# What does the symbol $W_t$ mean in the SHA-256 specification?

$$W_t = \begin{cases} M_t^{(i)} & 0 \leq t \leq 15 \\ &\\ \sigma_1^{256}(W_{t-2}) + W_{t-7} + \sigma_0^{256}(W_{t-15}) + W_{t-16} & 16 \leq t \leq 63 \end{cases}$$

The $$W_t$$ variable: What's the value of $$W_t$$? How do I get that value?

I'm still confused because of the explanation using the English language. I'm a beginner guys, any help from you will be a plus point for me about hashing.

It is in SHA-256 message schedule (NIST-FIPS 180-4);

The message $$M$$ with length $$l$$ is first padded as the usual way;

• append 1 to the end of the message,
• then, add $$k$$ zero bits such that $$l+1+k \equiv 448 \mod 512$$
• finally, add the length of the message in 64-bit. Now, the total padded length is divisible by 512.

After padding, the padded message parsed into $$M^{1},\ldots,M^{N}$$ where each has size 512-bit.

The sub index $$t$$ represents 32-bits in 512 bits. Thus, The $$M_t^{(i)}$$ is the $$t$$-th 32-bit in the $$M^{(i)}$$ for $$0 \leq t \leq 15$$

The $$W_t$$ is defined as your equation;

$$W_t = \begin{cases} M_t^{(i)} & 0 \leq t \leq 15 \\ &\\ \sigma_1^{256}(W_{t-2}) + W_{t-7} + \sigma_0^{256}(W_{t-15}) + W_{t-16} & 16 \leq t \leq 63 \end{cases}$$

Where $$\sigma_1^{256}(x) = \operatorname{ROTR}^{17}(x) \oplus \operatorname{ROTR}^{19}(x) \oplus \operatorname{SHR^{10}(x)}$$ $$\sigma_0^{256}(x) = \operatorname{ROTR}^{7}(x) \oplus \operatorname{ROTR}^{18}(x) \oplus \operatorname{SHR^{3}(x)}$$

$$\sigma_1^{256}(x)$$ and $$\sigma_0^{256}(x)$$ operate on 32 bits and produce 32 bits.

Note: the 256 above the $$\sigma$$ represents the 256 in SHA-256. Similarly, there is $$\sigma_0^{512}(x)$$ and $$\sigma_1^{512}(x)$$ for SHA-512.

• so you mean Wt = M16.....Mn ? Mar 8, 2019 at 10:44
• No. $W_1 = M_1^{(i)}, W_2 = M_2^{(i)}, \ldots, W_{15} = M_{15}^{(i)}$ Mar 8, 2019 at 11:00
• @OntaSs there is no $M_{16}$, the 16 words of the input block are numbered 0 to 15 only Mar 9, 2019 at 0:50
• @kelalaka Wt = M1 u mean (Wt-2) = (M1-2) ? sorry bro i learn self-taught about this. i got little confused if read mathematic formulas Mar 9, 2019 at 2:25
• @RichieFrame yeah i got it. How about W16....W63 ? what is the value ? Mar 9, 2019 at 3:27