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

Benny Chor; Oded Goldreich; Shafi Goldwasser

doi:10.1007/3-540-39799-X_35

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

Benny Chor, Oded Goldreich, Shafi Goldwasser

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

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.

Original language	English
Title of host publication	Advances in Cryptology — CRYPTO 1985 - Proceedings
Editors	Hugh C. Williams
Publisher	Springer Verlag
Pages	448-457
Number of pages	10
ISBN (Print)	9783540164630
DOIs	https://doi.org/10.1007/3-540-39799-X_35
State	Published - 1986
Externally published	Yes
Event	5th Annual International Cryptology Conference, CRYPTO 1985 - Santa Barbara, United States Duration: 18 Aug 1985 → 22 Aug 1985

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	218 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	5th Annual International Cryptology Conference, CRYPTO 1985
Country/Territory	United States
City	Santa Barbara
Period	18/08/85 → 22/08/85

Access to Document

10.1007/3-540-39799-X_35

Cite this

Chor, B., Goldreich, O., & Goldwasser, S. (1986). The Bit Security of Modular Squaring given Partial Factorization of the Modulos. In H. C. Williams (Ed.), Advances in Cryptology — CRYPTO 1985 - Proceedings (pp. 448-457). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 218 LNCS). Springer Verlag. https://doi.org/10.1007/3-540-39799-X_35

Chor, Benny ; Goldreich, Oded ; Goldwasser, Shafi. / The Bit Security of Modular Squaring given Partial Factorization of the Modulos. Advances in Cryptology — CRYPTO 1985 - Proceedings. editor / Hugh C. Williams. Springer Verlag, 1986. pp. 448-457 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

@inproceedings{81f8f83f2f2c4660bfcbb0572aa2e2cb,

title = "The Bit Security of Modular Squaring given Partial Factorization of the Modulos",

abstract = "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.",

author = "Benny Chor and Oded Goldreich and Shafi Goldwasser",

note = "Publisher Copyright: {\textcopyright} 1986, Springer-Verlag Berlin Heidelberg.; 5th Annual International Cryptology Conference, CRYPTO 1985 ; Conference date: 18-08-1985 Through 22-08-1985",

year = "1986",

doi = "10.1007/3-540-39799-X_35",

language = "אנגלית",

isbn = "9783540164630",

series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",

publisher = "Springer Verlag",

pages = "448--457",

editor = "Williams, {Hugh C.}",

booktitle = "Advances in Cryptology — CRYPTO 1985 - Proceedings",

}

Chor, B, Goldreich, O & Goldwasser, S 1986, The Bit Security of Modular Squaring given Partial Factorization of the Modulos. in HC Williams (ed.), Advances in Cryptology — CRYPTO 1985 - Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 218 LNCS, Springer Verlag, pp. 448-457, 5th Annual International Cryptology Conference, CRYPTO 1985, Santa Barbara, United States, 18/08/85. https://doi.org/10.1007/3-540-39799-X_35

The Bit Security of Modular Squaring given Partial Factorization of the Modulos. / Chor, Benny; Goldreich, Oded; Goldwasser, Shafi.
Advances in Cryptology — CRYPTO 1985 - Proceedings. ed. / Hugh C. Williams. Springer Verlag, 1986. p. 448-457 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 218 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

TY - GEN

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

AU - Chor, Benny

AU - Goldreich, Oded

AU - Goldwasser, Shafi

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

M3 - ???researchoutput.researchoutputtypes.contributiontobookanthology.conference???

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 -

Chor B, Goldreich O, Goldwasser S. The Bit Security of Modular Squaring given Partial Factorization of the Modulos. In Williams HC, editor, Advances in Cryptology — CRYPTO 1985 - Proceedings. Springer Verlag. 1986. p. 448-457. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/3-540-39799-X_35

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

Abstract

Publication series

Conference

Access to Document

Other files and links

Fingerprint

Cite this