KCDSA

KCDSA (Korean Certificate-based Digital Signature Algorithm) is a digital signature algorithm created by a team led by the Korea Internet & Security Agency (KISA). It is an ElGamal variant, similar to the Digital Signature Algorithm and GOST R 34.10-94. The standard algorithm is implemented over ${\displaystyle GF(p)}$, but an elliptic curve variant (EC-KCDSA) is also specified.

KCDSA requires a collision-resistant cryptographic hash function that can produce a variable-sized output (from 128 to 256 bits, in 32-bit increments). HAS-160, another Korean standard, is the suggested choice.

• ${\displaystyle p}$: a large prime such that ${\displaystyle |p|=512+256i}$ for ${\displaystyle i=0,1,\dots ,6}$.
• ${\displaystyle q}$: a prime factor of ${\displaystyle p-1}$ such that ${\displaystyle |q|=128+32j}$ for ${\displaystyle j=0,1,\dots ,4}$.
• ${\displaystyle g}$: a base element of order ${\displaystyle q}$ in ${\displaystyle \operatorname {GF} (p)}$.

The revised version of the spec additional requires either that ${\displaystyle (p-1)/(2q)}$ be prime or that all of its prime factors are greater than ${\displaystyle q}$.

• ${\displaystyle x}$: signer's private signature key such that ${\displaystyle 0<x<q}$.
• ${\displaystyle y}$: signer's public verification key computed by ${\displaystyle y=g^{\bar {x}}{\pmod {p}},}$ where ${\displaystyle {\bar {x}}=x^{-1}{\pmod {q}}}$.
• ${\displaystyle z}$: a hash-value of Cert Data, i.e., ${\displaystyle z=h({\text{Cert Data}})}$.

The 1998 spec is unclear about the exact format of the "Cert Data". In the revised spec, z is defined as being the bottom B bits of the public key y, where B is the block size of the hash function in bits (typically 512 or 1024). The effect is that the first input block corresponds to y mod 2^B.

• ${\displaystyle z}$: the lower B bits of y.

• ${\displaystyle h}$: a collision resistant hash function with |q|-bit digests.

To sign a message ${\displaystyle m}$:

• Signer randomly picks an integer ${\displaystyle 0<k<q}$ and computes ${\displaystyle w=g^{k}\mod {p}}$
• Then computes the first part: ${\displaystyle r=h(w)}$
• Then computes the second part: ${\displaystyle s=x(k-r\oplus h(z\parallel m)){\pmod {q}}}$
• If ${\displaystyle s=0}$, the process must be repeated from the start.
• The signature is ${\displaystyle (r,s)}$

The specification is vague about how the integer ${\displaystyle w}$ be reinterpreted as a byte string input to hash function. In the example in section C.1 the interpretation is consistent with ${\displaystyle r=h(I2OSP(w,|q|/8))}$ using the definition of I2OSP from PKCS#1/RFC3447.

To verify a signature ${\displaystyle (r,s)}$ on a message ${\displaystyle m}$:

• Verifier checks that ${\displaystyle 0\leq r<2^{|q|}}$ and ${\displaystyle 0<s<q}$ and rejects the signature as invalid if not.
• Verifier computes ${\displaystyle e=r\oplus h(z\parallel m)}$
• Verifier checks if ${\displaystyle r=h(y^{s}\cdot g^{e}\mod {p})}$. If so then the signature is valid; otherwise it is not valid.

EC-KCDSA is essentially the same algorithm using Elliptic-curve cryptography instead of discrete log cryptography.

The domain parameters are:

• An elliptic curve ${\displaystyle E}$ over a finite field.
• A point ${\displaystyle G}$ in ${\displaystyle E}$ generating a cyclic subgroup of prime order ${\displaystyle q}$. (${\displaystyle q}$ is often denoted ${\displaystyle n}$ in other treatments of elliptic-curve cryptography.)

The user parameters and algorithms are essentially the same as for discrete log KCDSA except that modular exponentiation is replaced by point multiplication. The specific differences are:

• The public key is ${\displaystyle Y={\bar {x}}G}$
• In signature generation, ${\displaystyle r=h(W_{x}||W_{y})}$ where ${\displaystyle W=kG}$
• In signature verification, the verifier tests whether ${\displaystyle r=h(sY+eG)}$

• KCDSA specification and analysis