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 Peter Schwabe committed Jun 20, 2019 1 \section{Proving equivalence of X25519 in C and Coq}  Benoit Viguier committed Jul 03, 2019 2 \label{C-Coq}  Benoit Viguier committed Jun 21, 2019 3 4 5 6  In this section we describe techniques used to prove the equivalence between the Clight description of TweetNaCl and Coq functions producing similar behaviors.  Benoit Viguier committed Jul 10, 2019 7 8 9  \todo{SUBSECTION?}  Benoit Viguier committed Jul 12, 2019 10 11 12 13 14 \begin{figure}[h] \include{tikz/proof} \caption{Overview construction of the proof} \label{tk:ProofOverview} \end{figure}  Benoit Viguier committed Jul 10, 2019 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60  Verifying \texttt{crypto\_scalarmult} also implies to verify all the functions subsequently called: \texttt{unpack25519}; \texttt{A}; \texttt{Z}; \texttt{M}; \texttt{S}; \texttt{car25519}; \texttt{inv25519}; \texttt{set25519}; \texttt{sel25519}; \texttt{pack25519}. We prove that the implementation of Curve25519 is \textbf{sound} \ie \begin{itemize} \item absence of access out-of-bounds of arrays (memory safety). \item absence of overflows/underflow on the arithmetic. \end{itemize} We also prove that TweetNaCl's code is \textbf{correct}: \begin{itemize} \item Curve25519 is correctly implemented (we get what we expect). \item Operations on \texttt{gf} (\texttt{A}, \texttt{Z}, \texttt{M}, \texttt{S}) are equivalent to operations ($+,-,\times,x^2$) in \Zfield. \item The Montgomery ladder does compute a scalar multiplication between a natural number and a point. \end{itemize} In order to prove the soundness and correctness of \texttt{crypto\_scalarmult}, we first create a skeleton of the Montgomery ladder with abstract operations which can be instanciated over lists, integers, field elements... A high level specification (over a generic field $\K$) allows use to prove the correctness of the ladder with respect to the curves theory. This high specification does not rely on the parameters of Curve25519. By instanciating $\K$ with $\Zfield$, and the parameters of Curve25519 ($a = 486662, b = 1$), we define a middle level specification. Additionally we also provide a low level specification close to the \texttt{C} code (over lists of $\Z$). We show this specification to be equivalent to the \textit{semantic version} of C (\texttt{CLight}) with VST. This low level specification gives us the soundness assurance. By showing that operations over instances ($\K = \Zfield$, $\Z$, list of $\Z$) are equivalent we bridge the gap between the low level and the high level specification with Curve25519 parameters. As such we prove all specifications to equivalent (Fig.\ref{tk:ProofStructure}). This garantees us the correctness of the implementation. \begin{figure}[h] \include{tikz/specifications} \caption{Structural construction of the proof} \label{tk:ProofStructure} \end{figure}  Benoit Viguier committed Jul 03, 2019 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 \subsection{Correctness Specification} We show the soundness of TweetNaCl by proving the following specification matches a pure Coq function. This defines the equivalence between the Clight representation and a Coq definition of the ladder. \begin{CoqVST} Definition crypto_scalarmult_spec := DECLARE _crypto_scalarmult_curve25519_tweet WITH v_q: val, v_n: val, v_p: val, c121665:val, sh : share, q : list val, n : list Z, p : list Z (*------------------------------------------*) PRE [ _q OF (tptr tuchar), _n OF (tptr tuchar), _p OF (tptr tuchar) ] PROP (writable_share sh; Forall (fun x => 0 <= x < 2^8) p; Forall (fun x => 0 <= x < 2^8) n; Zlength q = 32; Zlength n = 32; Zlength p = 32) LOCAL(temp _q v_q; temp _n v_n; temp _p v_p; gvar __121665 c121665) SEP (sh [{ v_q }] <<(uch32)-- q; sh [{ v_n }] <<(uch32)-- mVI n; sh [{ v_p }] <<(uch32)-- mVI p; Ews [{ c121665 }] <<(lg16)-- mVI64 