# learning ρ and λ method for ECDLP

2018/10/01 Yasunori Ushiro (Kanagawa University)

2018/12/22 Addition for Learning λ method

**Learning λ method is improved version of learning ρ method and parallel learning efficiency is good.**

I discovered that the ρ algorithm can reduce the number of iterations by learning.

This is the discovery of Columbus egg.

### 1. Concept of learning ρ and λ method

Elliptic curve cryptography (ECC) consists of two different points P and Q on the same elliptic curve.

ECC: y^{2}=x^{3}+ax+b (mod p). a,b,x,y are integers. r is a order. p and r are primes.

I discovered the learning methods where the trajectory groups do not depend on P, Q in ECC points

Decipherment of the 60-bit encryption 44 cases (equivalent to 70 bits).

It takes 10 hours by ρ algorithm. The learning ρ method shortened to about 30 seconds.

The learning λ method shortened to about 3.6 seconds. It is 10,000 times faster than the ρ method.

### 2. Numerical experiment result of ECDLP by learning λ method on SSD device

Learning ρ method Result on SSD

### 3. Numerical experiment result by learning λ method

Learning λ method Result

### 4. Numerical experiment result by learning ρ method

Learning ρ method Result

### 5. AI posibility for Learning λ Method

AI posibility

### 6. Details of algorithm for Learning ρ and λ method

It is scheduled to be released after publication in newspapers etc.