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**2017/403**

**Title:**

Condition on composite numbers easily factored with elliptic curve method

**Authors:**
Masaaki Shirase

**Abstract:**

For a composite integer N that we would like to factor, we consider a condition for the elliptic curve method using N as a scalar value to succeed and show that if N has a prime factor p such that $p=(DV^2+1)/4,\ V \in {\mathbb Z},\ D\in {3, 11, 19, 35, 43, 51, 67, 91, 115, 123, 163, 187, 235, 267, 403, 427}$, we can find a non-trivial divisor of N (multiple of p) in a short time. In the authorsâ€™ implementation on PARI/GP, a 1024-bit N was factored in a few seconds when p was 512 bits.

**ePrint:**
https://eprint.iacr.org/2017/403

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