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A Survey on the Algebraic Surface Cryptosystems

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A Survey on the Algebraic Surface Cryptosystems

Koichiro Akiyama ( TOSHIBA Corporation )

Joint work with Prof. Yasuhiro Goto

2013/03/02

- Introduction
Public key cryptosystem, Motivation

- Section Finding Problem
A Computational Hard Problem on Algebraic Surface

- Algebraic Surface Public-key Cryptosystem
Encryption/Decryption/Key Generation Algorithms

- Known Attacks
－Rational Point Attack

－Ideal Factorization Attack

- Conclusion and Future Research

A Survey on the Algebraic Surface Cryptosystems

- Introduction
Public key cryptosystem, Motivation

- Section Finding Problem
A Computational Hard Problem on Algebraic Surface

- Algebraic Surface Public-key Cryptosystem
Encryption/Decryption/Key Generation Algorithms

- Known Attacks
－Rational Point Attack

－Ideal Factorization Attack

- Conclusion and Future Research

A Survey on the Algebraic Surface Cryptosystems

Hello World

Hello World

Sender

A

Receiver

B

Computational

Hard Problem

B’s Public Key

B’s Secret Key

Ex. Integer Factorization

sHueLjOl8k7

sHueLjOl8k7

Security of public key cryptosystem relies on the

the problem which is hard to compute.

A Survey on the Algebraic Surface Cryptosystems

Our target is an algebraic surface

- Want to construct public-key cryptosystems having following features
- Resistant against known attacks by quantum computer.
( Not based on the factorization or discrete logarithm problems. )

- Fast in process time & compact in size.
- Based on a hard problem in algebraic geometry.

- Resistant against known attacks by quantum computer.

A Survey on the Algebraic Surface Cryptosystems

Algebraic Surface

Cryptosystem

Public key size

Public key size

Fast & compact

: number of valuables

(1) Short Public key

(2) Higher Dimensional

Equations

Multivariate

Cryptosystems

higher degree (>3) equations

Quadratic equations

RSA

Elliptic Curve

Cryptosystem

A Survey on the Algebraic Surface Cryptosystems

Factoring Problem

Hardness

Security requirement

Hard

Easy

Secure parameter

Size of the parameter

The Section Finding Problem

Next talk

Algebraic

Surface

Hard

Easy

Section

RSA Cryptosystem

Algebraic Surface Cryptosystem

This talk

Improvement

Attack Success!

Call for Attack

Design

Encryption

Algorithm

Selection of

Hard

Problem

Start

Security Proof

Define

the secure parameters

Elementary

Algorithm

Optimized

Algorithms

Practical

implementation

Hard even for Quantum Computer

Easy for Quantum Computer

A Survey on the Algebraic Surface Cryptosystems

- Introduction
Public key cryptosystem, Motivation

- Section Finding Problem
A Computational Hard Problem on Algebraic Surface

- Algebraic Surface Public-key Cryptosystem
Encryption/Decryption/Key Generation Algorithms

- Known Attacks
－Rational Point Attack

－Ideal Factorization Attack

- Conclusion and Future Research

A Survey on the Algebraic Surface Cryptosystems

An algebraic surface (we use) is

a 2-dimensional affine algebraic variety with fibration.

We consider algebraic surfaces defined over a finite field .

where is small enough to calculate,

but need not be 2.

A Survey on the Algebraic Surface Cryptosystems

Algebraic Surface

easy

section

Algebraic Surface

hard

A Survey on the Algebraic Surface Cryptosystems

The SFP is reduced to multivariable equations

To solve the SFP, we put the section as follows:

(are variables )

Substitute into , we obtain

A Survey on the Algebraic Surface Cryptosystems

- Introduction
Public key cryptosystem, Motivation

- Section Finding Problem
A Computational Hard Problem on Algebraic Surface

- Algebraic Surface Public-key Cryptosystem
Encryption/Decryption/Key Generation Algorithms

- Known Attacks
－Rational Point Attack

－Ideal Factorization Attack

- Conclusion and Future Research

A Survey on the Algebraic Surface Cryptosystems

- System parameters
- Size of finite field : prime
- Degree of section :

- Public key
- Algebraic surface
- Form of the plaintext polynomial
- Form of the divisor polynomial

- Secret key
- Section

(example)

( are given ）

（ are given ）

A Survey on the Algebraic Surface Cryptosystems

and

are designated.

