof in ....

Michael O. Rabin and Jeffrey O. Shallit. Randomized Algorithms in Number Theory. Communications in Pure and Applied Mathematics, 39:239--256, 1986.

....completeness error. In the current paper, we propose a very different range proof for [0; 1) Our proof system bases on the well known result of Lagrange that every nonnegative integer is a sum of four squares and on the algorithm of Rabin and Shallit that computes these squares efficiently [RS86]. On the other hand, a negative integer cannot be represented as such a sum. With realistic parameters, our proof system requires about 20 more communication than Boudot s proof system but is perfectly complete. Furthermore, one can use the same methodology as a novel framework to prove that a ....

....An integer can be represented as = 2 1 2 2 2 3 2 4 with integer i iff 0. Moreover, if 0 then the representation ( 1 ; 2 ; 3 ; 4 ) can be computed efficiently. 5 Proof. If 0, such i exist by a well known result of Lagrange from 1770. Rabin and Shallit [RS86] proposed a probabilistic polynomial time algorithm for computing the representation. On the other hand, no negative number is a sum of four squares. ut Briefly, during our proof system for [0; 1) prover first uses the Rabin Shallit algorithm to represent as 2 1 2 2 2 3 2 4 . ....

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Michael O. Rabin and Jeffrey O. Shallit. Randomized Algorithms in Number Theory. Communications in Pure and Applied Mathematics, 39:239--256, 1986.

....completeness error. In the current paper, we propose a very different range proof for [0, #) Our proof system bases on the well known result of Lagrange that every nonnegative integer is a sum of four squares and on the algorithm of Rabin and Shallit that computes these squares efficiently [RS86]. On the other hand, a negative integer cannot be represented as such a sum. With realistic parameters, our proof system requires about 20 more communication than Boudot s proof system but is perfectly complete. Furthermore, one can use the same methodology as a novel framework to prove that a ....

....Theorem 1. An integer can be represented as = 2 1 2 2 2 3 2 4 with integer i iff # 0. Moreover, if # 0 then the representation ( 1 , 2 , 3 , 4 ) can be computed efficiently. 5 Proof. If # 0, such i exist by a well known result of Lagrange from 1770. Rabin and Shallit [RS86] proposed a probabilistic polynomial time algorithm for computing the representation. On the other hand, no negative number is a sum of four squares. ## Briefly, during our proof system for [0, #) prover first uses the Rabin Shallit algorithm to represent as 2 1 2 2 2 3 2 4 . ....

[Article contains additional citation context not shown here]

Michael O. Rabin and Jeffrey O. Shallit. Randomized Algorithms in Number Theory. Communications in Pure and Applied Mathematics, 39:239--256, 1986.

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Michael O. Rabin and Je#ery O. Shallit. Randomized algorithms in number theory. Communications on Pure and Applied Mathematics, 39:239--256, 1986.

No context found.

Michael O. Rabin and Jerey O. Shallit. Randomized Algorithms in Number Theory. Communications in Pure and Applied Mathematics, 39:239{ 256, 1986.

No context found.

M. O. Rabin and J. O. Shallit, Randomized algorithms in number theory, Communications on Pure and Applied Mathematics 39 (1986), 239--256.

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