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Application of Quasigroup Fields in Designing Efficient Hash Functions\251 Czeslaw Koscielny 2003 University of Zielona Gora, Poland Institute of Control and Computation Engineering Academy of Management in Legnica, Poland Faculty of Computer Science c.koscielny@issi.uz.zgora.pl

Already four years ago, two eminent specialists on quasigroups, D\351nes and Keedwell [3], augured the advent of a new era in cryptology, consisting in the application of non-associative algebraic systems [1, 2, 4, 5, 6]. Nevertheless, at present very few researchers use these tools and in many cases it seems unreasonable and reckless. For example, constructing one-way functions, algorithms are usually designed, in which computations are performed using regular algebraic systems as groups, rings and fields, simple boolean operations, modular arithmetic, and cyclic permutations. Such an approach may simplify cryptanalysis. However, computations of the value of cryptographic one-way function should be easy, but taking into account the security, an algorithm describing the hash function should involve rather an algebraic system, which is strongly recalcitrant, and the behaviour of which is unforeseeable. A quasigroup field, as an algebraic system, has such favorable properties. Thus quasigroup fields can be easily applied in designing both unkeyed and keyed hash functions, iterated or not-iterated as well [7, p. 321], that is why in this contribution it will be shown how to do it.

Consider an algebraic system <NiNJIlFHNiI=, +, .>, consisting of a finite set of elements NiNJIlFHNiI= in which two internal binary operations, called addition and multiplication, respectively, are defined. Let NiMvLUkkYWJzR0kqcHJvdGVjdGVkR0YmNiNJIlFHNiJJInFHRik= where NiNJInFHNiI= denotes an arbitrary positive integer >2. The NiNJInFHNiI=-element quasigroup field, denoted by NiMtSSNRRkc2IjYjSSJxR0Yl is the above system, satisfying the axioms:<NiNJIlFHNiI=, +> and <NiNJIlFHNiI=, .> are quasigroups,c NiMtSSNpbkc2IjYkSSJhR0YlSSJRR0Yl d- NiMtSSNpbkc2IjYkSSJhR0YlSSJRR0Ylc NiMtSSNpbkc2IjYkSSJhR0YlSSJRR0Yl d- NiMtSSNpbkc2IjYkKUkiYUdGJSwkIiIiISIiSSJRR0Ylc NiM2JUkiYUc2IkkiYkdGJS1JI2luR0YlNiRJImNHRiVJIlFHRiU= NiM3Iy8tSSJhRzYiNiMsJkkiYkdGJyIiIkkiY0dGJ0YrLCZJI2FiR0YnRitJI2FjR0YnRis= o[ NiMqJiwmSSJhRzYiIiIiSSJiR0YmRidGJ0kiY0dGJkYn = NiMsJkkjYWNHNiIiIiJJI2JjR0YlRiY=]This means that multiplication is distributive under addition, and that there exist in NiMtSSNRRkc2IjYjSSJxR0Yl formal additive and multiplicative inverses, which can be functions many-to-one as well as one-to-one, both having nothing to do with addition and multiplication. From the last but one axiom it follows that the division by 0 is possible. For practical purposes, it is also assumed, that NiNJIlFHNiI= ={0, 1, ..., NiMsJkkicUc2IiIiIkYmISIi}. Since <NiNJIlFHNiI=, +>and <NiNJIlFHNiI=, .> are quasigroups, addition and multiplication need be neither commutative nor associative.