For example,

Form described the formula as fllows:

indicates an element of

A Survey on the Algebraic Surface Cryptosystems

In the case of

So the plaintext described as

In the case of plaintext must be divided into 2bits block

Therefore m embedded to m(x,y,t) as coefficients

A Survey on the Algebraic Surface Cryptosystems

Randomize

（operations）

message

Public Key：algebraic surface

embed

Message poly.

Random polynomial

Divisor polynomial

Cipher text

A Survey on the Algebraic Surface Cryptosystems

Cipher

Plaintext

Random

Random

Public key

Section substitute

Secret key: Section

factoring

message polynomial

message

Solve linear equations

A Survey on the Algebraic Surface Cryptosystems

Coefficients other than constant term

Secret key : section

Select randomly

Select randomly

Public key: algebraic surface

Calculate the constant term

A Survey on the Algebraic Surface Cryptosystems

- Introduction
Public key cryptosystem, Motivation

- Section Finding Problem
A Computational Hard Problem on Algebraic Surface

- Algebraic Surface Public-key Cryptosystem
Encryption/Decryption/Key Generation Algorithms

- Known Attacks
－Rational Point Attack

－Ideal Factorization Attack

- Conclusion and Future Research

A Survey on the Algebraic Surface Cryptosystems

where

subtract

Remove the plaintext

polynomial

A Survey on the Algebraic Surface Cryptosystems

=

substitution

rational points

construct

Solve

Linear Equation

extract

Success!

factoring

A Survey on the Algebraic Surface Cryptosystems

=

=

rational points

substitution

Solve linear

equations

reconstruct

A Survey on the Algebraic Surface Cryptosystems

are in the same form

and

This is also another solution

=

is a solution, there exists polynomial

If

which is in the same form of and satisfy .

For arbitrary which is in the same form of ,

We can avoid the attack, when we select the form of

which has enough polynomials not to be able to identify the correct one.

A Survey on the Algebraic Surface Cryptosystems

Cipher text

Ideal Factoring

where

Solve

Linear Eq.

A Survey on the Algebraic Surface Cryptosystems

Jan 2004 1st version was proposed in domestic conference

May 2006 1st version was presented

in international conference PQC2006

Jintai Ding pointed out a flaw in our system

Oct 2006 2nd version was presented in AMS conference.

.

Jan 2007 Shigenori Uchiyama proposed an attack against 2nd version

.

Apr2007 Felipe Voloch proposed another attack against 2nd version

Jan 2008 3rd version was proposed in domestic conference.

Mar 2009 3rdwas presented

in international conference PKC2009

May 2010 Jean-Charles Faugere( INRIA )

proposed an attack against 3rd version.

NowWe are preparing 4th version

whose security is equivalent to SFP.

A Survey on the Algebraic Surface Cryptosystems

- Introduction
Public key cryptosystem, Motivation

- Section Finding Problem
A Computational Hard Problem on Algebraic Surface

- Algebraic Surface Public-key Cryptosystem
Encryption/Decryption/Key Generation Algorithms

- Known Attacks
－Rational Point Attack

－Ideal Factorization Attack

- Conclusion and Future Research

A Survey on the Algebraic Surface Cryptosystems

- We showed a new type of public-key cryptosystem using an algebraic surface.
- We showed the algorithm for encryption, decryption and key generation.

- Our contributions are
- The public key size is O(n).
- Our cryptosystem is associated higher general equations than multivariate cryptosystems. ( contains equation which degree is more than 3)

A Survey on the Algebraic Surface Cryptosystems

Construct a secure algorithm

We try to construct a provable secure cryptosystem

Determine the recommendable parameter size

We developed an efficient algorithm to solve the SFP.

Now we estimate computational complexity by computational experimentation.

Next Talk

A Survey on the Algebraic Surface Cryptosystems

The Algebraic Surface Cryptosystem and its security