c_121665) (*------------------------------------------*) POST [ tint ] PROP (Forall (fun x => 0 <= x < 2^8) (CSM n p); Zlength (CSM n p) = 32) LOCAL(temp ret_temp (Vint Int.zero)) SEP (sh [{ v_q }] <<(uch32)-- mVI (CSM n p); sh [{ v_n }] <<(uch32)-- mVI n; sh [{ v_p }] <<(uch32)-- mVI p; Ews [{ c121665 }] <<(lg16)-- mVI64 c_121665 \end{CoqVST} In this specification we state as preconditions: \begin{itemize} \item[] \VSTe{PRE}: \VSTe{_p OF (tptr tuchar)}\\ The function \texttt{crypto\_scalarmult} takes as input three pointers to arrays of unsigned bytes (\VSTe{tptr tuchar}) \VSTe{_p}, \VSTe{_q} and \VSTe{_n}. \item[] \VSTe{LOCAL}: \VSTe{temp _p v_p}\\ Each pointer represent an address \VSTe{v_p}, \VSTe{v_q} and \VSTe{v_n}. \item[] \VSTe{SEP}: \VSTe{sh [{ v_p $\!\!\}\!\!]\!\!\!$ <<(uch32)-- mVI p}\\ In the memory share \texttt{sh}, the address \VSTe{v_p} points to a list of integer values \VSTe{mVI p}. \item[] \VSTe{PROP}: \VSTe{Forall (fun x => 0 <= x < 2^8) p}\\ In order to consider all the possible inputs, we assumed each elements of the list \texttt{p} to be bounded by $0$ included and $2^8$ excluded. \item[] \VSTe{PROP}: \VSTe{Zlength p = 32}\\ We also assumed that the length of the list \texttt{p} is 32. This defines the complete representation of \TNaCle{u8[32]}. \end{itemize} As Post-condition we have: \begin{itemize} \item[] \VSTe{POST}: \VSTe{tint}\\ The function \texttt{crypto\_scalarmult} returns an integer. \item[] \VSTe{LOCAL}: \VSTe{temp ret_temp (Vint Int.zero)}\\ The returned integer has value $0$. \item[] \VSTe{SEP}: \VSTe{sh [{ v_q $\!\!\}\!\!]\!\!\!$ <<(uch32)-- mVI (CSM n p)}\\ In the memory share \texttt{sh}, the address \VSTe{v_q} points to a list of integer values \VSTe{mVI (CSM n p)} where \VSTe{CSM n p} is the result of the \VSTe{crypto_scalarmult} over \VSTe{n} and \VSTe{p}. \item[] \VSTe{PROP}: \VSTe{Forall (fun x => 0 <= x < 2^8) (CSM n p)}\\ \VSTe{PROP}: \VSTe{Zlength (CSM n p) = 32}\\ We show that the computation for \VSTe{CSM} fits in \TNaCle{u8[32]}. \end{itemize} This specification shows that \VSTe{crypto_scalarmult} in C computes the same result as \VSTe{CSM} in Coq provided that inputs are within their respective bounds. By converting those array of 32 bytes into their respective little-endian value we prove the correctness of \VSTe{crypto_scalarmult} (Theorem \ref{CSM-correct}) in Coq (for the sake of simplicity we do not display the conversion in the theorem). \begin{theorem} \label{CSM-correct} For all $n \in \N, n < 2^{255}$ and where the bits 1, 2, 5 248, 249, 250 are cleared and bit 6 is set, for all $P \in E(\F{p^2})$, for all $p \in \F{p}$ such that $P.x = p$, there exists $Q \in E(\F{p^2})$ such that $Q = nP$ where $Q.x = q$ and $q$ = \VSTe{CSM} $n$ $p$. \end{theorem} A more complete description in Coq of Theorem \ref{CSM-correct} with the associated conversions is as follow: \begin{lstlisting}[language=Coq] Theorem Crypto_Scalarmult_Correct: forall (n p:list Z) (P:mc curve25519_Fp2_mcuType), Zlength n = 32 -> Zlength p = 32 -> Forall (fun x => 0 <= x /\ x < 2^8) n -> Forall (fun x => 0 <= x /\ x < 2^8) p -> Fp2_x (ZUnpack25519 (ZofList 8 p)) = P#x0 -> ZofList 8 (Crypto_Scalarmult n p) = (P *+ (Z.to_nat (Zclamp (ZofList 8 n)))) _x0. \end{lstlisting}  Benoit Viguier committed Jun 21, 2019 161 162 \subsection{Number Representation and C Implementation}  Benoit Viguier committed Jul 10, 2019 163 As described in Section \ref{preliminaries:B}, numbers in \TNaCle{gf} are represented  Benoit Viguier committed Jun 21, 2019 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 in base $2^{16}$ and we use a direct mapping to represent that array as a list integers in Coq. However in order to show the correctness of the basic operations, we need to convert this number as a full integer. \begin{definition} Let \Coqe{ZofList} : $\Z \rightarrow \texttt{list} \Z \rightarrow \Z$, a parametrized map by $n$ betwen a list $l$ and its it's little