Addition table in 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Multiplication table in 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Additive inverse in 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U2RMa1RhblxcQE9kU2NrWURdeHJVVEtdSEhNR0RPZnBDT05PX0RQbnBGT089Sz9vYGJabFhfc3lbbmFdTGFrOmZpXUZnOko6YG5cXHROXFx0VFs8UDsyOw==Multiplicative inverse in 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 that we want to multiply a vector NiMvSSJ1RzYiLUknTWF0cml4RzYkSSpwcm90ZWN0ZWRHRilJKF9zeXNsaWJHRiU2IzcjNyRJImFHRiVJImJHRiU= by a matrix NiMvSSJtRzYiLUknTWF0cml4RzYkSSpwcm90ZWN0ZWRHRilJKF9zeXNsaWJHRiU2IzckNyRJIkFHRiVJIkJHRiU3JEkiQ0dGJUkiREdGKA==over NiMtSSNRRkc2IjYjIiM1. Thus NiMvSSJ2RzYiKiZJInVHRiUiIiJJIk1HRiVGKA== = NiMqJi1JJ01hdHJpeEc2JEkqcHJvdGVjdGVkR0YnSShfc3lzbGliRzYiNiM3IzckSSJhR0YpSSJiR0YpIiIiLUYlNiM3JDckSSJBR0YpSSJCR0YpNyRJIkNHRilJIkRHRiZGLw== = NiMtSSdNYXRyaXhHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYjNyM3JCwmKiZJImFHRigiIiJJIkFHRihGL0YvKiZJImJHRihGL0kiQ0dGKEYvRi8sJiomRi5GL0kiQkdGKEYvRi8qJkYyRi9JIkRHRiVGL0Yv Assuming that, for example, NiMvSSJhRzYiIiIh, NiMvSSJiRzYiIiIi, NiMvSSJBRzYiIiIj, NiMvSSJCRzYiIiIk, NiMvSSJDRzYiIiIl, NiMvSSJERzYkSSpwcm90ZWN0ZWRHRiZJKF9zeXNsaWJHNiIiIiY=, we may compute the components of the vector NiNJInZHNiI= using operation tables in NiMtSSNRRkc2IjYjIiM1: NiNJInZHNiI= = NiMtSSdNYXRyaXhHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYjNyM3JCwmKiYiIiEiIiIiIiNGL0YvKiZGL0YvIiIlRi9GLywmKiZGLkYvIiIkRi9GLyomRi9GLyIiJkYvRi8= =NiMtSSdNYXRyaXhHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYjNyM3JCIiJSIiIQ== But to determine NiNJInVHNiI= knowing NiNJInZHNiI= we must solve the following system of linear equations NiMvLCYqJkkiYUc2IiIiIiIiI0YoRigqJkkiYkdGJ0YoIiIlRihGKEYsNiMvLCYqJkkiYUc2IiIiIiIiJEYoRigqJkkiYkdGJ0YoIiImRihGKCIiIQ== over NiMtSSNRRkc2IjYjIiM1 by substituting for a and b all possible elements of NiMtSSNRRkc2IjYjIiM1. The reader can verify that the following four solutions are possible:NiMvSSJhRzYiIiIh, NiMvSSJiRzYiIiIi,NiMvSSJhRzYiIiIl , NiMvSSJiRzYiIiIq,NiMvSSJhRzYiIiIn, NiMvSSJiRzYiIiIh,NiMvSSJhRzYiIiIp, NiMvSSJiRzYiIiIl,This task is much easier over an arbitrary ordinary field if a matrix NiNJIk1HNiI= is not singular: one only need to multiply NiNJInZHNiI= by NiMpSSJNRzYiLCQiIiIhIiI= . In the presented example of computing in NiMtSSNRRkc2IjYjIiM1 addition and multiplication are only used. It is evident that NiMtSSNRRkc2IjYjIiM1-based one-way functions have better cryptographic properties if algorithms for determining their values involve all four operations in a quasigroup field.

