The Digital Signature Scheme MQQ-SIG

Transcription

1 The Dgtal Sgature Scheme MQQ-SIG Itellectual Property Statemet ad Techcal Descrpto Frst publshed: 10 October 2010, Last update: 20 December 2010 Dalo Glgorosk 1 ad Rue Stesmo Ødegård 2 ad Rue Erled Jese 2 ad Ludovc Perret 4 ad Jea-Charles Faugère 5 ad Sve Joha Kapskog 2 ad Smle Markovsk 3 1 Departmet of Telematcs, Faculty of Iformato Techology, Mathematcs ad Electrcal Egeerg, The Norwega Uversty of Scece ad Techology (NTNU), O.S.Bragstads plass 2E, N-7491 Trodhem, NORWAY, dalog@tem.tu.o 2 Norwega Uversty of Scece ad Techology Cetre for Quatfable Qualty of Servce Commucato Systems. O.S. Bragstads plass 2E, N-7491 Trodhem, NORWAY, kapskog@q2s.tu.o, rue.odegard@q2s.tu.o, rueerle@stud.tu.o 3 Ss Cyrl ad Methodus Uversty, Faculty of Natural Sceces ad Mathematcs, Isttute of Iformatcs, P.O.Box 162, 1000 Skopje, MACEDONIA, smle@.edu.mk 4 Perre ad Mare Cure Uversty - Pars, Laboratory of Computer Sceces, Pars 6, 104 aveue du Présdet Keedy Pars FRANCE, ludovc.perret@lp6.fr 5 UPMC, Uversté Pars 06, LIP6 INRIA, Cetre Pars-Rocquecourt, SALSA Project-team CNRS, UMR 7606, LIP6 4, place Jusseu Pars, Cedex 5, FRANCE jea-charles.faugere@ra.fr Abstract: Ths documet cotas the Itellectual Property Statemet ad the techcal descrpto of the MQQ-SIG - a ew publc key dgtal sgature scheme. The complete scetfc publcato coverg the desg ratoale ad the securty aalyss wll be gve a separate publcato. MQQ- SIG cossts of 4 quadratc polyomals wth Boolea varables where = 160, 192, 224 or 256. Keywords: Publc Key Cryptosystems, Fast sgature geerato, Multvarate Quadratc Polyomals, Quasgroup Strg Trasformatos, Multvarate Quadratc Quasgroup 1 Itellectual Property Statemet We, the seve ames gve the ttle of ths documet ad udersged o ths statemet, the authors ad desgers of MQQ-SIG dgtal sgature scheme, do hereby agree to grat ay terested party a rrevocable, royalty free lcece to practce, mplemet ad use MQQ-SIG dgtal sgature scheme, provded our roles as authors ad desgers of the MQQ-SIG dgtal sgature scheme are recogzed by the terested party as authors ad desgers of the MQQ-SIG dgtal sgature scheme. Name Sgature Place Date 1. Dalo Glgorosk Trodhem 2. Sve Joha Kapskog Trodhem 3. Smle Markovsk Skopje 4. Rue Stesmo Ødegård Trodhem 5. Rue Erled Jese Trodhem 6. Ludovc Perret Pars 7. Jea-Charles Faugère Pars

