TY - GEN

T1 - The Bit Security of Modular Squaring given Partial Factorization of the Modulos

AU - Chor, Benny

AU - Goldreich, Oded

AU - Goldwasser, Shafi

N1 - Publisher Copyright:
© 1986, Springer-Verlag Berlin Heidelberg.

PY - 1986

Y1 - 1986

N2 - It is known that given a composite integer N = p 1 p 2 (such that p 1 ≡ p 2 ≡ 3 (mod 4)), and q a quadratic residue modulo N, guessing the least significant bit of a square root of q with any non-negligible advantage is as hard as factoring N. In this paper we extend the above result to multi-prime numbers N = p 1 p 2..p l (such that p 1 ≡ p 2 ≡.. ≡ p l ≡ 3 (mod 4)). We show that given N and q 1 a quadratic residue mod N, guessing the least significant bit of a square root of q is as hard as completely factoring N. Furthermore, the difficulty of guessing the least significant bit of the square root of q remains unchanged even when all but two of the prime factors of N, p 3,..,p l, are known. The result is useful in designing multi-party cryptographic protocols.

AB - It is known that given a composite integer N = p 1 p 2 (such that p 1 ≡ p 2 ≡ 3 (mod 4)), and q a quadratic residue modulo N, guessing the least significant bit of a square root of q with any non-negligible advantage is as hard as factoring N. In this paper we extend the above result to multi-prime numbers N = p 1 p 2..p l (such that p 1 ≡ p 2 ≡.. ≡ p l ≡ 3 (mod 4)). We show that given N and q 1 a quadratic residue mod N, guessing the least significant bit of a square root of q is as hard as completely factoring N. Furthermore, the difficulty of guessing the least significant bit of the square root of q remains unchanged even when all but two of the prime factors of N, p 3,..,p l, are known. The result is useful in designing multi-party cryptographic protocols.

UR - http://www.scopus.com/inward/record.url?scp=85034659465&partnerID=8YFLogxK

U2 - 10.1007/3-540-39799-X_35

DO - 10.1007/3-540-39799-X_35

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AN - SCOPUS:85034659465

SN - 9783540164630

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 448

EP - 457

BT - Advances in Cryptology — CRYPTO 1985 - Proceedings

A2 - Williams, Hugh C.

PB - Springer Verlag

T2 - 5th Annual International Cryptology Conference, CRYPTO 1985

Y2 - 18 August 1985 through 22 August 1985

ER -