### Abstract

We consider some attacks on multi-prime RSA (MPRSA) with a modulus N = p1 p2 . . . pr (r ≥ 3). It is believed that the small private exponent attack on the MPRSA is less effective than that on RSA (see Hinek et al.'s work at SAC 2003), which means smaller private exponents can be used in the MPRSA to speed up the decryption process. Our work shows that even if a private exponent is significantly beyond Hinek et al.'s bound, it still may be insecure if the prime difference Δ (Δ = pr - p1 = Nγ, supposing p1 < p2 < · · · < pr ) is small, i.e. 0 < γ < 1/r. Specifically, by taking full advantage of prime properties, our small private exponent attack reveals that the MPRSA is insecure when δ < 1 - √ 1 + 2γ - 3/r (if γ ≥ 3 2r - 1+d/4 ) or δ ≤ 3r - 1 4 -2 (if γ < 3 2r - 1+δ/4 ), where δ is the exponential of the private exponent d with base N, i.e., d = Nδ. In addition, we present a Fermat-like factoring attack which factors N efficiently when Δ < N1/r^{2} . These proposed attacks surpass previous works (e.g. Bahig et al.'s at ICICS 2012), and are proved effective in practice.

Original language | English |
---|---|

Pages (from-to) | 1533-1541 |

Number of pages | 9 |

Journal | IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences |

Volume | E97-A |

Issue number | 7 |

DOIs | |

Publication status | Published - Jan 1 2014 |

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### All Science Journal Classification (ASJC) codes

- Signal Processing
- Computer Graphics and Computer-Aided Design
- Electrical and Electronic Engineering
- Applied Mathematics

### Cite this

*IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences*,

*E97-A*(7), 1533-1541. https://doi.org/10.1587/transfun.E97.A.1533

**Improved attacks on multi-prime rsa with small prime difference.** / Zhang, Hui; Takagi, Tsuyoshi.

Research output: Contribution to journal › Article

*IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences*, vol. E97-A, no. 7, pp. 1533-1541. https://doi.org/10.1587/transfun.E97.A.1533

}

TY - JOUR

T1 - Improved attacks on multi-prime rsa with small prime difference

AU - Zhang, Hui

AU - Takagi, Tsuyoshi

PY - 2014/1/1

Y1 - 2014/1/1

N2 - We consider some attacks on multi-prime RSA (MPRSA) with a modulus N = p1 p2 . . . pr (r ≥ 3). It is believed that the small private exponent attack on the MPRSA is less effective than that on RSA (see Hinek et al.'s work at SAC 2003), which means smaller private exponents can be used in the MPRSA to speed up the decryption process. Our work shows that even if a private exponent is significantly beyond Hinek et al.'s bound, it still may be insecure if the prime difference Δ (Δ = pr - p1 = Nγ, supposing p1 < p2 < · · · < pr ) is small, i.e. 0 < γ < 1/r. Specifically, by taking full advantage of prime properties, our small private exponent attack reveals that the MPRSA is insecure when δ < 1 - √ 1 + 2γ - 3/r (if γ ≥ 3 2r - 1+d/4 ) or δ ≤ 3r - 1 4 -2 (if γ < 3 2r - 1+δ/4 ), where δ is the exponential of the private exponent d with base N, i.e., d = Nδ. In addition, we present a Fermat-like factoring attack which factors N efficiently when Δ < N1/r2 . These proposed attacks surpass previous works (e.g. Bahig et al.'s at ICICS 2012), and are proved effective in practice.

AB - We consider some attacks on multi-prime RSA (MPRSA) with a modulus N = p1 p2 . . . pr (r ≥ 3). It is believed that the small private exponent attack on the MPRSA is less effective than that on RSA (see Hinek et al.'s work at SAC 2003), which means smaller private exponents can be used in the MPRSA to speed up the decryption process. Our work shows that even if a private exponent is significantly beyond Hinek et al.'s bound, it still may be insecure if the prime difference Δ (Δ = pr - p1 = Nγ, supposing p1 < p2 < · · · < pr ) is small, i.e. 0 < γ < 1/r. Specifically, by taking full advantage of prime properties, our small private exponent attack reveals that the MPRSA is insecure when δ < 1 - √ 1 + 2γ - 3/r (if γ ≥ 3 2r - 1+d/4 ) or δ ≤ 3r - 1 4 -2 (if γ < 3 2r - 1+δ/4 ), where δ is the exponential of the private exponent d with base N, i.e., d = Nδ. In addition, we present a Fermat-like factoring attack which factors N efficiently when Δ < N1/r2 . These proposed attacks surpass previous works (e.g. Bahig et al.'s at ICICS 2012), and are proved effective in practice.

UR - http://www.scopus.com/inward/record.url?scp=84903699162&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84903699162&partnerID=8YFLogxK

U2 - 10.1587/transfun.E97.A.1533

DO - 10.1587/transfun.E97.A.1533

M3 - Article

AN - SCOPUS:84903699162

VL - E97-A

SP - 1533

EP - 1541

JO - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

JF - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

SN - 0916-8508

IS - 7

ER -