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CS252: Systems Programming Ninghui Li Topic 4: Programming in a FIZ and FLIZ: Simple Functional Programming Language What is FIZ FIZ F is for functional programming I is for integer (we only use integer data type) Z is for zero, denoting the simplicity of the language In Functional Programming, one defines functions writes expressions Syntax: instead of writing f(a, b, c); we write (f a b c) slide 2 Basic Features of FIZ non-negative integer evaluates to its value (inc exp) evaluates to value of exp + 1 (dec exp) evaluates to halt if value of exp is 0 otherwise evaluates to value of exp - 1 (ifz cexp texp fexp) evaluates to value of texp when cexp evaluates to 0 and to value of fexp when cexp evaluates to none 0 slide 3 Examples (inc 3) (inc (dec (inc 1))) (ifz (dec 1) 2 3) 4 2 2 slide 4 Defining New Functions (define (name arguments) function-body ) Examples: (define (add2 x) (inc (inc x))) (define (add x y) (ifz y x (add (inc x) (dec y)))) (name arguments) E.g., (add 2 3); (add (inc 2) (dec 3)) slide 5 Dealing with Errors (halt) stops the program Whenever the evaluation leads to non-integer or negative number, use (halt) Note: FIZ has no assignment, except in function invocation. FIZ has no loop, except in recursion slide 6 An Example: Add (define (add x y) (ifz y x (add (inc x) (dec y)))) ; if y==0, x+y=x ; otherwise x+y = (x+1) + (y-1) We assume lazy evaluation, i.e., in (ifz cond t_exp e_exp) We evaluate t_exp only when cond is 0, and evaluate e_exp only when cond is not 0 Lazy versus Eager Evaluation In Eager evaluation In Lazy evaluation (add 3 0) (add 3 0) becomes becomes (ifz 0 3 (add (inc 3) (dec 0))) (ifz 0 3 (add (inc 3) (dec 0))) We do not evaluate the last We need to evaluate the last argument in the above function argument yet, and figures out call, i.e., that we do not need it (add (inc 3) (dec 0)), Which becomes (add 4 -1) This then becomes infinite loop A similar issue in C: In “If (cond_1 && func(a,b))”, if cond_1 is false, would func(a,b) be called? More Examples: Multiplication (define (mul x y) (ifz y 0 ; when y=0, answer is 0 (add x (mul x (dec y)))) ; otherwise, answer is x + x*(y-1) Key is to find a base case, and then figure out how to reach the base case. Multiplication base case: When y=0, x*y=0 Recursive step: x*y = x + x*(y-1) More Examples: Substraction ; Evalutes x – y What is the base case? • When y is 0, result is x. What is the recursive step? • Computer (dec x) – (dec y) (define (sub x y) (ifz y x (ifz x (halt) ; if y==0, x-y = x ; else result is negative (sub (dec x) (dec y))))) ; else x-y = (x-1) - (y-1) More Examples: Less Than ; (define (lt x y) ; Evaluates to 0 if x < y, and 1 otherwise What are the base cases? • One of x and y is 0. What is the recursive step? • Compare x-1 with y-1 (define (lt x y) (ifz y 1 ; if y==0, x<y cannot be true (ifz x 0 ; else if x==0, x<y must be true (lt (dec x) (dec y))))) ; else x<y if and only if x-1 < y-1 Last FIZ Example: Remainder ; Evaluates x mod y (i.e., x % y) What are the base cases? • When x < y, x mod y = x What is the recursive step? • Otherwise, x mod y = (x-y) mod y (define (rem x y) (ifz y (halt) (ifz (lt x y) x ; Division by 0 ; if x<y, x mod y = x (rem (sub x y) y)))) ; else x mod y = (x-y) mod y Concept of Public Key Encryption In Traditional Cryptography, encryption key = decryption key, and must be kept secret, and key distribution/agreement is a difficult problem to solve In public key encryption, each party has a pair (K, K-1) of keys: K is the public key, and used for encryption K-1 is the private key, and used for decryption Satisfies DK-1[EK[M]] = M Knowing K, it is computationally expensive to find K-1 The public key K may be made publicly available, e.g., in a publicly available directory Many can encrypt, only one can decrypt Public-key systems aka asymmetric crypto systems RSA Algorithm Invented in 1978 by Ron Rivest, Adi Shamir and Leonard Adleman Published as R L Rivest, A Shamir, L Adleman, "On Digital Signatures and Public Key Cryptosystems", Communications of the ACM, vol 21 no 2, pp120126, Feb 1978 Security relies on the difficulty of factoring large composite numbers Essentially the same algorithm was discovered in 1973 by Clifford Cocks, who works for the British intelligence RSA Public Key Crypto System Key generation: 