As it is known, hash function NiMvLUkiaEc2IjYjSSJNR0YmSSJIR0Ym maps NiNJImtHNiI=-element message strings NiNJIk1HNiI= of arbitrary length to fixed NiNJInJHNiI=-elements strings NiNJIkhHNiI=. The NiNJInJHNiI=-element of hash value NiNJIkhHNiI= represents compactly and uniquely the message (usually NiMqJkkickc2IiIiIkkia0dGJUYm ). For designing quasigroup field-based hash functions plenty diverse algorithms can be applied. In this application a simple algorithm which computes NiMqJilJInhHNiJJInJHRiYiIiItSSJtR0YmNiNGJUYo (mod NiMtSSJnRzYiNiNJInhHRiU= )over an arbitrary NiMtSSNRRkc2IjYjSSJxR0Yl, making use of all four operations, has been chosen.LetNiMvSSJNRzYiJkkibUdGJTYjIiIi, NiMmSSJtRzYiNiMiIiM=, ..., NiMmSSJtRzYiNiNJImtHRiU=, NiMtSSNpbkc2IjYkJkkibUdGJTYjSSJpR0YlLUkjUUZHRiU2I0kicUdGJQ==be an input message, associated with the polynomial of degree NiMsJkkia0c2IiIiIkYmISIi over NiMtSSNRRkc2IjYjSSJxR0YlNiMvLUkibUc2IjYjSSJ4R0YmLCYqJiZGJTYjIiIiRi0pRigsJkkia0dGJkYtRi0hIiJGLUYtKiYmRiU2IyIiI0YtKUYoNyMsJkYwRi1GNUYxRi1GLQ== +...+ NiMsJiomJkkibUc2IjYjLCZJImtHRiciIiJGKyEiIkYrSSJ4R0YnRitGKyZGJjYjRipGKw==Further letNiMvLUkiZ0c2IjYjSSJ4R0YmLCYqJiZGJTYjLCZJInJHRiYiIiJGL0YvRi8pRihGLkYvRi8qJiZGJTYjRi5GLylGKDcjLCZGLkYvRi8hIiJGL0Yv+...+NiMsJiomJkkiZ0c2IjYjIiIjIiIiSSJ4R0YnRipGKiZGJjYjRipGKg==be an arbitrary polynomial of degree NiNJInJHNiI= over NiMtSSNRRkc2IjYjSSJxR0Yl Then, using operations in NiMtSSNRRkc2IjYjSSJxR0Yl we can determine d(x) = NiMqJilJInhHNiJJInJHRiYiIiItSSJtR0YmNiNGJUYo (mod NiMtSSJnRzYiNiNJInhHRiU= ) = NiMsJiomJkkiaEc2IjYjSSJyR0YnIiIiKUkieEdGJywmRilGKkYqISIiRipGKikmRiY2I0YtLCZGKUYqIiIjRi5GKg==+...+ NiMsJiomJkkiaEc2IjYjIiIjIiIiSSJ4R0YnRipGKiZGJjYjRipGKg==and the computed hash value will be NiMvSSJIRzYiJkkiaEdGJTYjIiIi, NiMmSSJoRzYiNiMiIiM=, ..., NiMmSSJoRzYiNiNJInJHRiU=, NiMtSSNpbkc2IjYkJkkiaEdGJTYjSSJpR0YlLUkjUUZHRiU2I0kicUdGJQ== The above algorithm can be implemented by means of hardware in the form of feedback shift register over NiMtSSNRRkc2IjYjSSJxR0Yl shown in Fig. 1. MFNWtKUb<ob<R=MDLCdNNZ[m:dK>H:\\rK`qms:<O`Lo\\jyyyykV`mVHocHOn\\PndQwdq?ir?asFasMYtMMUTMUZ=VZmV`mveYweAxjAXpmXpQytQyyyyyK:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::bTqYNmE:PK[aN\\@Nd\\Qgqxh`X;EUWEK`\\wZqXYyyctoYXKlpv^<jDPKDLSP<mdxMmLOJ`TqMkHuN:=nN@J:Qj[Er?>_VBb[M:Ev:N[ABDs:;:KCA:<:\\<>:>?:<:\\<>:>?:<:\\<>:>?:<:\\<>:>?:<VR\\m_UmAarYLWBDLtTMQhOciNexY[ljmTRb<pSAQEhObhxcdyMLKUxUu`Qstkx@yMPlYuY\\UpkMJ\\=aJY]=ip_ihfOqR?pBn]mOfDA\\=yjpYhN^aP@sNN[yp`RfZIhflfj<I:>nj;;Z:>?B:[SL=YBAE][bksTAGXTUGq;dAQdROE]?XIqgyYyt[cMQUsGbvkd>aXj[fYydLkeess_MY<MdHQsG[vkSG`WXswG?iWwiTHcgOWEBYVC]SGGc[arE=CkMCWeihosDi[V@mHqvXQbKv[vfm[Fsm>no`c[Ip^nn@A_vafe_iT^degv@:=FfEyqMStEICexoMitSGq;dAagRos_ICBcgKCrN?Xnsej?ingtQkDxaDUovDi[YXxRnoTYZEqj^A]HqnHQox`oUwgiNf@vkyNwkFmDvflojHvePioaHsFGvkV^`X\\EqbXPrYxl_>ds`c;YcWOmPAcjfjAyepAmffjqo_R>odg\\;qs_xxn^_[?oOI[m`fdFjA`]?xxlOd[apXin_f\\iHasqtC^trgiifr]OdNnyH@lnwnw?oVPofo_gocRh^MGav^fb^ywV]Cqauw_\\ghGnnOI[ml>YR<PryDXKPwmxSKMUO]LY<r=AmETO`=POhNUTSsTtdUNMatVMVeTP:mP`avblnjdVYPngypv\\nVUt^`RMxs@\\RbeNLtP?xr:IrWHMXTl>YrLAtrPYUutJDWSTr=Am`Qw]Ivvqr=QtZ<XcmQ]mjnqy=hm:lnYDY>lWpYMIHwiyKaXLiyPs@rcPjpTtaajsPPgYOytOYtPGpv[eShLPmYqbEuGhr`UNimxA\\Q`esGmQMaTHqy@tKr]SYiy[aTY=NJhYPLps]R=xq\\IPAULOEpmdm;IWmTWXTlldVNeLItwdTNyAQ;tKAulEAoXqKTHM\\PRp@oqmvFUjUYjJIuNLpK]xVIwpdn>is\\eQXTkDMSfLYNqYtEm<YvUpv\\XpwewiPM?xWHEt;`VlemNYUtxU^\\VOTOUUWLaLXtQjdTKaU=ELjXkier`@OvxsaTq=xYZIV:qLEhjsDq\\anl\\SpDKkXKU`omMluPOtHlI\\QpEKc\\umHrc=YuHqNmmcemWdy?@t^ynLiOOlOEUKkpSREVwlU[PTndTP@quMRlMXVinQXsdDKoHocQs=`rxiNtdoFms=xWGinnaSUamVAQxtSxPMK`lfdVQQkJ]lMms>UvhmysLvF`lXHuseolUnwDUYLRXTJDtO]IJ<`w=eXEIQoIxZDk=mxhHSe`XpyPQPTPLltmKclVfymxxV[tSGMXuAnp]NXULc<kR]xwtvDQqjEkUEqlHqEpUXTOQ=kBAN?XWgmopus;YLNPVlERdDwDPoUxUXHqeLMBATlyka=YshS\\MUc\\SYMsUAxZtvwqOjYV\\mOhes_Yu[xvcPQQTwXXnYEvHHMDtwPhoEyYHPLUUSmEqpqTnUroaMWQy\\hOItWkAU_\\sbin@qoGipl`wm=vdPseQP[DvQpm]UYWPTkUpYmsQIKA\\KWhrgXXQinSUl=TtITrLIkAHqC]lt]Q==OeEjimyCQov]JI`Oi<sgDLQYY?yvgUXsuV\\immLo:YV\\`WU]slHUPYTTlUNhxyqtgljNUn<YYUupBYXqTSC`V?xXQ<mkQscePAEMODWlpSOapQuQIhm?IqruLdxU[ax=UX_]nFPT<]ne\\q`IvK`m>EpyaPS`WOxOoyk^InfXqf\\YWDvbHM`xV\\TRDTKuepDiTbqvgtkN]sV]l`xqctndmoODTrLuF<UJlTSqXbAY[xVimNWdTBmWYUmY@mU`nDMT<TQI\\qvHsRiT>AxOMq_ixDXk@DtstSMapX`k\\Ms]=yWxRXDqHhsmaU?\\QxeWhLWQho;ll^ukWQQtUOrTqCEu^iuIIPv]QIHWQUlbaoyuLMUkNpyyHJupyREYaPpVLTx`UoiVDyTfEt:YV\\qvLEvktmkejiPknPloxUJpVTetMduXTnYIxR@Yw`l[aJ`xXSAxmmjc]UM@srejeAVDYSqhuEDuVXNKpLsltwTyyxu^IxgPr_]QeuQ[qWMdN:QyayNNEQ[moFEmGQrjdUOEYyYmEIVY=xR`Q@xKnUMMyx^LpnawQUoMlVCax;xrpuudXwTTr=lXC<YLxx?tKDeprakCMYHhsBdjUTr=Ax`QMLEr]iob`SyYppYNXHxKhvcTN>hr`INjIwmixcHLvas=PWStwCQXSArDIoKLlZmjfxN[]JMEYv\\VfpN;aps]WZevdljj]WPLJqljc`QXll@loLyNeakJlLUXj:is[MOGmPPPYh`ShiOHLydtPJxrv@mDEv[TwgtNNulAquhtUh@XWMyIxWn<PceJkUNRPS\\INiiSO]TMprXatZ`x?YK`@xR<MtDo;\\S_@s@PTO`UOaqypSeev@LxrYuXTltqQGyNPMsXMRTivKdMtPP?xrIds?uVIYRpYQKxWNqmxPoSTpJeMouY^IV^ijXTTRpShTM_xwoxndiQ_xnlUJs]ySukKMoDpOOAweLWBpXFxqDusFDQKtN^IRO`n?aTaxoFEYgevVmTShURxrfLJ:TS@\\LH<mChLNlKfLMs\\LH\\_lXy<xdJ?aj@r^fnbaiEI[Nn[fv^>XlNN[AnlDhoRG`cve;Gv:Qj=?ynFbVN[A^[>qoxwhIOqxiqbVeCixeHivP[vykjAm=VtUQoQgviqrZQowyu\\Ixaaa@`_?