DSS数字签名标准

签名过程:

graph TD;

id2 -.Signature.-> id12

subgraph Signature Generation;

id0(Message/Data)-.-> id1(Hash Function)

id1 -.Message Digest.-> id2(Sinature Generation)

id3(Private Key) -.-> id2

end

subgraph Signature Verification;

id10(Message/Data)-.-> id11(Hash Function)

id11 -.Message Digest.-> id12(Sinature Verification)

id13(Public Key) -.-> id12

id12 -.-> id14(Valid/Invalid)

end初始化设置:

Obtain Domain Parameter;

Obtain Assurance of Domain Parameter Validity;

Obtain DS Key Pair;

Obtain Assurance of Public Key Validity;

Obtain Assurance of Possession of the DS Private Key;

Register the Public Key and Identify with a TTP(Optional);

数字签名生成:

Generate a Message Digest;

Obtain Additional Information for the Digital Signature Process;

Generate a Digital Signature;

Verify the Digital Signature(Optional);

数字签名的验证和确认:

graph LR;

subgraph Actions;

id1(Get the Claimed Signatory's Identifiers) -.-> id2(Obtain the Domain Parameters and Public Key)

id2 -.-> id3(Generate a Message Digest)

id3 -.-> id4(Verify the Digital Signature)

end

subgraph Assurance;

aid1(Obtain assurance of the Claimed Signatory's Identity)

aid2(Obtain Assurance of Domain Parameter Validity) -.-> aid3(Obtain Assurance of the Validity of the Owner's Public Key)

aid2 -.-> aid4(Obtain Assurance that the Owner Possesses the Private Key)

end

subgraph ValidationComplete;

vid1(Digital Signature Validation Complete)

end

aid1 -.-> vid1

aid3 -.-> vid1

aid4 -.-> vid1

id4 -.-> vid1公钥\(y=g^x\mod p\);

私钥\(x\in [1,q-1]\);

素数\(p\), 位长度为\(L\), \(p\in (2^{L-1}, 2^L)\);

和\(p-1\)互质的素数\(q\), 位长度记为\(N\), \(q\in (2^{N-1}, 2^N)\);

乘法群\(GF(p)\)中阶为\(q\)的子群的生成子\(g\), \(g \in (1,p)\);

伪随机整数\(k\), \(k\in [1,q-1]\);

\(p, q, g\);

可选的domain_parameter_seed/counter, 用于\(p,q\)的生成;

规范指定的(L,N)长度选择:

L = 1024, N = 160;

L = 2048, N = 224;

L = 2048, N = 256;

L = 3072, N = 256;

哈希函数的选择要满足其安全强度大于\(min(L,N)\);

记Hash函数的输出位字符串的位长度为\(outlen\);

记truncate_l(bit_str, len)表示取位字符串bit_str的最左边的len位;

\(k^{-1}\)表示关于随机数\(k\)的模\(q\)的逆, 即\((k^{-1}\cdot k)\mod q = 1\);

签名\((r,s)\)的计算如下:

\[\begin{aligned}

& r = (g^k \mod p) \mod q \\

& z = truncate_l(Hash(M), min(N, outlen)) \\

& s = (k^{-1}(z+x\cdot r))\mod q;

\end{aligned}

\]

假设认证者已经确认了域参数和公钥;

记接受者收到了消息\(M'\), 和签名\((r', s')\), 则签名验证如下;

签名需满足\(0\lt r' \lt q\), \(0 \lt s' \lt q\);

\(r'\)需满足\(r'=v\):

\(w = (s')^{-1}\mod q\);

\(z = truncate_l(Hash(M'), min(N, outlen))\);

\(u1 = (z\cdot w)\mod q\);

\(u2 = (r' \cdot w)\mod q\);

\(v = ((g^{u1}\cdot y^{u2})\mod p) \mod q\);

公钥\((n, e)\);

私钥\((n, d)\);

相关的标准有:

ANS X9.31;

相关标准:

其它内容待补充;

FIPS 186-4;

FIPS 186-5-draft;