1. Select 2 large prime numbers of about the same size, p and q Typically each p, q has between 512 and 2048 bits 2. Compute n = pq, and (n) = (q-1)(p-1) 3. Select e, 1<e< (n), s.t. gcd(e, (n)) = 1 Typically e=3 or e=65537 4. Compute d, 1< d< (n) s.t. ed 1 mod (n) Knowing (n), d easy to compute. Public key: (e, n) Private key: d RSA Description (cont.) Encryption Given a message M, 0 < M < n M Zn {0} use public key (e, n) compute C = Me mod n C Zn {0} Decryption Given a ciphertext C, use private key (d) Compute Cd mod n = (Me mod n)d mod n = Med mod n = M RSA Example p = 41, q = 23, n = 943, (n) = 40*22= 880 Choosing e = 3 Then d = 587 because 587 * 3 % 880 = 1761 % 880=1 Let M = 15. Then C Me mod n C 153 (mod 943) = 546 M Cd mod n M 546587 (mod 943) = 15 Topic 6: Public Key Encrypption and Digital Signatures 17 How to do modular exponentiation in FIZ. ; How to compute xe mod y? What is the base case? • When e = 0, xe mod y = 1 What is the recursive step? • Otherwise, xe mod y = (x * (xe-1 mod y)) mod y (define (msexp x e y) (ifz e 1 (rem (mul (msexp x (dec e) y) x) y))) Topic 6: Public Key Encrypption and Digital Signatures 18 A Faster Way to Do modular exponentiation. ; How to compute x^e mod y? How to improve the recursive step? • When e is even, xe mod y = ((xe/2 mod y)2 mod y) • When e is odd, xe mod y = (x*(xe-1 mod y) mod y) (define (mexp x e y) (ifz e 1 (ifz (rem e 2) (rem (square (mexp x (div e 2) y)) y) (rem (mul x (mexp x (dec e) y)) y)))) If e==0, then result is 1 Else if e is even, result is (x^(e/2) mod y) ^ 2 mod y Else if e is odd, result is (x * (x^(e-1) mod y)) mod y Topic 6: Public Key Encrypption and Digital Signatures 19 The FLIZ Language There are the following constant expressions • 123 ; a number • [] ; the empty list • [c c … c] ; a list of one of more constant expressions Examples: 123 ; 123 [1 2 3] ; [ 1 2 3 ] [] ; [ ] [1 [2 3] [4 5]] ; [ 1 [ 2 3 ] [ 4 5] ] [1 [] 2 [[3 4]]]; [ 1 [ ] 2 [ [ 3 4 ] ] ] slide 20 The FLIZ Language: Builtin Functions (list exph expt) a list obtained by inserting the value of exph as first element to the value of expt (head exp) first element of the list exp (tail exp) a list obatined by removing the first element (ifa cexp texp fexp) when cexp evaluates to a number, evaluates to texp, otherwise evaluates to fexp (ifn cexp texp fexp) evaluates to value of texp when cexp evaluates to [] and to value of fexp when cexp evaluates not to [] slide 21 Programming in FLIZ ; append two lists (append x y) • What is the base case? • When x is the empty list, result is y • What is the recursive step? • Otherwise, result is a list with head being the head of x, and tail being that from appending tail of x and y (define (append x y) (ifn x y (list (head x) (append (tail x) y)))) Programming in FLIZ ; flatten a list (flatten x) • What is the base case? • When x is the empty list, result is []; or when x is an atomic value (i.e., a number), result is [x] • What is the recursive step? • Flatten head and tail respectively, and then append them. ((define (flatten x) (ifn x x (ifa x (list x []) (append (flatten (head x)) (flatten (tail x)))))) Another FLIZ Example Program ; flatten a list (define (flatten x) (ifn x x (ifa x (list x []) (append (flatten (head x)) (flatten (tail x)))))) FIZ and FLIZ are based on LISP/Scheme LISP/Scheme is a functional programming language Uses the list data structure extensively Program/Data equivalence Heavy use of recursion Garbage-collected, heap-allocated Usually interpreted, but good compilers exist 25 History of Function Programming Lisp was created in 1958 by John McCarthy at MIT Stands for LISt Processing Scheme developed in 1975 A dialect of Lisp Racket ML OCaml Haskell 26 Application Areas Artificial Intelligence expert systems planning Simulation, modeling Rapid prototyping 27 Functional Languages Imperative Languages Ex. Fortran, Algol, Pascal, Ada based on von Neumann computer model Functional Languages Ex. Scheme, Lisp, ML, Haskell Based on mathematical model of computation and lambda calculus 28 Review Be able to write simple programs in FIZ and FLIP. Know the concept of lazy evaluation versus eager evaluation. How RSA work is not required. 29 Upcoming Attraction How to write an interpreter/compiler frontend? Topic 5: Regular expressions & Lexical Analyzer Topic 6: Context-free grammar & Parser 30