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Bit Operations Refresher

SHA-256 is a flurry of bit operations on 32-bit unsigned integers.

Op	Symbol	What it does
AND	`a & b`	bits set in both
OR	`a	b`
XOR	`a ^ b`	bits differing
NOT	`~a`	flip all bits
Left shift	`a << n`	shift bits left, fill with 0s
Right shift	`a >> n`	shift bits right, fill with 0s
Rotate right	`ROTR(a,n)`	bits that fall off the right come around

In SHA-256, ROTR is critical:

ROTR(x, n) = (x >> n) | (x << (32 - n))    (mod 2^32)

Two derived functions used in the spec:

Σ0(x) = ROTR(x,2)  ^ ROTR(x,13) ^ ROTR(x,22)
Σ1(x) = ROTR(x,6)  ^ ROTR(x,11) ^ ROTR(x,25)
σ0(x) = ROTR(x,7)  ^ ROTR(x,18) ^ (x >> 3)
σ1(x) = ROTR(x,17) ^ ROTR(x,19) ^ (x >> 10)

Plus two "majority/choice" functions:

Ch(x,y,z)  = (x & y) ^ (~x & z)        # if x then y else z, bitwise
Maj(x,y,z) = (x & y) ^ (x & z) ^ (y & z)  # majority of three bits

All arithmetic is mod 2^32 — wrap on overflow.

next: Practice

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