endian representation with a base $2^n$. \end{definition} We define it in Coq as: \begin{lstlisting}[language=Coq] Fixpoint ZofList {n:Z} (a:list Z) : Z := match a with | [] => 0 | h :: q => h + (pow 2 n) * ZofList q end. \end{lstlisting} We define a notation where $n$ is $16$. \begin{lstlisting}[language=Coq] Notation "Z16.lst A" := (ZofList 16 A). \end{lstlisting} We also define a notation to do the modulo, projecting any numbers in $\Zfield$. \begin{lstlisting}[language=Coq] Notation "A :GF" := (A mod (2^255-19)). \end{lstlisting} Remark that this representation is different from \Coqe{Zmodp}. However the equivalence between operations over $\Zfield$ and $\F{p}$ is easily proven. Using these two definitions, we proved intermediates lemmas such as the correctness of the multiplication \Coqe{M} where \Coqe{M} replicate the computations and steps done in C. \begin{lemma} For all list of integers \texttt{a} and \texttt{b} of length 16 representing $A$ and $B$ in $\Zfield$, the number represented in $\Zfield$ by the list \Coqe{(M a b)} is equal to $A \times B \bmod \p$. \end{lemma} And seen in Coq as follows: \begin{Coq} Lemma mult_GF_Zlength : forall (a:list Z) (b:list Z), Zlength a = 16 -> Zlength b = 16 -> (Z16.lst (M a b)) :GF = (Z16.lst a * Z16.lst b) :GF. \end{Coq} \subsection{Inversions in \Zfield}  Benoit Viguier committed Jul 10, 2019 209 We define a Coq version of the inversion mimicking  Benoit Viguier committed Jun 21, 2019 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 the behavior of \TNaCle{inv25519} over \Coqe{list Z}. \begin{lstlisting}[language=Ctweetnacl] sv inv25519(gf o,const gf a) { gf c; int i; set25519(c,a); for(i=253;i>=0;i--) { S(c,c); if(i!=2 && i!=4) M(c,c,a); } FOR(i,16) o[i]=c[i]; } \end{lstlisting} We specify this with 2 functions: a recursive \Coqe{pow_fn_rev} to to simulate the for loop and a simple \Coqe{step_pow} containing the body. Note the off by one for the loop. \begin{lstlisting}[language=Coq] Definition step_pow (a:Z) (c g:list Z) : list Z := let c := Sq c in if a <>? 1 && a <>? 3 then M c g else c. Function pow_fn_rev (a:Z) (b:Z) (c g: list Z) {measure Z.to_nat a} : (list Z) := if a <=? 0 then c else let prev := pow_fn_rev (a - 1) b c g in step_pow (b - 1 - a) prev g. \end{lstlisting} This \Coqe{Function} requires a proof of termination. It is done by proving the  Benoit Viguier committed Jul 10, 2019 243 well-foundness of the decreasing argument: \Coqe{measure Z.to_nat a}. Calling  Benoit Viguier committed Jun 21, 2019 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 \Coqe{pow_fn_rev} 254 times allows us to reproduce the same behavior as the \texttt{Clight} definition. \begin{lstlisting}[language=Coq] Definition Inv25519 (x:list Z) : list Z := pow_fn_rev 254 254 x x. \end{lstlisting} Similarily we define the same function over $\Z$. \begin{lstlisting}[language=Coq] Definition step_pow_Z (a:Z) (c:Z) (g:Z) : Z := let c := c * c in if a <>? 1 && a <>? 3 then c * g else c. Function pow_fn_rev_Z (a:Z) (b:Z) (c:Z) (g: Z) {measure Z.to_nat a} : Z := if (a <=? 0) then c else let prev := pow_fn_rev_Z (a - 1) b c g in step_pow_Z (b - 1 - a) prev g. Definition Inv25519_Z (x:Z) : Z := pow_fn_rev_Z 254 254 x x. \end{lstlisting} And prove their equivalence in $\Zfield$. \begin{lstlisting}[language=Coq] Lemma Inv25519_Z_GF : forall (g:list Z), length g = 16 -> (Z16.lst (Inv25519 g)) :GF = (Inv25519_Z (Z16.lst g)) :GF. \end{lstlisting} In TweetNaCl, \TNaCle{inv25519} computes an inverse in $\Zfield$. It uses the Fermat's little theorem by doing an exponentiation to $2^{255}-21$. This is done by applying a square-and-multiply algorithm. The binary representation of $p-2$ implies to always do a multiplications aside for bit 2 and 4, thus the if case. To prove the correctness of the result we can use multiple strategies such as: \begin{itemize} \item Proving it is special case of square-and-multiply algorithm applied to a