>ep>njc<IJNl;<J:Ev:N[ABDJ:d:K;<J:;\\<>ZDJNZ:>tEvr>N\\]xe\\QcQhs]oaRvytie_q_U_hFv[SVvG?rRX[XVk]Oai@]xNxFq]V>av`nbpyuIuVAibQjnwkcxwoakRgc<HslYdrooTh_MXt:Ntr?wFNv@@iF@foFk;?hUwi;YfS_cKVrUpeQx^\\gj^@o[fkApuvfrFvk[njIv[Co]r__LH_Nn_jOpGA]FWm<YfC_c[@xe?ct>jAQ`SPb[QZH>ifnlcnbTXnqom;AqC@]JaeRFg<bfcRO]I\\uuVYIk_UXsYJYVHQd_iYyyS;YviOYFIUroicey;UVP[WDscSCHwODS_bAWh^EbMqTMGHtyhs]ySucSGeAwbHSeFiSOWgbyXCeec[INQCeEUXSH^iUfuSu[vXgTxaTXSYXIcLGRccec[InAr>scKGgwkVbaUDmwFetVUwuyTpsUXKS<=iKkEeeChstM?YFaS?wwlSYdsH_SSZIfL=vqmfNugAqW>mVUwbimu>KBl]yk?It[iy]TIYbGKgVEX=;txIUGcfXSF=IYRoeAwF;Qgi[URaHn]Ue[SAadQqGhghAgUTcs]IXiehrAsNoYeIHise:YVqMFnydrkgm]fSwFx=doawZielAv<qROwdakUM]EUctXMxr]CKiSeOXoYfWwCFuun_eqqs]Uw=qGsiu]uC>miQGgXaIvCITIr`WGSWXIcTIwT[=yJiFwcUEWfMWxBKRq;EEYbwQTsODFWVTIwdqCIGhKyg]wwckWL?wd_vXQrKGRxeXNsuwQTVwt;EdLyUDYXmGTCaEgIvK;IAMVsuDc_IbYtX[xoOhIsIkKtmYDWCTsywXSCIaxuaXtWcByGQkHG_i`GUBwVwwXAOWCUbEyt_?YOwIEOycOubuyUscFyvLuSoCxRkEi?uEovbArjaGh_EH]vkKdxAid?rhaShaRbMyUwSN=GviuCcwr_iUUYh]bV]ewGU[qrUyY=YSfOCkgUaaFxKcpQSUIbZuy[miZGv;Uyr]vv_HesfMWxrwTxIfVIgxMfW_Seys]_IPeERIwk?wNcVYmhVeysuyLIevABUUw?kHi;Smax?AxWIIZyIlsFXaTsMsjIeykUVEdwcc_qBFWe@?xZ_ISshXmGQiS<YvPEiXoBbKhkYFGqt;gUXSfKMV_IFk]IE_gMITXSURASm;u`uvHIfoeHaOes]FuccdegGuXkuEM?r;_IpOFMEeVwRYevD]xwigAQyH[WvkDhYRQ_D[MRLaiNOs=AxRkiwsfWyfbWVCgxtkT?wv`_Vr]CYaULqgWmIAKb;_IpudZAIwskDiT]YWIhpI\\wlAUcQt]DKlmlKLmwuwkpPKxMNMSG=qG<wwATSHtKUqydwlUS`qJViSyqoZiUVlwQyRo\\Qnqp;aTXTXf`lvUtluO>mYmAu\\um\\UqpaWxaP<Qu?qWxPsoYxRxkAASpYwTAXxaNv\\kIHv;Tq>lTvlPL\\qrhjKLP`LvxpVmMRWySMijLAYbQLranXTuhIwBqsTmkAqVJhnvupuYn]mjsHsNlMtQVrQJwlVUHviMQsqwnhxQhKhPrNlMtQuZEXXTUhYlBaloPXEektPP?xrYexRQjQUlbeuBhS;YvIQYfEseinRXnbLKIhY]LXfPw@<JJ=QjTTf@nj=VKAs[@XZEdKNp?TZEiCMSKEcVKCAtP>XlNLKAf\\UGxjpixPj<Ij?vjjIGkC?WB@KfSuW_cUyqUNgSaAw:[Ue_egkbs_wpAx>Eyueb`]uwSW@OhR[DhSu]MxXYx`]h<IB:Me<IdAWBEqBNkC:Ev:;B?WB:[DH<nj?Z?X::E>Z:;:fJ:<>:j<;BJ:^_I^wU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1. If the operations in NiMtSSNRRkc2IjYjSSJxR0Yl and the polynomial NiMtSSJnRzYiNiNJInhHRiU= are generally known, the hash function based on the presented algorithm is unkeyed. But one can easily convert this function to a keyed hash function NiMvLUkiaEc2IjYkSSJNR0YmSSJLR0YmSSJIR0Ym, when the secret key NiNJIktHNiI= is represented by some information concerning coefficients of the polynomial NiMtSSJnRzYiNiNJInhHRiU=, operations in NiMtSSNRRkc2IjYjSSJxR0Yl and the initial state of the register memory cells, containing the computed value of hash function. It should be noticed that the presented Maple procedure for computing the value of hash function exactly simulates the operation of feedback shift register, shown in Fig. 1. Furthermore, the function is not-iterated, so inconvenient operations such as the division of messages into blocks of fixed length, and padding, are eliminated. To show how the presented hash function based on NiMtSSNRRkc2IjYjSSJxR0Yl works, the following procedures have been implemented in Maple interpreter and written into the file qfmesdig.m: S := proc(a, b::nonnegint) ... end proc: returns