2 2 Descrpto of the MQQ-SIG dgtal sgature scheme A geerc descrpto for our scheme ca be expressed as a 3 4 trucato of a typcal multvarate quadratc system: S P S : {0, 1} {0, 1} where S = S x + v (.e. S s a bjectve affe trasformato), S s a osgular lear trasformato, ad P s a bjectve multvarate quadratc mappg o {0, 1}. The bjectve multvarate quadratc mappg P : {0, 1} {0, 1} s defed Table 1. Bjectve multvarate quadratc mappg P (x) Iput: A vector x = (f 1,..., f ) of lear Boolea fuctos of varables. We mplctly suppose that a multvarate quadratc quasgroup s prevously defed, ad that = 32k, k {5, 6, 7, } s also prevously determed. Output: lear expressos P (x 1,..., x ), = 1,..., ad multvarate quadratc polyomals P (x 1,..., x ), = 9,..., 1. Represet a vector x = (f 1,..., f ) of lear Boolea fuctos of varables x 1,..., x, as a strg x = X 1... X where X are vectors of dmeso ; 2. Compute y = Y 1... Y where: Y 1 = X 1, Y j+1 = X j X j+1, for eve j = 2, 4,..., ad Y j+1 = X j+1 X j, for odd j = 3, 5, Output: y. Table 1. Defto of the bjectve multvarate quadratc mappg P : {0, 1} {0, 1} The algorthm for geeratg the publc ad prvate key s defed the Table 2. Algorthm for geeratg Publc ad Prvate key for the MQQ-SIG scheme Iput: Iteger, where = 32 k ad k {5, 6, 7, }. Output: Publc key P: 4 multvarate quadratc polyomals P(x1,..., x), = 1+ 4,...,, Prvate key: Two permutatos σ 0 0 ad σ1 0 of the umbers {1,..., }, ad 1 bytes for ecodg a quasgroup. 1. Geerate a MQQ accordg to equatos (1)... (4). 2. Geerate a osgular Boolea matrx S ad affe trasformato S accordg to equatos (5),..., (). 3. Compute y = S(P (S (x))), where x = (x 1,..., x ). 4. Output: The publc key s y as 4 multvarate quadratc polyomals P (x 1,..., x ) = 1 + 4,...,, ad the prvate key s the tuple (σ0 0, σ1 0, ). Table 2. Geeratg the publc ad prvate key The algorthm for sgg by the prvate key (σ 0 0, σ 1 0, ) s defed Table 3. Algorthm for dgtal sgature wth the prvate key (σ 0 0, σ1 0, ) Iput: A documet M to be sged. Output: A sgature sg = (x 1,..., x ). 1. Compute y = (y 1,..., y ) = H(M), where M s the message to be sged, H() s a stadardzed cryptographc hash fucto such as SHA-1, or SHA-2, wth a hash output of ot less tha bts. The otato H(M) deotes the least sgfcat bts from the hash output H(M). 2. Set y = S 1 (y). 3. Represet y as y = Y 1... Y where Y are Boolea vectors of dmeso. 4. By usg the left ad rght parastrophes \ ad / of the quasgroup compute x = X 1... X, such that: X 1 = Y 1, X j = X j 1 \Y j, for eve j = 2, 4,..., ad X j = Y j/x j 1, for odd j = 3, 5, Compute x = S 1 (x ) + v = (x 1,..., x ). 6. The MQQ-SIG dgtal sgature of the documet M s the vector sg = (x 1,..., x ). Table 3. Dgtal sgg

3 The algorthm for sgature verfcato wth the publc key P = {P (x 1,..., x ) = 1+ 4,..., } s gve Table 4. Algorthm for sgature verfcato wth a publc key P = {P (x 1,..., x ) = 1 + 4,..., } Iput: A documet M ad ts sgature sg = (x 1,..., x ). Output: TRUE or FALSE. 1. Compute y = (y 1+,..., y ) = H(M), where M s the sged message, H() s a stadardzed 4 4 cryptographc hash fucto such as SHA-1, or SHA-2, wth a hash output of ot less tha bts, ad the otato H(M) 4 deotes the least sgfcat 4 bts from the hash output H(M). 2. Compute z = (z 1+,..., z 4 ) = P(sg). 3. If z = y the retur TRUE, else retur FALSE. Table 4. Dgtal verfcato 3 Multvarate Quadratc Quasgroups A Multvarate Quadratc Quasgroup (MQQ) of order 2 d used ths verso of MQQ-SIG ca be descrbed shortly by the followg expresso: x y B U(x) A 2 y + B A 1 x + c (1) where x = (x 1,..., x d ), y = (y 1,..., y d ), the matrces A 1, A 2 ad B are osgular GF (2), of sze d d, the vector c s a radom d-dmesoal vector wth elemets GF (2) ad all of them are geerated by a uformly radom process. The matrx U(x) s a upper tragular matrx wth all dagoal elemets equal to 1, ad the elemets above the ma dagoal are lear expressos of the varables of x = (x 1,..., x d ). It s computed by the followg expresso: d 1 U(x) = I + U A 1 x, (2) =1 where the matrces U have all elemets 0 except the elemets the rows from {1,..., } that are strctly above the ma dagoal. Those elemets ca be ether 0 or 1. Oce we have a multvarate quadratc quasgroup vv (x 1,..., x d, y 1,..., y d ) = (f 1 (x 1,..., x d, y 1,..., y d ),..., f d (x 1,..., x d, y 1,..., y d )) we wll be terested those quasgroups that wll satsfy the followg codtos: {1,..., d}, Rak(B f ) 2d 4, (3a) j {1,..., d}, Rak(B fj ) = 2d 2 (3b) where matrces B f are 2d 2d Boolea matrces defed from the expressos f as B f = [b j,k ], b j,d+k = b d+k,j = 1, ff x j y k s a term f. (4) Proposto 1. For d =, a multvarate quadratc quasgroup that satsfes the codtos (1),..., (4) ca be ecoded a uque way wth 1 bytes.