specific number and then show that this number is indeed $2^{255}-21$. \item Unrolling the for loop step-by-step and applying the equalities $x^a \times x^b = x^{(a+b)}$ and $(x^a)^2 = x^{(2 \times a)}$. We prove that the resulting exponent is $2^{255}-21$. \end{itemize} We use the second method for the benefits of simplicity. However it requires to apply the unrolling and exponentiation formulas 255 times. This can be automated in Coq with tacticals such as \Coqe{repeat}, but it generates a proof object which will take a long time to verify. \subsection{Speeding up with Reflections} In order to speed up the verification, we use a technique called reflection. It provides us with flexibility such as we don't need to know the number of times nor the order in which the lemmas needs to be applied (chapter 15 in \cite{CpdtJFR}). The idea is to \textit{reflect} the goal into a decidable environment. We show that for a property $P$, we can define a decidable boolean property $P_{bool}$ such that if $P_{bool}$ is \Coqe{true} then $P$ holds. $$reify\_P : P_{bool} = true \implies P$$ By applying $reify\_P$ on $P$ our goal become $P_{bool} = true$. We then compute the result of $P_{bool}$. If the decision goes well we are left with the tautology $true = true$. To prove that the \Coqe{Inv25519_Z} is computing $x^{2^{255}-21}$, we define a Domain Specific Language. \begin{definition} Let \Coqe{expr_inv} denote an expression which is either a term; a multiplication of expressions; a squaring of an expression or a power of an expression. And Let \Coqe{formula_inv} denote an equality between two expressions. \end{definition} \begin{lstlisting}[language=Coq] Inductive expr_inv := | R_inv : expr_inv | M_inv : expr_inv -> expr_inv -> expr_inv | S_inv : expr_inv -> expr_inv | P_inv : expr_inv -> positive -> expr_inv. Inductive formula_inv := | Eq_inv : expr_inv -> expr_inv -> formula_inv. \end{lstlisting} The denote functions are defined as follows: \begin{lstlisting}[language=Coq] Fixpoint e_inv_denote (m:expr_inv) : Z := match m with | R_inv => term_denote | M_inv x y => (e_inv_denote x) * (e_inv_denote y) | S_inv x => (e_inv_denote x) * (e_inv_denote x) | P_inv x p => pow (e_inv_denote x) (Z.pos p) end. Definition f_inv_denote (t : formula_inv) : Prop := match t with | Eq_inv x y => e_inv_denote x = e_inv_denote y end. \end{lstlisting} All denote functions also take as an argument the environment containing the variables. We do not show it here for the sake of readability. Given that an environment, \Coqe{term_denote} returns the appropriate variable. With such Domain Specific Language we have the equality between: \begin{lstlisting}[backgroundcolor=\color{white}] f_inv_denote (Eq_inv (M_inv R_inv (S_inv R_inv)) (P_inv R_inv 3)) = (x * x^2 = x^3) \end{lstlisting} On the right side, \Coqe{(x * x^2 = x^3)} depends on $x$. On the left side, \texttt{(Eq\_inv (M\_inv R\_inv (S\_inv R\_inv)) (P\_inv R\_inv 3))} does not depend on $x$. This allows us to use computations in our decision precedure. We define \Coqe{step_inv} and \Coqe{pow_inv} to mirror the behavior of \Coqe{step_pow_Z} and respectively \Coqe{pow_fn_rev_Z} over our DSL and we prove their equality. \begin{lstlisting}[language=Coq] Lemma step_inv_step_pow_eq : forall (a:Z) (c:expr_inv) (g:expr_inv), e_inv_denote (step_inv a c g) = step_pow_Z a (e_inv_denote c) (e_inv_denote g). Lemma pow_inv_pow_fn_rev_eq : forall (a:Z) (b:Z) (c:expr_inv) (g:expr_inv), e_inv_denote (pow_inv a b c g) = pow_fn_rev_Z a b (e_inv_denote c) (e_inv_denote g). \end{lstlisting} We then derive the following lemma. \begin{lemma} \label{reify} With an appropriate choice of variables, \Coqe{pow_inv} denotes \Coqe{Inv25519_Z}. \end{lemma} In order to prove formulas in \Coqe{formula_inv}, we have the following a decidable procedure. We define \Coqe{pow_expr_inv}, a function which returns the power of an expression. We then compare the two values and decide