Obtaining shared secret as scalar multiplication (continuing GooglePay theme)

[Миша Винник]

Hello!

How can we initialize field over CryptoPP::ASN1::secp256r1();?

Grabbing x and y from accepted ephemeral key, and priv_int - CryptoPP::Integer representation of our private.

CryptoPP::ECP field;

CryptoPP::ECPPoint p_result = field.ScalarMultiply(p_eph, priv_int);

Now it ends up in error - and it is obviuous, because not know how to init field properly.

Thank you very much.

any help is welcome!

[Миша Винник]

Question seems to be closed. secp256r1 Cryptopp representation not allows scalar multiplication for not-odd modulus. And from standard it is not-odd!

Some code if anybody wants to test for self. Params from here: http://www.secg.org/SEC2-Ver-1.0.pdf .

CryptoPP::Integer m, a, b;
CryptoPP::byte m_str[] = "FFFFFFFF00000001000000000000000000000000FFFFFFFFFFFFFFFFFFFFFFFF";
CryptoPP::byte a_str[] = "FFFFFFFF00000001000000000000000000000000FFFFFFFFFFFFFFFFFFFFFFFC";
CryptoPP::byte b_str[] = "5AC635D8AA3A93E7B3EBBD55769886BC651D06B0CC53B0F63BCE3C3E27D2604B";


m.Decode(&(m_str[0]), sizeof(m_str) / sizeof(m_str[0]),CryptoPP::Integer::UNSIGNED);
a.Decode(&(a_str[0]), sizeof(a_str) / sizeof(a_str[0]), CryptoPP::Integer::UNSIGNED);
b.Decode(&(b_str[0]), sizeof(b_str) / sizeof(b_str[0]), CryptoPP::Integer::UNSIGNED);
bool is_odd = m.IsOdd();
CryptoPP::ECP field(m,a,b);
//CryptoPP::DL_GroupPrecomputation<CryptoPP::ECP>::Element group(field);
//multiplying over this field

CryptoPP::ECPPoint p_result = field.ScalarMultiply(p_eph, priv_int);

Why so? it is a standart curve, why not allowing scalar multiplication?!

[Миша Винник]

No, it should be odd. Standard say so. Maybe anybody knows how to properly parse it from string?

I have a second week elliptic curve maraphon, help is very welcome!

[Jeffrey Walton]

On Tue, Mar 27, 2018 at 11:27 AM, Миша Винник <m.vin...@gmail.com> wrote:
> Hello!
>
> How can we initialize field over CryptoPP::ASN1::secp256r1();?
>
> Grabbing x and y from accepted ephemeral key, and priv_int -
> CryptoPP::Integer representation of our private.

https://www.cryptopp.com/wiki/Elliptic_Curve_Diffie-Hellman#Ephemeral_Key_as_.28x.2Cy.29

Jeff

[Миша Винник]

Yeah, wiki is back! Thanks, be crawling on it for now

[Миша Винник]

Hmmm.. It is answer to a bit another question.
Question is about creating field from constructor, as here: https://groups.google.com/forum/#!topic/cryptopp-users/o2YiAC2emvY

At fact, field is created, but because of incorrect parsing hex from string, modulus is incorrect and ScalarMultiply method is unreachable

[Миша Винник]

Giving the same result as dh.Agree(...). Beautiful.

For strangers who wants to obtain the same shared from field theory:

unsigned long modulus,a_element,b_element;

CryptoPP::Integer m, a, b;

string m_str("FFFFFFFF00000001000000000000000000000000FFFFFFFFFFFFFFFFFFFFFFFF");
vector<unsigned char> m_vec = hex_to_bytes(m_str);
m.Decode(&(m_vec.at(0)), m_vec.size(), CryptoPP::Integer::UNSIGNED);
bool is_odd = m.IsOdd();


string a_str("FFFFFFFF00000001000000000000000000000000FFFFFFFFFFFFFFFFFFFFFFFC");
vector<unsigned char> a_vec = hex_to_bytes(a_str);
a.Decode(&(a_vec.at(0)), a_vec.size(), CryptoPP::Integer::UNSIGNED);


string b_str("5AC635D8AA3A93E7B3EBBD55769886BC651D06B0CC53B0F63BCE3C3E27D2604B");
vector<unsigned char> b_vec = hex_to_bytes(b_str);
b.Decode(&(b_vec.at(0)), b_vec.size(), CryptoPP::Integer::UNSIGNED);


//field init
CryptoPP::ECP field(m,a,b);
//multiplying over this field google_ephemeral_point * our_privatekey_int

CryptoPP::ECPPoint p_result = field.ScalarMultiply(p_eph, priv_int);

//obtaining shared from result:
CryptoPP::byte x_res[32];
CryptoPP::byte y_res[32];
p_result.x.Encode(&(x_res[0]), 32,CryptoPP::Integer::UNSIGNED);
p_result.y.Encode(&(y_res[0]), 32, CryptoPP::Integer::UNSIGNED);

[Миша Винник]

Shared secret obtained, further things are describing in main theme on it: https://groups.google.com/forum/#!topic/cryptopp-users/IFvwPWcbxCI

[Jeffrey Walton]

On Wednesday, March 28, 2018 at 3:37:25 AM UTC-4, Миша Винник wrote:

Shared secret obtained, further things are describing in main theme on it: https://groups.google.com/forum/#!topic/cryptopp-users/IFvwPWcbxCI

Congrats man, good work.

You are the first person I am aware to take it to completion.

[Миша Винник]

Thanks, Jeff. For now working on verifying signature, if have some time, will try to do it also using math background of issue.