4 4 Nosgular Boolea matrces MQQ-SIG I MQQ-SIG the osgular matrces S are defed by the followg expresso: 16 S 1 = =0 I σ I σ 1, (5) =0 where I σ 0, = {0, 1, 2,..., 16 } ad I σ 1, = {0, 1, 2,..., } are permutato matrces of sze, the operato s a btwse exclusve or of the elemets the permutato matrces ad permutatos σ 0 ad σ1 are permutatos o elemets. They are defed by the followg expressos: σ0 0 radom permutato o {1, 2,... }, σ 0 = RotateLeft(σ0 1, ), for = 1,..., 16, σ0 1 (6) radom permutato o {1, 2,... }, σ 1 = RotateLeft(σ1 1, ), for = 1,..., , We chose the permutatos σ 0 0 ad σ 1 0 utl we obta a o-sgular matrx S 1. Oce we have a osgular matrx S 1 we wll compute ts verse obtag S = (S 1 ) 1 ad from there we wll obta the affe trasformato S (x) = S x + v, (7) where the vector v s dmesoal Boolea vector defed from the values of the permutato σ 1 0 = (s 1, s 2,..., s ) by the followg expresso: v = (v 1, v 2,..., v ), where v = (( s 1+ 1 ) 2 ( ) mod ) mod ( s65+ ) 1 2 ( ) mod mod 2. () I words: we costruct the bts of the vector v by costructg two arrays. The frst array s costructed by takg the four least sgfcat bts of the values s 1,..., s ad each of them s shfted by four postos to the left. The secod array s just smple extracto of the values s 65,..., s 65+. Fally we XOR correspodgly those two arrays of values order to produce the vector v of bts. Proposto 2. The lear trasformato S 1 ca be ecoded a uque way wth 2 bytes. 5 Characterstcs of the MQQ-SIG dgtal sgature scheme The ma characterstcs of our MQQ-SIG dgtal sgature scheme ca be brefly summarzed as follows: there s o message expaso; the legth of the sgature s bts where ( = 160, 192, 224 or 256); ts cojectured securty level s 2 2 ; ts verfcato speed s comparable to the speed of other multvarate quadratc PKCs; software ts sgg speed s the rage of 300 7,000 tmes faster tha RSA ad ECC schemes; hardware ts sgg or verfcato speed s more tha 10,000 tmes faster tha RSA ad ECC schemes; t s also well suted for producg short sgatures smart cards ad RFIDs;

5 5.1 The sze of the publc ad the prvate key The sze of the publc key s 0.75 (1 + (+1) 2 ) bts. The prvate key of our scheme s the tuple (σ 0 0, σ 1 0, ). The correspodg memory sze eeded for storage of the prvate key s bytes. I Table 5 we gve the sze of the publc key ( KBytes) ad the sze of the prvate key ( bytes) for {160, 192, 224, 256}. Sze of the Sze of the publc key (KBytes) prvate key (bytes) Table 5. Memory sze KBytes for the publc key ad bytes for the prvate key 5.2 Performace of the software ad hardware mplemetato of the MQQ-SIG algorthm We have mplemeted MQQ-SIG C for the SUPERCOP bechmarkg system yp.to/supercop.html ad tested t together wth the correspodg RSA ad ECC. I Table 6 we gve the comparso of MQQ-SIG wth RSA ad ECC 64-bt mode of operato o Itel Core 7 920X mache rug at 2 GHz. The umbers the table represet CPU cycles. Although, our C code s ot yet optmzed for the key geerato part, we expect that the performace of key geerato part would be the most tme cosumg part of our algorthm. O the other had, from the Table 6 t s clear that sgg of 59 bytes MQQ-SIG s faster tha RSA the rage from 565 up to 636 tmes, ad s faster tha ECC the rage from 325 up to 517 tmes. The verfcato speed our code s ot so dstctvely faster tha the correspodg RSA ad ECC sce t s programmed for oe core. We expect that the hgh parallelzable ature of MQQ-SIG ca be used to acheve much hgher speeds multcore systems (CPUs or GPUs). Securty bts Algorthm KeyGe Sgg of 59 bytes Verfcato of a sgature of 59 bytes RSA ,69,553 2,230,4 61,116 0 ECC160 1,201,1 1,24,00 1,476,196 MQQSIG160 1,062,12,500 3,440 97,644 RSA ,324,721 7,346, , ECC192 1,799,24 1,95,752 2,242,9 MQQSIG192 1,2,301,276 4,260 72,60 RSA204 76,466,59 14,15, , ECC224 2,022,96 2,10,556 2,501,10 MQQSIG224 2,539,322,544 4,160 92,960 RSA3072 2,719,353,53 31,941, , ECC256 2,296,976 2,41,96 2,33,56 MQQSIG256 4,96,642,44 4,932 13,14 Table 6. Comparso betwee performace of RSA, ECC ad MQQ-SIG CPU cycles 64-bt mode of operato o Itel Core 7 920X mache rug at 2 GHz.