Encryption / Decryption VHDL Core Final Presentation Part A

1. Technion - Israel institute of technology department of Electrical Engineering הטכניון - מכון טכנולוגי לישראלהפקולטה להנדסת חשמל Encryption / Decryption VHDL CoreFinal Presentation Part A Instructor : MonyOrbach Semester : Winter-Spring 2013 Performed By: Watad Duna, WatadEsam Duration : Year

2. Project’s Goals • Implementing RSA Encryption Decryption Core. • Enhancing encryption and decryption processing time. • Achieving standard safety level

3. Project(Part a) Goals • Golden Matlab model. • VHDL RSA Encryption\Decryption Core. • Encryption\Decryption Core Simulation.

4. RSA Theory - Reminder • Key Generation Algorithm • Generate two large random primes, p and q • Compute n = pq and (phi) φ = (p-1)(q-1) • Choose an integer e, 1<e<phi, such that gcd(e,phi) = 1 • Compute the secret exponent d, 1 < d < phi, such that (e x d) ≡ 1 (mod phi) • The public key is (n, e) and the private key (d, p, q)

5. RSA Theory - Reminder(Cont.) • Encryption Sender A does the following: • Obtains the recipient B's public key (n, e) • Represents message as a positive integer m, 1< m< n • Computes c = me mod n • Sends the c to B

6. RSA Theory - Reminder(Cont.) • Decryption Recipient B does the following: • Uses his private key (n, d) to compute m = cd mod n • Extracts the message from the representative m

7. Golden Model • Java.math.BigInteger • M = java.math.BigInteger(m); • E = java.math.BigInteger(e); • N = java.math.BigInteger(n); • C = m.modPow(e,n);

8. System Launching Operations • RSA algorithm needs parameters (p,q) as input • This inputs will determine another parameter • N,Phi,e,d • The following code finds all these parameters

9. System Launching Operations-cont. function res = FindPrimeBiggerThan( input ) myres= input; a = java.math.BigInteger('1'); primeWereFounded = 0; while ( primeWereFounded ==0) myres = myres.add(a) primeWereFounded = myres.isProbablePrime(1012456875); end res = myres; end function res = FindE( input,p,q) one = java.math.BigInteger('1'); mp = p.subtract(one); mq = q.subtract(one); phi = mp.multiply(mq); tmp = java.math.BigInteger('2'); startFrom = tmp.pow(255); while(startFrom.gcd(phi)!=1) startFrom = startFrom.add(one); end res = startFrom; end function res = IsRealyPrime( input ) a = java.math.BigInteger('2'); %b = java.math.BigInteger('260'); zero = java.math.BigInteger('0'); one = java.math.BigInteger('1'); b = a.pow(260); res = 1; myinput = input; while(a.compareTo(b)==(-1)) if(myinput.mod(a).compareTo(zero) == 0) res = 0; break; end a = a.add(one); end res = res; end function res = FindDForE( e , phi ) one = java.math.BigInteger('1'); tmp = java.math.BigInteger('2'); d = tmp.pow(256); while(d.multiply(e).mod(phi).compareTo(one)) d= d.add(one) end res = d end

10. System Launching Operations- Results P = 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084171 Q= 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298168510365734290848462028012946095045414468772940216441841386573 N = PxQ phi = 179769313486231590772930519078902473361797697894230657273430081157732675805500963132708477322429081537772856017757052452219066775828510814325211150767994823723247172937861228229027691397192757191152180662345359209725607627810522086248050040267465404662290479408319732043037440995227714070917226519336939765240 E = 57896044618658097711785492504343953926634992332820282019728792003956564819971 D =

11. Enc./Dec Unit Design reg Mux M reg %n X reg Mux control e

12. The Enc./Dec. Unit – The Algorithm • To calculate c = m^e%n • Represent e by it’s binary base ( e= e1e2e3e4… ek) • c = m%n • For i=2 to i=k • If ei=0 then , C  (C x C) % n • If ei=1 then , C ( ((C x C) %n) x m%n ) %n • Return C. • Example e=149 ( e = 10010101) • i=2 , m^2%n= c (m^1%n x m^1%n ) % n • i=3 , m^4%n= c (m^2%n x m^2%n ) % n • i=4 , m^9%n= c (( (m^4%n x m^4%n ) % n ) x m^1%n) %n • i=5 , m^18%n= c (m^9%n x m^9%n ) % n • i=6 , m^37%n= c (( (m^18%n x m^18%n ) % n ) x m^1%n) %n • i=7 , m^74%n= c (m^37%n x m^37%n ) % n • i=8 , m^149%n= c (( (m^74%n x m^74%n ) % n ) x m^1%n) %n

13. The Enc./Dec- Mult a Adder Controller 1024 Entries b 2048 bits

14. Multiply Unit - Controller • For each bit in b • Say the index of the bit is i : • If b(i)=0 then the i-output will be 1024 bits vector with value 0 • Else the i-output will be 1024 bits vector • Bits i-0 with value 0 • Bits i-i+1024 will a bits • Bit 2048 – i+1024 with value 0 • Simple example : • 1010 x 0101 = 00001010+00000000+00101000+00000000 = 01101010 = 50 = 5 x 10

15. The Enc./Dec-Mod -Inputs . In1: Abinary number (2048 bits). . In2=N : A binary number (1024 bits). . Select ( technical problem) • Outputs . Ready : Indicate that the computing was finished.. Result : The final result In1%In2 .

16. Mod Unit Design Result Subtractor Reg MUX N M U X Comp.0 In1 Comp.1 In2=N Conc.Unit Ready Comp_R ‘1’

17. Mod Unit- Conc.Unit • The unit concatenates zeros . -Input . A binary number (1024 bits) N -Outputs . 1025 binary numbers (2048 bits). Example :

18. Mod Unit- NMux • The output is the data which has been selected by the select vector. The index of the last bit of value ‘1’ is the index of the data that would be sent to the output. -Input .select: vector of 1025 bits. .Data :1025 binary number (2048 bits). -Output . A binary number (2048 bits). Comp comp

19. Summary • Acquired a lot of knowledge about Encryption especially RSA Algorithm • Achieved standard safety level • Top-Down design and encapsulation are very useful • We adopt abstraction as a style of coping with the big challenge of the large numbers

20. Planning Ahead • Completing simulation successfully. • Synthesis On FPGA. • Timing & Performance enhancements