over their equality. \begin{Coq} Fixpoint pow_expr_inv (t:expr_inv) : Z := match t with | R_inv => 1 (* power of a term is 1. *) | M_inv x y => (pow_expr_inv x) + (pow_expr_inv y) (* power of a multiplication is the sum of the exponents. *) | S_inv x => 2 * (pow_expr_inv x) (* power of a squaring is the double of the exponent. *) | P_inv x p => (Z.pos p) * (pow_expr_inv x) (* power of a power is the multiplication of the exponents. *) end. Definition decide_e_inv (l1 l2:expr_inv) : bool := (pow_expr_inv l1) ==? (pow_expr_inv l2). Definition decide_f_inv (f:formula_inv) : bool := match f with | Eq_inv x y => decide_e_inv x y end. \end{Coq} We prove our decision procedure correct. \begin{lemma} \label{decide} For all formulas $f$, if the decision over $f$ returns \Coqe{true}, then the denoted equality by $f$ is true. \end{lemma} Which is formalized as: \begin{Coq} Lemma decide_formula_inv_impl : forall (f:formula_inv), decide_f_inv f = true -> f_inv_denote f. \end{Coq} By reification to over DSL (lemma \ref{reify}) and by applying our decision (lemma \ref{decide}). we proved the following theorem. \begin{theorem} \Coqe{Inv25519_Z} computes an inverse in \Zfield. \end{theorem} \begin{Coq} Theorem Inv25519_Z_correct : forall (x:Z), Inv25519_Z x = pow x (2^255-21). \end{Coq}  Benoit Viguier committed Jul 10, 2019 432 From \Coqe{Inv25519_Z_GF} and \Coqe{Inv25519_Z_correct}, we conclude the  Benoit Viguier committed Jun 21, 2019 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 functionnal correctness of the inversion over \Zfield. \begin{corollary} \Coqe{Inv25519} computes an inverse in \Zfield. \end{corollary} \begin{Coq} Corollary Inv25519_Zpow_GF : forall (g:list Z), length g = 16 -> Z16.lst (Inv25519 g) :GF = (pow (Z16.lst g) (2^255-21)) :GF. \end{Coq} \subsection{Packing and other Applications of Reflection} We prove the functional correctness of \Coqe{Inv25519} with reflections. This technique can also be used where proofs requires some computing or a small and finite domain of variable to test e.g. for all $i$ such that $0 \le i < 16$. Using reflection we prove that we can split the for loop in \TNaCle{pack25519} into two parts. \begin{lstlisting}[language=Ctweetnacl] for(i=1;i<15;i++) { m[i]=t[i]-0xffff-((m[i-1]>>16)&1); m[i-1]&=0xffff; } \end{lstlisting} The first loop is computing the substraction while the second is applying the carrying. \begin{lstlisting}[language=Ctweetnacl] for(i=1;i<15;i++) { m[i]=t[i]-0xffff } for(i=1;i<15;i++) { m[i]=m[i]-((m[i-1]>>16)&1); m[i-1]&=0xffff; } \end{lstlisting} This loop separation allows simpler proofs. The first loop is seen as the substraction of a number in \Zfield.  Benoit Viguier committed Jul 10, 2019 468 We then prove that with the iteration of the second loop, the number represented in $\Zfield$ stays the same.  Benoit Viguier committed Jun 21, 2019 469 470 471 472 473 474 475 476 477 This leads to the proof that \TNaCle{pack25519} is effectively reducing mod $\p$ and returning a number in base $2^8$. \begin{Coq} Lemma Pack25519_mod_25519 : forall (l:list Z), Zlength l = 16 -> Forall (fun x => -2^62 < x < 2^62) l -> ZofList 8 (Pack25519 l) = (Z16.lst l) mod (2^255-19). \end{Coq}  Benoit Viguier committed Jul 10, 2019 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576  \subsection{Using the Verifiable Software Toolchain} The Verifiable Software Toolchain uses a strongest postcondition strategy. The user must first write a formal specification of the function he wants to verify in Coq. This should be as close as possible to the C implementation behavior. This will simplify the proof and help with stepping throught the CLight version of the software. With the range of inputes defined, VST steps mechanically through each instruction and ask the user to verify auxiliary goals such as array bound access, or absence of overflows/underflows. We call this specification a low level specification. A user will then