# Step 5 output after all n 1024 bit blocks have been

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Step 5Output. After all N 1024-bit blocks have been processed, the output fromthe Nthstage is the 512-bit message digest.We can summarize the behavior of SHA-512 as follows:H0= IVHi= SUM64(Hi-1, abcdefghi)MD = HNwhereIV= initial value of the abcdefgh buffer, defined in step 3abcdefghi= the output of the last round of processing of the ithmessage blockN= the number of blocks in the message (including padding andlength fields)SUM64=addition modulo264 performed separately on each word of thepair of inputsMD=final message digest valueSHA-512 Round FunctionLet us look in more detail at the logic in each of the 80 steps of theprocessing of one 512-bit block (Figure below).
Each round is defined by the following set of equations:T1= h + Ch(e, f, g) ++ Wt+ KtT2=+ Maj(a, b, c)h=gg = ff = ee = d + T1d = cc = bb = aa = T1+ T2wheret= step number; 0<=t<=79;Ch(e, f, g)= (e AND f)(NOT e AND g)the conditional function: If e then felse gMaj(a, b, c)= (a AND b)(a AND c)(b AND c)the function is true only of the majority (two or three) of thearguments are true
Wt= a 64-bit word derived from the current 1024-bit input blockKt= a 64-bit additive constant+= addition modulo 264Two observations can be made about the round function.1. Six of the eight words of the output of the round function involve simplypermutation (b, c, d, f, g, h) by means of rotation. This is indicated by shading inFigure above.2. Only two of the output words (a, e) are generated by substitution. Wordeis afunction of input variables (d, e, f, g, h), as well as the round word Wtand theconstant Kt. Wordais a function of all of the input variables exceptd, as wellas the round word Wtand the constant Kt.It remains to indicate how the 64-bit word values Wtare derived from the1024-bit message. Figure below illustrates the mapping.The first 16 values of Wtare taken directly from the 16 words of the current block.The remaining values are defined as
Thus, in the first 16 steps of processing, the value of Wtis equal to thecorresponding word in the message block. For the remaining 64 steps, the value ofWtconsists of the circular left shift by one bit of the XOR of four of the precedingvalues of Wt, with two of those values subjected to shift and rotate operations.The SHA-512 algorithm has the property that every bit of the hash code is afunction of every bit of the input. The complex repetition of the basic function F
produces results that are well mixed; that is, it is unlikely that two messageschosen at random, even if they exhibit similar regularities, will have the same hashcode.HMAC(HASH-BASED MESSAGE AUTHENTICATION CODE)Interest in developing a MAC, derived from a cryptographic hash code hasbeen increasing mainly because hash functions are generally faster and are also notlimited by export restrictions unlike block ciphers. Additional reason also wouldbe that the library code for cryptographic hash functions is widely available. Theoriginal proposal is for incorporation of a secret key into an existing hash

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