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;------------------------------------------
;
; Author: Diana Toma
; TIMA-VDS, Grenoble, France
; March 2003
; ACL2 formalization of SHAs
; Padding function for SHA-384 and SHA-512
;------------------------------------------
;I strongly recommend after charging the book to do :comp t in order to accelerate the computation
(IN-PACKAGE "ACL2")
(include-book "bv-op-defthms")
;---padding
;for sha-512 and sha-384
;Let M be a message of length len bits. The purpose of padding is to extend M to a multiple of 1024 bits. To obtain the padded message, append the bit 1 to the end of message M, followed by k zero bits, where k is the smallest, non-negative solution to the equation (len+1+k) mod 1024 = 896. Then append the 128-bit binary representation of number len.
;For example, the (8-bit ASCII) message ``abc'' has the length 8*3=24, so the message is padded with one bit, then 896-(24+1)=871 zero bits, and then the message length, to become the 1024-bit padded message:
;ACL2 !>(padding-512 ' (0 1 1 0 0 0 0 1 0 1 1 0 0 0 1 0 0 1 1 0 0 0 1 1 ))
;(0 1 1 0
; 0 0 0 1 0 1 1 0 0 0 1 0 0 1 1 0 0 0 1 1
; 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0)
(local
(defthm 2n<i
(implies (and (integerp n) (< 0 n) (integerp i)
(< n i) (integerp (* i (/ n))) )
(<= 2 (* i (/ n) )))))
(local
(defthm n<=i
(implies (and (integerp n) (< 0 n) (integerp i)
(equal (mod i n) 0) (< 0 i) )
(<= n i) )))
(defun padding-512 (m)
(if (and (bvp m)
(< (len m) (expt 2 128)))
(if (<= (mod (1+ (len m)) 1024) 896)
(append m (list 1)
(make-list (- 896 (mod (1+ (len m)) 1024))
:initial-element 0)
(bv-to-n (int-bv-big-endian (len m)) 128))
(append m (list 1)
(make-list (- 1920 (mod (1+ (len m)) 1024))
:initial-element 0 )
(bv-to-n (int-bv-big-endian (len m)) 128)))
nil))
(defthm bvp-padding-512
(bvp (padding-512 m)))
(defthm len-padding-512
(implies (and (bvp m)
(< (len m) (expt 2 128)))
(<= 1024 (len (padding-512 m))))
:hints
(("Goal"
:in-theory (disable mod MOD-ZERO ASSOCIATIVITY-OF-+ ))
("subgoal 2"
:use (:instance simplify-mod-+-mod1 (w (+ 1921 (len m)))
(x (+ 1 (len m))) (y 1024) (z 1024) ))
("subgoal 1"
:use (:instance simplify-mod-+-mod1 (w (+ 897 (len m)))
(x (+ 1 (len m))) (y 1024) (z 1024) ))))
(defthm len-padding-512-mod-1024=0
(implies (bvp m)
(equal (mod (len (padding-512 m)) 1024) 0))
:hints
(("Goal"
:in-theory (disable MOD-ZERO int-bv-big-endian ))
("subgoal 2"
:use (:instance simplify-mod-+-mod1 (w (+ 2049 (len m)))
(x (+ 1 (len m))) (y 1024) (z 1024) ))
("subgoal 1"
:use (:instance simplify-mod-+-mod1 (w (+ 1025 (len m)))
(x (+ 1 (len m))) (y 1024) (z 1024) ))))
(local
(defthm last-512-aux
(implies (and (BVP M)
(< (LEN M) 340282366920938463463374607431768211456)
(< 896 (MOD (+ 1 (LEN M)) 1024))
(<= (NFIX (+ 1 (LEN M)
1920 (- (MOD (+ 1 (LEN M)) 1024))))
(LEN M)))
(<= 1921 (MOD (+ 1 (LEN M)) 1024) ))))
(defthm last128-padding-512=length
(implies (and (bvp m) (< (len m) (expt 2 128)))
(equal (bv-int-big-endian
(nthcdr (- (len (padding-512 m)) 128)
(padding-512 m)))
(len m)))
:hints
(("Goal"
:in-theory (disable bv-int-big-endian int-bv-big-endian ))
("subgoal 2.2" :use last-512-aux)))
(defthm end-message-padding-512
(implies (and (bvp m) (< (len m) (expt 2 128)))
(equal (nth (len m) (padding-512 m)) 1))
:hints
(("Goal"
:in-theory (disable bv-int-big-endian int-bv-big-endian ))))
(defthm first-padding-512=message
(implies (and (bvp m) (< (len m) (expt 2 128)))
(equal (firstn ( len m) (padding-512 m)) m))
:hints
(("Goal"
:in-theory (disable bv-int-big-endian int-bv-big-endian nthcdr ))))
(defthm 0-fill-padding-512
(implies (and (bvp m) (< (len m) (expt 2 128)))
(equal (segment (1+ (len m))
(- (len (padding-512 m)) 128)
(padding-512 m))
(make-list (- (len (padding-512 m)) (+ 129 (len m)))
:initial-element 0))))
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