the sum of two arbitrary elements NiNJImFHNiI=,NiMtSSNpbkc2IjYkSSJiR0YlLUkjUUZHRiU2I0kicUdGJQ== .P := proc(a, b::nonnegint) ... end proc: returns the product of two arbitrary elements NiNJImFHNiI=,NiMtSSNpbkc2IjYkSSJiR0YlLUkjUUZHRiU2I0kicUdGJQ==.Ai := proc(a::nonnegint) ... end proc: returns the additive inverse of an arbitrary element NiMtSSNpbkc2IjYkSSJhR0YlLUkjUUZHRiU2I0kicUdGJQ==.Mi := proc(a::nonnegint) ... end proc: returns the multiplicative inverse of an arbitrary element NiMtSSNpbkc2IjYkSSJhR0YlLUkjUUZHRiU2I0kicUdGJQ==. qf10init := proc() ... end proc: initializes computations in NiMtSSNRRkc2IjYjIiM1. qf256init := proc() ... end proc: initializes computations in NiMtSSNRRkc2IjYjIiRjIw==. qsha := proc(m, g, h0::array) ... end proc: a procedure which computes and returns the value of a hash function. The parameters m, g, h0 represent a message, a polynomial NiMtSSJnRzYiNiNJInhHRiU=, and an initial state of the register in which the value of a hash function is stored, respectively. qshaf256 := proc(mn::string, h0, g::array) ... end proc: a procedure, similarly as qsha returning the value of of hash function, but for use only if the order of applied quasigroup field equals to 256 and if the message of arbitrary length is stored in a disk file, named mn. hadis := proc(md1, md2::array) ... end proc: a procedure, returning Hamming distance between two q-nary r-tuples md1, md2 over NiMtSSNRRkc2IjYjSSJxR0Yl, of the same arbitrary length r. hadisbin := proc(md1, md2::array, q::posint) ... end proc: converts q-nary r-tuples md1, md2 to binary n-tuples and returns the Hamming distance between the obtained binary n-tuples. The contribution enables the reader to observe the properties and to test the behaviour of the presented NiNJI1FGRzYi-based hash function using two quasigroup fields, having 10 and 256 elements. The computations in quasigroup fields of order 10 or 256 using the files s10, p10, mi10, ai10, and s256, p256, mi256, ai256, containing suitable operation tables, are have performed. The reader may, of course, use his own similar files, in which the tables of operations in a quasigroup field of arbitrary order are stored. In this case the reader should also change the routines initializing computations.

At the beginning let us consider the more interesting case from the practical point of view, and let us use NiMtSSNRRkc2IjYjIiRjIw==restart; read "qfmesdig.m"; qf256init();Further assuming that a hash value of 100 bytes, that is 800 bits will be long, we may choose at random the parameters g and h0, used by the procedure qshag := []: h0 := []: _randseed := 1000: for i to 101 do g := [op(g), rand(256)()] end do: for i to 100 do h0 := [op(h0), rand(256)()] end