have an easier time to prove that his low level specification matches a simpler higher level one. In order to further speed-up the verification process, it has to be know that to prove \VSTe{crypto_scalarmult}, a user only need the specification of e.g. \VSTe{M}. This provide with multiple advantages: the verification by the Coq kernel can be done in parallel and multiple users can work on proving different functions at the same time. For the sake of completeness we proved all intermediate functions. Memory aliasing is the next point a user should pay attention to. The way VST deals with the separation logic is similar to a consumer producer problem. A simple specification of \texttt{M(o,a,b)} will assume three distinct memory share. When called with three memory share (\texttt{o, a, b}), the three of them will be consumed. However assuming this naive specification when \texttt{M(o,a,a)} is called (squaring), the first two memory shares (\texttt{o, a}) are consumed and VST will expect a third memory share where the last \texttt{a} is pointing at which does not \textit{exist} anymore. Examples of such cases are summarized in Fig \ref{tk:MemSame}. \begin{figure}[h] \include{tikz/memory_same_sh} \caption{Aliasing and Separation Logic} \label{tk:MemSame} \end{figure} This forces the user to either define multiple specifications for a single function or specify in his specification which aliasing version is being used. For our specifications of functions with 3 arguments, named here after \texttt{o, a, b}, we define an additional parameter $k$ with values in $\{0,1,2,3\}$: \begin{itemize} \item if $k=0$ then \texttt{o} and \texttt{a} are aliased. \item if $k=1$ then \texttt{o} and \texttt{b} are aliased. \item if $k=2$ then \texttt{a} and \texttt{b} are aliased. \item else there is no aliasing. \end{itemize} This solution allows us to make cases analysis over possible aliasing. \subsection{Verifiying \texttt{for} loops} Final state of \texttt{for} loops are usually computed by simple recursive functions. However we must define invariants which are true for each iterations. Assume we want to prove a decreasing loop where indexes go from 3 to 0. Define a function $g : \N \rightarrow State \rightarrow State$ which takes as input an integer for the index and a state and return a state. It simulate the body of the \texttt{for} loop. Assume it's recursive call: $f : \N \rightarrow State \rightarrow State$ which iteratively apply $g$ with decreasing index: \begin{equation*} f ( i , s ) = \begin{cases} s & \text{if } s = 0 \\ f( i - 1 , g ( i - 1 , s )) & \text{otherwise} \end{cases} \end{equation*} Then we have : \begin{align*} f(4,s) &= g(0,g(1,g(2,g(3,s)))) % \\ % f(3,s) &= g(0,g(1,g(2,s))) \end{align*} To prove the correctness of $f(4,s)$, we need to prove that intermediate steps $g(3,s)$; $g(2,g(3,s))$; $g(1,g(2,g(3,s)))$; $g(0,g(1,g(2,g(3,s))))$ are correct. Due to the computation order of recursive function, our loop invariant for $i\in\{0;1;2;3;4\}$ cannot use $f(i)$. To solve this, we define an auxiliary function with an accumulator such that given $i\in\{0;1;2;3;4\}$, it will compute the first $i$ steps of the loop. We then prove for the complete number of steps, the function with the accumulator and without returns the same result. We formalized this result in a generic way as follows: \begin{Coq} Variable T : Type. Variable g : nat -> T -> T. Fixpoint rec_fn (n:nat) (s:T) := match n with | 0 => s | S n => rec_fn n (g n s) end. Fixpoint rec_fn_rev_acc (n:nat) (m:nat) (s:T) := match n with | 0 => s | S n => g (m - n - 1) (rec_fn_rev_acc n m s) end. Definition rec_fn_rev (n:nat) (s:T) := rec_fn_rev_acc n n s. Lemma Tail_Head_equiv : forall (n:nat) (s:T), rec_fn n s = rec_fn_rev n s. \end{Coq} Using this formalization, we prove that the 255 steps of the montgomery ladder in C provide the same computations are the one defined in Algorithm \ref{montgomery-double-add}.