### Abstract

We describe a parallel algorithm for modular exponentiation y ≡ x^{k} mod n. Then we discuss timing attacks against an implementation of the proposed parallel algorithm for modular exponentiation. When we have two processors, which perform modular exponentiation, an exponent k is scattered into two partial exponents k^{(0)} and k^{(1)}, where k^{(0)} and k^{(1)} are derived by bitwise AND operation from k such that k^{(0)} = k ∧ (0101⋯01)_{2} and k^{(1)} = k ∧(1010 ⋯10)_{2}. Two partial modular exponentiations y_{0} ≡ x^{k(0)} mod n and y_{1} ≡ x^{k(1)} mod n are performed in parallel using two processors. Then we can obtain y by computing y ≡ y_{0}y_{1} mod n. In general, the hamming weight of k^{(0)} and k^{(1)} are smaller than that of k. Thus fast computation of modular exponentiation y ≡ x^{k} mod n can be achieved. Moreover we show a timing attack against an implementation of this algorithm. We perform a software simulation of the attack and analyze security of the parallel implementation.

Original language | English |
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Pages (from-to) | 319-330 |

Number of pages | 12 |

Journal | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Volume | 2846 |

Publication status | Published - Dec 1 2003 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

**Timing attack against implementation of a parallel algorithm for modular exponentiation.** / Sakai, Yasuyuki; Sakurai, Kouichi.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Timing attack against implementation of a parallel algorithm for modular exponentiation

AU - Sakai, Yasuyuki

AU - Sakurai, Kouichi

PY - 2003/12/1

Y1 - 2003/12/1

N2 - We describe a parallel algorithm for modular exponentiation y ≡ xk mod n. Then we discuss timing attacks against an implementation of the proposed parallel algorithm for modular exponentiation. When we have two processors, which perform modular exponentiation, an exponent k is scattered into two partial exponents k(0) and k(1), where k(0) and k(1) are derived by bitwise AND operation from k such that k(0) = k ∧ (0101⋯01)2 and k(1) = k ∧(1010 ⋯10)2. Two partial modular exponentiations y0 ≡ xk(0) mod n and y1 ≡ xk(1) mod n are performed in parallel using two processors. Then we can obtain y by computing y ≡ y0y1 mod n. In general, the hamming weight of k(0) and k(1) are smaller than that of k. Thus fast computation of modular exponentiation y ≡ xk mod n can be achieved. Moreover we show a timing attack against an implementation of this algorithm. We perform a software simulation of the attack and analyze security of the parallel implementation.

AB - We describe a parallel algorithm for modular exponentiation y ≡ xk mod n. Then we discuss timing attacks against an implementation of the proposed parallel algorithm for modular exponentiation. When we have two processors, which perform modular exponentiation, an exponent k is scattered into two partial exponents k(0) and k(1), where k(0) and k(1) are derived by bitwise AND operation from k such that k(0) = k ∧ (0101⋯01)2 and k(1) = k ∧(1010 ⋯10)2. Two partial modular exponentiations y0 ≡ xk(0) mod n and y1 ≡ xk(1) mod n are performed in parallel using two processors. Then we can obtain y by computing y ≡ y0y1 mod n. In general, the hamming weight of k(0) and k(1) are smaller than that of k. Thus fast computation of modular exponentiation y ≡ xk mod n can be achieved. Moreover we show a timing attack against an implementation of this algorithm. We perform a software simulation of the attack and analyze security of the parallel implementation.

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M3 - Article

AN - SCOPUS:0242340685

VL - 2846

SP - 319

EP - 330

JO - Lecture Notes in Computer Science

JF - Lecture Notes in Computer Science

SN - 0302-9743

ER -