do: g := g; h0 := h0; g := convert(g, array): h0 := convert(h0, array):The statement _randseed := 1000: is not necessary - it causes that the reader will obtain identical results as the author. Taking one byte size messages, differing in one bit, we will havem1 := convert([0], array): m2 := convert([1], array): md1 := qsha(m1, g, h0); md2 := qsha(m2, g, h0); dbyte := hadis(md1, md2); dbit := hadisbin(md1, md2, 256);The result is quite good. Next we will take, say, 5 times, two 1000 bytes long random messages, and we will compute their hashes and Hamming distance between message digestsm1 := array(1..1000): m2 := array(1..1000): for k to 5 do for i to 1000 do m1[i] := rand(256)(); m2[i] := rand(256)() end do: md1 := qsha(m1, g, h0): md2 := qsha(m2, g, h0); dbyte := hadis(md1, md2); dbit := hadisbin(md1, md2, 256) end do; The result is also entirely satisfactory. Finally, we show how to determine message digest when a message of an arbitrary size in a disk file is stored. The files mike0.jpg and mike1.jpg of size 114362 bytes (906896 bits) seem to be identical, but inquiring reader may verify that the value of 10000-th byte in the first file equals to 177, while the value of the corresponding byte in the second file is equal to 176. Thus we may treat these files as strings of messages, differing in one bit (and, of course, in one byte). Let us determine message digests, or simply hashes, of these two messages, and Hamming distance between the computed hashes, assuming the same values of g and h0 as above (be patient, please, it will take several minutes) mf1 := qshaf256("mike0.jpg" ,g, h0); mf2 := qshaf256("mike1.jpg" ,g, h0); dbyte := hadis(mf1, mf2); dbit := hadisbin(mf1, mf2, 256);It is no surprise that obtained values of Hamming distance between mf1 and mf2 are similar as previously. To demonstrate that the presented cryptographic hash function can work over a quasigroup field with an arbitrary number of elements, now the above calculations (of course, with the exception of the case when a message is represented as a disk file) are repeated using NiMtSSNRRkc2IjYjIiM1qf10init(); g := []: h0 := []: _randseed := 1000: for i to 101 do g := [op(g), rand(10)()] end do: for i to 100 do h0 := [op(h0), rand(10)()] end do: g := g; h0 := h0; g := convert(g, array): h0 := convert(h0, array): m1 := convert([0], array): m2 := convert([1], array): md1 := qsha(m1, g, h0); md2 := qsha(m2, g, h0); dbyte := hadis(md1, md2); dbit := hadisbin(md1, md2, 10);m1 := array(1..1000): m2 := array(1..1000): for k to 5 do for i to 1000 do m1[i] := rand(10)(); m2[i] := rand(10)() end do: md1 := qsha(m1, g, h0): md2 :=qsha(m2, g, h0); dbyte := hadis(md1, md2); dbit := hadisbin(md1, md2, 10) end do;

Contemporary books and papers on cryptology generally consider that any hash function maps n-bit streams of messages of arbitrary length to strings of fixed length r bits, being unique representations of messages. The inputs of a hash function are processed using operations in groups, rings, fields, modular arithmetic, boolean logic, cyclic permutations. In the presented application more general, an unconventional approach to cryptographic hash functions, based on quasigroup field, has been discussed. This new approach consists in considering the inputs of a hash function as non-binary elements belonging to quasigroup field - these inputs are thus processed using quasigroup field operations. Since tables of operations in a quasigroup field are random and irregular, known advanced attacks on hash functions discussed here are rather destined to fail. So, quasigroup field-based hash functions are more secure than those, recommended and used until now. The problem of constructing quasigroup based hash functions has been only sketched here. However, the only one example, discussed in the application, suffices to show that the proposed method of designing hash functions deserves to be noticed by application researchers in cryptography, since the method allows to generate effective, flexible-in-software implementation and fast-in-hardware implementation, highly secure collision resistant hash algorithms, meeting avalanche criterion, and producing message digests of arbitrary lengths.

[1] D\351nes, J., Keedwell, A.D.: Latin Squares and their Applications, Budapest, Akademiai Kiad\363, 1974[2] D\351nes, J., Keedwell, A.D.: Latin Squares - New Developments in the Theory and Applications, Annals Disc. Math., Vol. 46, Amsterdam, North-Holland, 1991 [3] D\351nes J. and Keedwell A.D.: Some Applications of Non-Associative Algebraic Systems in Cryptology, Department of Mathematics and Statistics, University of Surrey, Technical Report 99/03, 1999 [4] Hsu, D. F.: Cyclic Neofields and Combinatorial Designs, Lecture Notes in Mathematics, Vol. 824, Springer-Verlag, Berlin, 1980 [5] Koscielny C.: Spurious Galois Fields, Appl. Math. and Comp. Sci., vol. 5, No. 1, pp. 169 -188, 1995 [6] Laywine C. F., Mullen G. L.: Discrete Mathematics Using Latin Squares, New York, Wiley, 1998[7] Menezes, A. J.,van Oorschot P., Vanstone S.: Handbook of Applied Cryptography, CRC Press, 1996[8] Markovski, S., Gligoroski, D., Bakeva, V., Stojcevska, B.: works of these authors, concerning also the application of quasigroups in message digest algorithms can be found at http://www.pmf.ukim.edu.mk/~danilo/ResearchPapers/Crypto/index.htm

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